Dimensional Reduction over Fuzzy Coset Spaces
Aschieri, P; Manousselis, P; Madore, J
2004-01-01
We examine gauge theories on Minkowski space-time times fuzzy coset spaces. This means that the extra space dimensions instead of being a continuous coset space S/R are a corresponding finite matrix approximation. The gauge theory defined on this non-commutative setup is reduced to four dimensions and the rules of the corresponding dimensional reduction are established. We investigate in particular the case of the fuzzy sphere including the dimensional reduction of fermion fields.
Dimensional Reduction over Fuzzy Coset Spaces
Aschieri, P.; Madore, J.; Manousselis, P.; Zoupanos, G.
2004-04-01
We examine gauge theories on Minkowski space-time times fuzzy coset spaces. This means that the extra space dimensions instead of being a continuous coset space S/R are a corresponding finite matrix approximation. The gauge theory defined on this non-commutative setup is reduced to four dimensions and the rules of the corresponding dimensional reduction are established. We investigate in particular the case of the fuzzy sphere including the dimensional reduction of fermion fields.
Coset space dimensional reduction of gauge theories
Energy Technology Data Exchange (ETDEWEB)
Kapetanakis, D. (Physik Dept., Technische Univ. Muenchen, Garching (Germany)); Zoupanos, G. (CERN, Geneva (Switzerland))
1992-10-01
We review the attempts to construct unified theories defined in higher dimensions which are dimensionally reduced over coset spaces. We employ the coset space dimensional reduction scheme, which permits the detailed study of the resulting four-dimensional gauge theories. In the context of this scheme we present the difficulties and the suggested ways out in the attempts to describe the observed interactions in a realistic way. (orig.).
Coset space dimensional reduction of gauge theories
International Nuclear Information System (INIS)
Kapetanakis, D.; Zoupanos, G.
1992-01-01
We review the attempts to construct unified theories defined in higher dimensions which are dimensionally reduced over coset spaces. We employ the coset space dimensional reduction scheme, which permits the detailed study of the resulting four-dimensional gauge theories. In the context of this scheme we present the difficulties and the suggested ways out in the attempts to describe the observed interactions in a realistic way. (orig.)
Dimensionality Reduction for Multivariate Phase Space Reconstruction
Siek, M. B.; Solomatine, D. P.
2009-04-01
In nonlinear chaotic modelling, the reconstructed phase space of a dynamical system often has a high and complex dimensional space. Although a suitable pair of embedding dimension and time delay are appropriately selected when performing the reconstruction, the phase space structure may consist of a number of irrelevant and redundant variables and noises. The fact of equidistance time delayed variables in the phase space reconstruction can be one of the reasons. In this paper, the univariate and multivariate phase space dimensionality reductions based on principal component analysis (PCA) are proposed to solve these issues by creating a compact and lower dimensional phase space of a dynamical system which can improve the accuracy of chaotic model predictions. The chaotic model is built using adaptive local models based on the dynamical neighbours in the reconstructed phase space of observed time series data. The ocean surge time series data along the Dutch coast which are characterized as deterministic chaos are used as good candidates for testing the proposed method. In practice, the chaotic model can serve as a reliable and accurate model to support decision-makers in operational ship navigation and flood forecasting.
Dimensional reduction from entanglement in Minkowski space
Brustein, Ram; yarom, Amos
2005-01-01
Using a quantum field theoretic setting, we present evidence for dimensional reduction of any sub-volume of Minkowksi space. First, we show that correlation functions of a class of operators restricted to a sub-volume of D-dimensional Minkowski space scale as its surface area. A simple example of such area scaling is provided by the energy fluctuations of a free massless quantum field in its vacuum state. This is reminiscent of area scaling of entanglement entropy but applies to quantum expectation values in a pure state, rather than to statistical averages over a mixed state. We then show, in a specific case, that fluctuations in the bulk have a lower-dimensional representation in terms of a boundary theory at high temperature.
High temperature dimensional reduction in Snyder space
Directory of Open Access Journals (Sweden)
K. Nozari
2015-11-01
Full Text Available In this paper, we formulate the statistical mechanics in Snyder space that supports the existence of a minimal length scale. We obtain the corresponding invariant Liouville volume which properly determines the number of microstates in the semiclassical regime. The results show that the number of accessible microstates drastically reduces at the high energy regime such that there is only one degree of freedom for a particle. Using the Liouville volume, we obtain the deformed partition function and we then study the thermodynamical properties of the ideal gas in this setup. Invoking the equipartition theorem, we show that 2/3 of the degrees of freedom freeze at the high temperature regime when the thermal de Broglie wavelength becomes of the order of the Planck length. This reduction of the number of degrees of freedom suggests an effective dimensional reduction of the space from 3 to 1 at the Planck scale.
High temperature dimensional reduction in Snyder space
Nozari, K.; Hosseinzadeh, V.; Gorji, M. A.
2015-11-01
In this paper, we formulate the statistical mechanics in Snyder space that supports the existence of a minimal length scale. We obtain the corresponding invariant Liouville volume which properly determines the number of microstates in the semiclassical regime. The results show that the number of accessible microstates drastically reduces at the high energy regime such that there is only one degree of freedom for a particle. Using the Liouville volume, we obtain the deformed partition function and we then study the thermodynamical properties of the ideal gas in this setup. Invoking the equipartition theorem, we show that 2/3 of the degrees of freedom freeze at the high temperature regime when the thermal de Broglie wavelength becomes of the order of the Planck length. This reduction of the number of degrees of freedom suggests an effective dimensional reduction of the space from 3 to 1 at the Planck scale.
Discrete symmetries and coset space dimensional reduction
International Nuclear Information System (INIS)
Kapetanakis, D.; Zoupanos, G.
1989-01-01
We consider the discrete symmetries of all the six-dimensional coset spaces and we apply them in gauge theories defined in ten dimensions which are dimensionally reduced over these homogeneous spaces. Particular emphasis is given in the consequences of the discrete symmetries on the particle content as well as on the symmetry breaking a la Hosotani of the resulting four-dimensional theory. (orig.)
Supersymmetry breaking by dimensional reduction over coset spaces
Manousselis, P.; Zoupanos, G.
2001-04-01
We study the dimensional reduction of a ten-dimensional supersymmetric E8 gauge theory over six-dimensional coset spaces. We find that the coset space dimensional reduction over a symmetric coset space leaves the four dimensional gauge theory without any track of the original supersymmetry. On the contrary the dimensional reduction over a non-symmetric coset space leads to a softly broken supersymmetric gauge theory in four dimensions. The SO7/SO6 and G2/SU(3) are used as representative prototypes of symmetric and non symmetric coset spaces, respectively.
On dimensional reduction over coset spaces
International Nuclear Information System (INIS)
Kapetanakis, D.; Zoupanos, G.
1990-01-01
Gauge theories defined in higher dimensions can be dimensionally reduced over coset spaces giving definite predictions for the resulting four-dimensional theory. We present the most interesting features of these theories as well as an attempt to construct a model with realistic low energy behaviour within this framework. (author)
Dimensional Reduction over Coset Spaces and Supersymmetry Breaking
Manousselis, Pantelis; Zoupanos, George
2002-03-01
We address the question of supersymmetry breaking of a higher dimensional supersymmetric theory due to coset space dimensional reduction. In particular we study a ten-dimensional supersymmetric E8 gauge theory which is reduced over all six-dimensional coset spaces. We find that the original supersymmetry is completely broken in the process of dimensional reduction when the coset spaces are symmetric. On the contrary softly broken four-dimensional supersymmetric theories result when the coset spaces are non-symmetric. From our analysis two promising cases are emerging which lead to interesting GUTs with three fermion families in four dimensions, one being non-supersymmetric and the other softly broken supersymmetric.
On the consistency of coset space dimensional reduction
Chatzistavrakidis, A; Prezas, N; Zoupanos, G
2007-01-01
In this letter we consider higher-dimensional Yang-Mills theories and examine their consistent coset space dimensional reduction. Utilizing a suitable ansatz and imposing a simple set of constraints we determine the four-dimensional gauge theory obtained from the reduction of both the higher-dimensional Lagrangian and the corresponding equations of motion. The two reductions yield equivalent results and hence they constitute an example of a consistent truncation.
On the consistency of coset space dimensional reduction
Chatzistavrakidis, A.; Manousselis, P.; Prezas, N.; Zoupanos, G.
2007-11-01
In this Letter we consider higher-dimensional Yang-Mills theories and examine their consistent coset space dimensional reduction. Utilizing a suitable ansatz and imposing a simple set of constraints we determine the four-dimensional gauge theory obtained from the reduction of both the higher-dimensional Lagrangian and the corresponding equations of motion. The two reductions yield equivalent results and hence they constitute an example of a consistent truncation.
Model Building by Coset Space Dimensional Reduction Scheme Using Ten-Dimensional Coset Spaces
Jittoh, T.; Koike, M.; Nomura, T.; Sato, J.; Shimomura, T.
2008-12-01
We investigate the gauge-Higgs unification models within the scheme of the coset space dimensional reduction, beginning with a gauge theory in a fourteen-dimensional spacetime where extra-dimensional space has the structure of a ten-dimensional compact coset space. We found seventeen phenomenologically acceptable models through an exhaustive search for the candidates of the coset spaces, the gauge group in fourteen dimension, and fermion representation. Of the seventeen, ten models led to {SO}(10) (× {U}(1)) GUT-like models after dimensional reduction, three models led to {SU}(5) × {U}(1) GUT-like models, and four to {SU}(3) × {SU}(2) × {U}(1) × {U}(1) Standard-Model-like models. The combinations of the coset space, the gauge group in the fourteen-dimensional spacetime, and the representation of the fermion contents of such models are listed.
Building a model by coset space dimensional reduction using 10 dimensional coset spaces
Jittoh, Toshifumi; Koike, Masafumi; Nomura, Takaaki; Sato, Joe; Shimomura, Takashi
2008-05-01
We investigate gauge-Higgs unification models within the scheme of the coset space dimensional reduction, beginning with a gauge theory in a fourteen-dimensional spacetime whose extra-dimensional space has a structure of a ten-dimensional compact coset space. We found seventeen phenomenologically acceptable models through an exhaustive search for the candidates of the coset spaces, the gauge group in fourteen dimension, and fermion representation. Of the seventeen, ten models led to SO(10)(×U(1)) GUT-like models after dimensional reduction, three models led to SU(5)×U(l) GUT-like models, and four to SU(3)×SU(2)×U(1)×U(1) Standard-Model-like models. The combinations of the coset space, the gauge group in the fourteen-dimensional spacetime, and the representation of the fermion contents of such models are listed.
Coset Space Dimensional Reduction of Einstein--Yang--Mills theory
Chatzistavrakidis, A.; Prezas, N.; Zoupanos, G.
2007-01-01
In the present contribution we extend our previous work by considering the coset space dimensional reduction of higher-dimensional Einstein--Yang--Mills theories including scalar fluctuations as well as Kaluza--Klein excitations of the compactification metric and we describe the gravity-modified rules for the reduction of non-abelian gauge theories.
Coset space dimensional reduction of Einstein-Yang-Mills theory
Chatzistavrakidis, A.; Manousselis, P.; Prezas, N.; Zoupanos, G.
2008-04-01
In the present contribution we extend our previous work by considering the coset space dimensional reduction of higher-dimensional Einstein--Yang--Mills theories including scalar fluctuations as well as Kaluza--Klein excitations of the compactification metric and we describe the gravity-modified rules for the reduction of non-abelian gauge theories.
Model building by Coset Space Dimensional Reduction scheme
Jittoh, Toshifumi; Koike, Masafumi; Nomura, Takaaki; Sato, Joe; Shimomura, Takashi
2009-04-01
We investigate the gauge-Higgs unification models within the scheme of the coset space dimensional reduction, beginning with a gauge theory in a fourteen-dimensional spacetime where extra-dimensional space has the structure of a ten-dimensional compact coset space. We found seventeen phenomenologically acceptable models through an exhaustive search for the candidates of the coset spaces, the gauge group in fourteen dimension, and fermion representation. Of the seventeen, ten models led to SO(10)(×U(1)) GUT-like models after dimensional reduction, three models led to SU(5)×U(1) GUT-like models, and four to SU(3)×SU(2)×U(1)×U(1) Standard-Model-like models. The combinations of the coset space, the gauge group in the fourteen-dimensional spacetime, and the representation of the fermion contents of such models are listed.
Dictionary-based output-space dimensionality reduction
Strasser, Pablo; Wang, Jung; Armand, Stéphane; Kalousis, Alexandros
2013-01-01
In this paper we propose a method for output dimensionality reduction based on dictionary learning. Our final goal is the prediction of complete time series from standard input vectorial data. To do so we formulate a single learning problem which on the one hand learns a new representation of the output space, using dictionary learning, and reduces its dimension, while on the other hand learns to predict from the input data the new output representation, using standard multi-output regression...
Effective decorrelation and space dimensionality reduction of multiscaling volatility
Capobianco, Enrico
2004-09-01
We consider an approach for modeling non-stationary and non-Gaussian curves which has a natural impact on financial time series analysis due to the characteristic features of volatility processes. Provided that one can approximate the signal of interest, in this case stock index returns, with a greedy approximation scheme based on wavelet-like functions, an effective space dimensionality reduction of the problem can be found by a decomposition technique which selects the scales according to an energy-based optimization scheme and finds the most informative sources of the underlying multiscaling volatility process.
Soft Supersymmetry Breaking from Coset Space Dimensional Reduction
Manousselis, P.; Zoupanos, G.
2002-03-01
A ten-dimensional supersymmetric E8 gauge theory is compactified over six-dimensional coset spaces. We find that the resulting four dimensional theory is a softly broken supersymmetric gauge theory in case the coset space is non-symmetric.
Dimensional Reduction of Conformal Tensors and Einstein-Weyl Spaces
Jackiw, Roman
2007-09-01
Conformal Weyl and Cotton tensors are dimensionally reduced by a Kaluza-Klein procedure. Explicit formulas are given for reducing from four and three dimensions to three and two dimensions, respectively. When the higher dimensional conformal tensor vanishes because the space is conformallly flat, the lower-dimensional Kaluza-Klein functions satisfy equations that coincide with the Einstein-Weyl equations in three dimensions and kink equations in two dimensions.
Model building by coset space dimensional reduction in eight-dimensions
Jittoh, Toshifumi; Koike, Masafumi; Nomura, Takaaki; Sato, Joe; Toyama, Yutsuki
2009-05-01
We investigate gauge-Higgs unification models in eight-dimensional spacetime where extra-dimensional space has the structure of a four-dimensional compact coset space. The combinations of the coset space and the gauge group in the eight-dimensional spacetime of such models are listed. After the dimensional reduction of the coset space, we identified SO (10), SO (10) × U (1) and SO (10) × U (1) × U (1) as the possible gauge groups in the four-dimensional theory that can accomodate the Standard Model and thus is phenomenologically promising. Representations for fermions and scalars for these gauge groups are tabulated.
Coset Space Dimensional Reduction approach to the Standard Model
International Nuclear Information System (INIS)
Farakos, K.; Kapetanakis, D.; Koutsoumbas, G.; Zoupanos, G.
1988-01-01
We present a unified theory in ten dimensions based on the gauge group E 8 , which is dimensionally reduced to the Standard Mode SU 3c xSU 2 -LxU 1 , which breaks further spontaneously to SU 3L xU 1em . The model gives similar predictions for sin 2 θ w and proton decay as the minimal SU 5 G.U.T., while a natural choice of the coset space radii predicts light Higgs masses a la Coleman-Weinberg
Douzas, George; Grammatikopoulos, Theodoros; Zoupanos, George
2009-02-01
We consider a mathcal{N}=1 supersymmetric E 8 gauge theory, defined in ten dimensions and we determine all four-dimensional gauge theories resulting from the generalized dimensional reduction à la Forgacs-Manton over coset spaces, followed by a subsequent application of the Wilson flux spontaneous symmetry-breaking mechanism. Our investigation is constrained only by the requirements that (i) the dimensional reduction leads to the potentially phenomenologically interesting, anomaly-free, four-dimensional E 6, SO10 and SU5 GUTs and (ii) the Wilson flux mechanism makes use only of the freely acting discrete symmetries of all possible six-dimensional coset spaces.
Energy Technology Data Exchange (ETDEWEB)
Krause, Josua; Dasgupta, Aritra; Fekete, Jean-Daniel; Bertini, Enrico
2016-10-23
Dealing with the curse of dimensionality is a key challenge in high-dimensional data visualization. We present SeekAView to address three main gaps in the existing research literature. First, automated methods like dimensionality reduction or clustering suffer from a lack of transparency in letting analysts interact with their outputs in real-time to suit their exploration strategies. The results often suffer from a lack of interpretability, especially for domain experts not trained in statistics and machine learning. Second, exploratory visualization techniques like scatter plots or parallel coordinates suffer from a lack of visual scalability: it is difficult to present a coherent overview of interesting combinations of dimensions. Third, the existing techniques do not provide a flexible workflow that allows for multiple perspectives into the analysis process by automatically detecting and suggesting potentially interesting subspaces. In SeekAView we address these issues using suggestion based visual exploration of interesting patterns for building and refining multidimensional subspaces. Compared to the state-of-the-art in subspace search and visualization methods, we achieve higher transparency in showing not only the results of the algorithms, but also interesting dimensions calibrated against different metrics. We integrate a visually scalable design space with an iterative workflow guiding the analysts by choosing the starting points and letting them slice and dice through the data to find interesting subspaces and detect correlations, clusters, and outliers. We present two usage scenarios for demonstrating how SeekAView can be applied in real-world data analysis scenarios.
Signal processing using methods of dimensionality reduction of representation space
Popovskiy, V. V.
2003-01-01
The universal method of processing of signals with the help of downturn of their dimensions is considered. This work has methodical, generalizing nature, and is aimed at drawing the attention of specialist and scientist to the unity of the definitions and solutions of the problems, connected with N-dimensional presentation and confluent presentations.
Jittoh, Toshifumi; Koike, Masafumi; Nomura, Takaaki; Sato, Joe; Shimomura, Takashi
2009-03-01
We investigate ten-dimensional gauge theories whose extra six-dimensional space is a compact coset space, S/R, and whose gauge group is a direct product of two Lie groups. We list candidates of the gauge group and embeddings of R into them. After dimensional reduction of the coset space, we find fermion and scalar representations of GGUT×U(1) with GGUT=SU(5), SO(10) and E6, which accommodate all of the standard model particles. We also discuss possibilities to generate distinct Yukawa couplings among the generations using representations with different dimensions for GGUT=SO(10) and E6 models.
A local approach to dimensional reduction. II. Conformal invariance in Minkowski space
Nikolov, Petko A.; Petrov, Nikola P.
2003-01-01
We consider the problem of obtaining conformally invariant differential operators in Minkowski space. We show that the conformal electrodynamics equations and the gauge transformations for them can be obtained in the frame of the method of dimensional reduction developed in the first part of the paper. We describe a method for obtaining a large set of conformally invariant differential operators in Minkowski space.
Euclidean sections of protein conformation space and their implications in dimensionality reduction.
Duan, Mojie; Li, Minghai; Han, Li; Huo, Shuanghong
2014-10-01
Dimensionality reduction is widely used in searching for the intrinsic reaction coordinates for protein conformational changes. We find the dimensionality-reduction methods using the pairwise root-mean-square deviation (RMSD) as the local distance metric face a challenge. We use Isomap as an example to illustrate the problem. We believe that there is an implied assumption for the dimensionality-reduction approaches that aim to preserve the geometric relations between the objects: both the original space and the reduced space have the same kind of geometry, such as Euclidean geometry vs. Euclidean geometry or spherical geometry vs. spherical geometry. When the protein free energy landscape is mapped onto a 2D plane or 3D space, the reduced space is Euclidean, thus the original space should also be Euclidean. For a protein with N atoms, its conformation space is a subset of the 3N-dimensional Euclidean space R(3N). We formally define the protein conformation space as the quotient space of R(3N) by the equivalence relation of rigid motions. Whether the quotient space is Euclidean or not depends on how it is parameterized. When the pairwise RMSD is employed as the local distance metric, implicit representations are used for the protein conformation space, leading to no direct correspondence to a Euclidean set. We have demonstrated that an explicit Euclidean-based representation of protein conformation space and the local distance metric associated to it improve the quality of dimensionality reduction in the tetra-peptide and β-hairpin systems. © 2014 Wiley Periodicals, Inc.
Phase space of a gravitating particle and dimensional reduction at the Planck scale
Starodubtsev, A. N.
2015-10-01
Several approaches to quantizing general relativity suggest that quantum gravity at very short distances behaves effectively as a two-dimensional theory. The mechanism of this dimensional reduction is not yet understood. We attempt to explain it by studying the phase space of a test particle coupled to a gravitational field. The general relativity constraints relate the particle energy-momentum to some curvature invariants taking values in a group manifold. Some directions in the resulting momentum space turn out to be compact, which leads to a kind of "inverse Kaluza-Klein reduction" at short distances.
Dimensionality reduction for density ratio estimation in high-dimensional spaces.
Sugiyama, Masashi; Kawanabe, Motoaki; Chui, Pui Ling
2010-01-01
The ratio of two probability density functions is becoming a quantity of interest these days in the machine learning and data mining communities since it can be used for various data processing tasks such as non-stationarity adaptation, outlier detection, and feature selection. Recently, several methods have been developed for directly estimating the density ratio without going through density estimation and were shown to work well in various practical problems. However, these methods still perform rather poorly when the dimensionality of the data domain is high. In this paper, we propose to incorporate a dimensionality reduction scheme into a density-ratio estimation procedure and experimentally show that the estimation accuracy in high-dimensional cases can be improved.
Aerts, Sven
2011-12-01
We introduce vector space based approaches to natural language processing and some of their similarities with quantum theory when applied to information retrieval. We explain how dimensional reduction is called for from both a practical and theoretical point of view and how this can be achieved through choice of product or through projectors onto subspaces.
The N=4 supersymmetric E8 gauge theory and coset space dimensional reduction
International Nuclear Information System (INIS)
Olive, D.; West, P.
1983-01-01
Reasons are given to suggest that the N=4 supersymmetric E 8 gauge theory be considered as a serious candidate for a physical theory. The symmetries of this theory are broken by a scheme based on coset space dimensional reduction. The resulting theory possesses four conventional generations of low-mass fermions together with their mirror particles. (orig.)
Dimensionality reduction and approximation via space extension and multilinear array decomposition
Demiralp, Metin; Demiralp, Emre
2012-12-01
Scientists often face the challenge of extracting meaningful patterns from large amounts of high dimensional data such as digital images, brain scans and stellar spectra. Previous research suggests that space extension methods such as kernel methods coupled with dimensionality reduction can extract patterns that might not be clearly evident in the original data. In this work we first increase the dimensionality of the data vector to produce a multilinear array, then we decompose this array into binary outer products via a new multilinear array decomposition method. Results from approximation of digital images are encouraging.
Combined Dimensionality Reduction in Search and Detection Spaces via Diffusion Mapping
Romanov, Dmitri; Smith, Stanley; Brady, John; Coifman, Ronald; Levis, Robert
2008-03-01
Strong-field control settings involve highly nonlinear processes. Typically, both search and detection spaces are high-dimensional (with dimension ˜100 each). This poses considerable problems to analysis and interpretation of the process-related data. Here, we use the recently developed nonlinear statistical method of diffusion mapping to effectively reduce the combined dimensionality of the search and detection space and to sample essential patterns in the lower-dimensional representation. The diffusion maps are constructed and analyzed for the case study of maximizing integrated intensity in a second harmonic generation experiment. The use of a sampling set of 1000 random pulses in the diffusion mapping is sufficient for effective dimensionality reduction and for revealing the inherent structure of the process-related data. Extrapolation of the low-dimensional diffusion-space pattern helps indicate the area in the search space that is most amenable to effective optimization. The diffusion-mapping algorithm is sufficiently fast and robust that may make it a valuable preprocessing tool for optimal pulse searching.
Frolov, V.; Sutton, P.; Zelnikov, A.
2000-01-01
In a wide class of D-dimensional spacetimes which are direct or semi-direct sums of a (D-n)-dimensional space and an n-dimensional homogeneous ``internal'' space, a field can be decomposed into modes. As a result of this mode decomposition, the main objects which characterize the free quantum field, such as Green functions and heat kernels, can effectively be reduced to objects in a (D-n)-dimensional spacetime with an external dilaton field. We study the problem of the dimensional reduction of the effective action for such spacetimes. While before renormalization the original D-dimensional effective action can be presented as a ``sum over modes'' of (D-n)-dimensional effective actions, this property is violated after renormalization. We calculate the corresponding anomalous terms explicitly, illustrating the effect with some simple examples.
Adaptive Metric Dimensionality Reduction
Gottlieb, Lee-Ad; Kontorovich, Aryeh; Krauthgamer, Robert
2013-01-01
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA for metric spaces: namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverag...
Soft supersymmetry breaking due to dimensional reduction over non-symmetric coset spaces
Manousselis, P.; Zoupanos, G.
2001-10-01
A ten-dimensional supersymmetric E8 gauge theory is compactified over six-dimensional coset spaces, establishing further our earlier conjecture that the resulting four-dimensional theory is a softly broken supersymmetric gauge theory in the case that the used coset space is non-symmetric. The specific non-symmetric six-dimensional spaces examined in the present study are Sp(4)/(SU(2)×U(1))non-max and SU(3)/U(1)×U(1).
Dimensional reduction in momentum space and scale-invariant cosmological fluctuations
Amelino-Camelia, Giovanni; Arzano, Michele; Gubitosi, Giulia; Magueijo, João
2013-11-01
We adopt a framework where quantum gravity’s dynamical dimensional reduction of spacetime at short distances is described in terms of modified dispersion relations. We observe that by subjecting such models to a momentum-space diffeomorphism one obtains a “dual picture” with unmodified dispersion relations, but a modified measure of integration over momenta. We then find that the UV Hausdorff dimension of momentum space which can be inferred from this modified integration measure coincides with the short-distance spectral dimension of spacetime. This result sheds light into why scale-invariant fluctuations are obtained if the original model for two UV spectral dimensions is combined with Einstein gravity. By studying the properties of the inner product we derive the result that it is only in two energy-momentum dimensions that microphysical vacuum fluctuations are scale invariant. This is true ignoring gravity, but then we find that if Einstein gravity is postulated in the original frame, in the dual picture gravity switches off, since all matter becomes conformally coupled. We argue that our findings imply that the following concepts are closely connected: scale invariance of vacuum quantum fluctuations, conformal invariance of the gravitational coupling, UV reduction to spectral dimension two in position space, and UV reduction to Hausdorff dimension two in energy-momentum space.
Dimensionality reduction methods:
Amenta, Pietro; D'Ambra, Luigi; Gallo, Michele
2005-01-01
In case one or more sets of variables are available, the use of dimensional reduction methods could be necessary. In this contest, after a review on the link between the Shrinkage Regression Methods and Dimensional Reduction Methods, authors provide a different multivariate extension of the Garthwaite's PLS approach (1994) where a simple linear regression coefficients framework could be given for several dimensional reduction methods.
Fermion masses from dimensional reduction
International Nuclear Information System (INIS)
Kapetanakis, D.; Zoupanos, G.
1990-01-01
We consider the fermion masses in gauge theories obtained from ten dimensions through dimensional reduction on coset spaces. We calculate the general fermion mass matrix and we apply the mass formula in illustrative examples. (orig.)
Fermion masses from dimensional reduction
Energy Technology Data Exchange (ETDEWEB)
Kapetanakis, D. (National Research Centre for the Physical Sciences Democritos, Athens (Greece)); Zoupanos, G. (European Organization for Nuclear Research, Geneva (Switzerland))
1990-10-11
We consider the fermion masses in gauge theories obtained from ten dimensions through dimensional reduction on coset spaces. We calculate the general fermion mass matrix and we apply the mass formula in illustrative examples. (orig.).
An alternative dimensional reduction prescription
International Nuclear Information System (INIS)
Edelstein, J.D.; Giambiagi, J.J.; Nunez, C.; Schaposnik, F.A.
1995-08-01
We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to drop the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular 2-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher dimensional space. (author). 12 refs
Phase space reduction and the instanton crossover in (1+1)-dimensional turbulence
Moriconi, L.; Dias, G. S.
2001-09-01
We study (1+1)-dimensional turbulence in the framework of the Martin-Siggia-Rose field theory formalism. The analysis is focused on the asymptotic behaviour at the right tail of the probability distribution function (pdf) of velocity differences, where shock waves do not contribute. A BRS-preserving scheme of phase space reduction, based on the smoothness of the relevant velocity fields, leads to an effective theory for a few degrees of freedom. The sum over fluctuations around the instanton solution is written as the expectation value of a functional of the time-dependent physical fields, which evolve according to a set of Langevin equations. A natural regularization of the fluctuation determinant is provided from the fact that the instanton dominates the action for a finite time interval. The transition from the turbulent to the instanton dominated regime is related to logarithmic corrections to the saddle-point action, manifested on their turn as multiplicative power law corrections to the velocity differences pdf.
Many Faces of Dimensional Reduction
Filippov, A. T.
2006-06-01
After a brief discussion of dimensional reductions leading to the 1+1 dimensional dilaton gravity theory we consider general properties of these theories and identify problems that arise in its further reductions to one dimensional theories - cosmological models, static states (in particular, black holes) and gravity-matter waves. To bypass shortcomings of the standard ('naive') reduction we propose to exploit more general ideas: 1. separating the space and time variables in generic models, 2. reductions of the moduli spaces in integrable models that may also be viewed as dimensional reductions. This allows us to clearly see a duality between static and cosmological solutions (that we call 'SC-duality') and to demonstrate a close relation of these objects to gravity-matter waves.
Dimensionality Reduction Ensembles
Farrelly, Colleen M.
2017-01-01
Ensemble learning has had many successes in supervised learning, but it has been rare in unsupervised learning and dimensionality reduction. This study explores dimensionality reduction ensembles, using principal component analysis and manifold learning techniques to capture linear, nonlinear, local, and global features in the original dataset. Dimensionality reduction ensembles are tested first on simulation data and then on two real medical datasets using random forest classifiers; results ...
Bayesian supervised dimensionality reduction.
Gönen, Mehmet
2013-12-01
Dimensionality reduction is commonly used as a preprocessing step before training a supervised learner. However, coupled training of dimensionality reduction and supervised learning steps may improve the prediction performance. In this paper, we introduce a simple and novel Bayesian supervised dimensionality reduction method that combines linear dimensionality reduction and linear supervised learning in a principled way. We present both Gibbs sampling and variational approximation approaches to learn the proposed probabilistic model for multiclass classification. We also extend our formulation toward model selection using automatic relevance determination in order to find the intrinsic dimensionality. Classification experiments on three benchmark data sets show that the new model significantly outperforms seven baseline linear dimensionality reduction algorithms on very low dimensions in terms of generalization performance on test data. The proposed model also obtains the best results on an image recognition task in terms of classification and retrieval performances.
Isometries, dimensional reduction, and superunification
International Nuclear Information System (INIS)
Mansouri, F.; Witten, L.
1984-01-01
Dimensional reduction is carried out for space-times, with or without torsion, which admit a group, G, of isometries. The spectrum and the field equations are derived directly from the higher dimensional theory. A method of probing the extra dimensions is suggested
Dimensionality Reduction with Adaptive Approximation
Kokiopoulou, Effrosyni; Frossard, Pascal
2007-01-01
In this paper, we propose the use of (adaptive) nonlinear approximation for dimensionality reduction. In particular, we propose a dimensionality reduction method for learning a parts based representation of signals using redundant dictionaries. A redundant dictionary is an overcomplete set of basis vectors that spans the signal space. The signals are jointly represented in a common subspace extracted from the redundant dictionary, using greedy pursuit algorithms for simultaneous sparse approx...
Dimensionality reduction in complex models
Boukouvalas, Alexis; Maniyar, Dharmesh M.; Cornford, Dan
2007-01-01
As a part of the Managing Uncertainty in Complex Models (MUCM) project, research at Aston University will develop methods for dimensionality reduction of the input and/or output spaces of models, as seen within the emulator framework. Towards this end this report describes a framework for generating toy datasets, whose underlying structure is understood, to facilitate early investigations of dimensionality reduction methods and to gain a deeper understanding of the algorithms employed, both i...
Dimensionality Reduction through Sub-Space Mapping for Nearest Neighbour Algorithms
Payne, Terry R.; Edwards, Peter
2000-01-01
Many learning algorithms make an implicit assumption that all the attributes present in the data are relevant to a learning task. However, several studies have demonstrated that this assumption rarely holds; for many supervised learning algorithms, the inclusion of irrelevant or redundant attributes can result in a degradation in classification accuracy. While a variety of different methods for dimensionality reduction exist, many of these are only appropriate for datasets which contain a sma...
Lyashenko, I A; Popov, V L
2015-02-16
An impact of an elastic sphere with an elastic half space under no-slip conditions (infinitely large coefficient of friction) is studied numerically using the method of dimensionality reduction. It is shown that the rebound velocity and angular velocity, written as proper dimensionless variables, are determined by a function of only the ratio of tangential and normal stiffness ("Mindlin-ratio"). The obtained numerical results can be approximated by a simple analytical expression.
Dimensional reduction of Dirac operator
Nikolov, Petko A.; Ruseva, Gergana R.
2006-07-01
We construct an explicit example of dimensional reduction of the free massless Dirac operator with an internal SU(3) symmetry, defined on a 12-dimensional manifold that is the total space of a principal SU(3)-bundle over a four-dimensional (nonflat) pseudo-Riemannian manifold. Upon dimensional reduction the free 12-dimensional Dirac equation is transformed into a rather nontrivial four-dimensional one: a pair of massive Lorentz spinor SU(3)-octets interacting with an SU(3)-gauge field with a source term depending on the curvature tensor of the gauge field. The SU(3) group is complicated enough to illustrate features of the general case. It should not be confused with the color SU(3) of quantum chromodynamics where the fundamental spinors, the quark fields, are SU(3) triplets rather than octets.
On the dimensional reduction procedure
Cognola, Guido; Zerbini, Sergio
2001-05-01
The issue related to the so-called dimensional reduction procedure is revisited within the Euclidean formalism. First, it is shown that for symmetric spaces, the local exact heat-kernel density is equal to the reduced one, once the harmonic sum has been successfully performed. In the general case, due to the impossibility to deal with exact results, the short t heat-kernel asymptotics is considered. It is found that the exact heat-kernel and the dimensionally reduced one coincide up to two non-trivial leading contributions in the short t expansion. Implications of these results with regard to dimensional-reduction anomaly are discussed.
Classification Constrained Dimensionality Reduction
Raich, Raviv; Costa, Jose A.; Damelin, Steven B.; Hero III, Alfred O.
2008-01-01
Dimensionality reduction is a topic of recent interest. In this paper, we present the classification constrained dimensionality reduction (CCDR) algorithm to account for label information. The algorithm can account for multiple classes as well as the semi-supervised setting. We present an out-of-sample expressions for both labeled and unlabeled data. For unlabeled data, we introduce a method of embedding a new point as preprocessing to a classifier. For labeled data, we introduce a method tha...
Nonlinear dimensionality reduction
Lee, John A
2007-01-01
Methods of dimensionality reduction provide a way to understand and visualize the structure of complex data sets. This book describes the methods to reduce the dimensionality of numerical databases. For each method, the description starts from intuitive ideas, develops the mathematical details, and ends by outlining the algorithmic implementation.
Ultrafast localization of the optic disc using dimensionality reduction of the search space.
Mahfouz, Ahmed Essam; Fahmy, Ahmed S
2009-01-01
Optic Disc (OD) localization is an important pre-processing step that significantly simplifies subsequent segmentation of the OD and other retinal structures. Current OD localization techniques suffer from impractically-high computation times (few minutes/image). In this work, we present an ultrafast technique that requires less than a second to localize the OD. The technique is based on reducing the dimensionality of the search space by projecting the 2D image feature space onto two orthogonal (x- and y-) axes. This results in two 1D signals that can be used to determine the x- and y- coordinates of the OD. Image features such as retinal vessels orientation and the OD brightness and shape are used in the current method. Four publicly-available databases, including STARE and DRIVE, were used to evaluate the proposed technique. The OD was successfully located in 330 images out of 340 images (97%) with an average computation time of 0.65 seconds.
A Convex Model for Nonnegative Matrix Factorization and Dimensionality Reduction on Physical Space
Esser, Ernie; Moller, Michael; Osher, Stanley; Sapiro, Guillermo; Xin, Jack
2012-07-01
A collaborative convex framework for factoring a data matrix $X$ into a non-negative product $AS$, with a sparse coefficient matrix $S$, is proposed. We restrict the columns of the dictionary matrix $A$ to coincide with certain columns of the data matrix $X$, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use $l_{1,\\infty}$ regularization to select the dictionary from the data and show this leads to an exact convex relaxation of $l_0$ in the case of distinct noise free data. We also show how to relax the restriction-to-$X$ constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in $X$. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.
Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation
Kalinay, Pavol; Percus, Jerome K.
2012-09-01
A point-like particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the 1D Fokker-Planck (Kramers) equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m→0, with a series of corrections expanded in powers of m/ γ, γ denotes the friction coefficient. The corrections are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.
Argatov, Ivan I.; Popov, Valentin L.
2016-08-01
The method of dimensionality reduction (MDR) is extended for the axisymmetric frictionless unilateral Hertz-type contact problem for a viscoelastic half-space and an arbitrary axisymmetric rigid indenter under the assumption that an arbitrarily evolving in time circular contact area remains singly connected during the whole process of indentation. In particular, the MDR is applied to study in detail the so-called rebound indentation problem, where the contact radius has a single maximum. It is shown that the obtained closed-form analytical solution for the rebound indentation displacement (recorded in the recovery phase, when the contact force vanishes) does not depend on the indenter shape.
Dimensionality Reduction by Weighted Connections between Neighborhoods
Directory of Open Access Journals (Sweden)
Fuding Xie
2014-01-01
Full Text Available Dimensionality reduction is the transformation of high-dimensional data into a meaningful representation of reduced dimensionality. This paper introduces a dimensionality reduction technique by weighted connections between neighborhoods to improve K-Isomap method, attempting to preserve perfectly the relationships between neighborhoods in the process of dimensionality reduction. The validity of the proposal is tested by three typical examples which are widely employed in the algorithms based on manifold. The experimental results show that the local topology nature of dataset is preserved well while transforming dataset in high-dimensional space into a new dataset in low-dimensionality by the proposed method.
Dimensionality Reduction Mappings
Bunte, Kerstin; Biehl, Michael; Hammer, Barbara
2011-01-01
A wealth of powerful dimensionality reduction methods has been established which can be used for data visualization and preprocessing. These are accompanied by formal evaluation schemes, which allow a quantitative evaluation along general principles and which even lead to further visualization
Spontaneous dimensional reduction?
Carlip, Steven
2012-10-01
Over the past few years, evidence has begun to accumulate suggesting that spacetime may undergo a "spontaneous dimensional reduction" to two dimensions near the Planck scale. I review some of this evidence, and discuss the (still very speculative) proposal that the underlying mechanism may be related to short-distance focusing of light rays by quantum fluctuations.
Dimensionality Reduction Mappings
Bunte, Kerstin; Biehl, Michael; Hammer, Barbara
2011-01-01
A wealth of powerful dimensionality reduction methods has been established which can be used for data visualization and preprocessing. These are accompanied by formal evaluation schemes, which allow a quantitative evaluation along general principles and which even lead to further visualization schemes based on these objectives. Most methods, however, provide a mapping of a priorly given finite set of points only, requiring additional steps for out-of-sample extensions. We propose a general vi...
Information visualization by dimensionality reduction: a review
Safa Najim
2014-01-01
Information visualization can be considered a process of transforming similarity relationships between data points to a geometric representation in order to see unseen information. High-dimensionality data sets are one of the main problems of information visualization. Dimensionality Reduction (DR) is therefore a useful strategy to project high-dimensional space onto low-dimensional space, which it can be visualized directly. The application of this technique has several benefits. First, DR c...
OBJECTIVE REDUCTION OF THE SPACE-TIME DOMAIN DIMENSIONALITY FOR EVALUATING MODEL PERFORMANCE
In the United States, photochemical air quality models are the principal tools used by governmental agencies to develop emission reduction strategies aimed at achieving National Ambient Air Quality Standards (NAAQS). Before they can be applied with confidence in a regulatory sett...
Feature Space Dimensionality Reduction for Real-Time Vision-Based Food Inspection
Directory of Open Access Journals (Sweden)
Mai Moussa CHETIMA
2009-03-01
Full Text Available Machine vision solutions are becoming a standard for quality inspection in several manufacturing industries. In the processed-food industry where the appearance attributes of the product are essential to customer’s satisfaction, visual inspection can be reliably achieved with machine vision. But such systems often involve the extraction of a larger number of features than those actually needed to ensure proper quality control, making the process less efficient and difficult to tune. This work experiments with several feature selection techniques in order to reduce the number of attributes analyzed by a real-time vision-based food inspection system. Identifying and removing as much irrelevant and redundant information as possible reduces the dimensionality of the data and allows classification algorithms to operate faster. In some cases, accuracy on classification can even be improved. Filter-based and wrapper-based feature selectors are experimentally evaluated on different bakery products to identify the best performing approaches.
The dimensional reduction in a multi-dimensional cosmology
International Nuclear Information System (INIS)
Demianski, M.; Golda, Z.A.; Heller, M.; Szydlowski, M.
1986-01-01
Einstein's field equations are solved for the case of the eleven-dimensional vacuum spacetime which is the product R x Bianchi V x T 7 , where T 7 is a seven-dimensional torus. Among all possible solutions, the authors identify those in which the macroscopic space expands and the microscopic space contracts to a finite size. The solutions with this property are 'typical' within the considered class. They implement the idea of a purely dynamical dimensional reduction. (author)
An Information Geometric Framework for Dimensionality Reduction
Carter, Kevin M.; Raich, Raviv; Hero III, Alfred O.
2008-01-01
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of learning tasks such as classification, clustering, and visualization, these methods have focused primarily on Riemannian manifolds in Euclidean space. While sufficient for many applications, there are many high-dimensional signals which have no straightforwar...
Rotational Invariant Dimensionality Reduction Algorithms.
Lai, Zhihui; Xu, Yong; Yang, Jian; Shen, Linlin; Zhang, David
2017-11-01
A common intrinsic limitation of the traditional subspace learning methods is the sensitivity to the outliers and the image variations of the object since they use the norm as the metric. In this paper, a series of methods based on the -norm are proposed for linear dimensionality reduction. Since the -norm based objective function is robust to the image variations, the proposed algorithms can perform robust image feature extraction for classification. We use different ideas to design different algorithms and obtain a unified rotational invariant (RI) dimensionality reduction framework, which extends the well-known graph embedding algorithm framework to a more generalized form. We provide the comprehensive analyses to show the essential properties of the proposed algorithm framework. This paper indicates that the optimization problems have global optimal solutions when all the orthogonal projections of the data space are computed and used. Experimental results on popular image datasets indicate that the proposed RI dimensionality reduction algorithms can obtain competitive performance compared with the previous norm based subspace learning algorithms.
Killing reduction of 5-dimensional spacetimes
Yang, Xuejun; Ma, Yongge; Shao, Jianbing; Zhou, Wei
2003-07-01
In a 5-dimensional spacetime (M,gab) with a Killing vector field ξa which is either everywhere time like or everywhere space like, the collection of all trajectories of ξa gives a 4-dimensional space S. The reduction of (M,gab) is studied in the geometric language, which is a generalization of Geroch’s method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on S is obtained by the reduction of vacuum Einstein’s equations on M, which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. In addition to the symmetry-reduced action from the Hilbert action on M, an alternative action of the fields on S is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on M.
Dimensionality Reduction Particle Swarm Algorithm for High Dimensional Clustering
Energy Technology Data Exchange (ETDEWEB)
Cui, Xiaohui [ORNL; ST Charles, Jesse Lee [ORNL; Potok, Thomas E [ORNL; Beaver, Justin M [ORNL
2008-01-01
The Particle Swarm Optimization (PSO) clustering algorithm can generate more compact clustering results than the traditional K-means clustering algorithm. However, when clustering high dimensional datasets, the PSO clustering algorithm is notoriously slow because its computation cost increases exponentially with the size of the dataset dimension. Dimensionality reduction techniques offer solutions that both significantly improve the computation time, and yield reasonably accurate clustering results in high dimensional data analysis. In this paper, we introduce research that combines different dimensionality reduction techniques with the PSO clustering algorithm in order to reduce the complexity of high dimensional datasets and speed up the PSO clustering process. We report significant improvements in total runtime. Moreover, the clustering accuracy of the dimensionality reduction PSO clustering algorithm is comparable to the one that uses full dimension space.
Dynamic dimensionality reduction for hyperspectral imagery
Safavi, Haleh; Liu, Keng-Hao; Chang, Chein-I.
2011-06-01
Data dimensionality (DR) is generally performed by first fixing size of DR at a certain number, say p and then finding a technique to reduce an original data space to a low dimensional data space with dimensionality specified by p. This paper introduces a new concept of dynamic dimensionality reduction (DDR) which considers the parameter p as a variable by varying the value of p to make p adaptive compared to the commonly used DR, referred to as static dimensionality reduction (SDR) with the parameter p fixed at a constant value. In order to materialize the DDR another new concept, referred to as progressive DR (PDR) is also developed so that the DR can be performed progressively to adapt the variable size of data dimensionality determined by varying the value of p. The advantages of the DDR over SDR are demonstrated through experiments conducted for hyperspectral image classification.
Kawata, Y.; Niki, N.; Ohmatsu, H.; Aokage, K.; Kusumoto, M.; Tsuchida, T.; Eguchi, K.; Kaneko, M.
2015-03-01
Advantages of CT scanners with high resolution have allowed the improved detection of lung cancers. In the recent release of positive results from the National Lung Screening Trial (NLST) in the US showing that CT screening does in fact have a positive impact on the reduction of lung cancer related mortality. While this study does show the efficacy of CT based screening, physicians often face the problems of deciding appropriate management strategies for maximizing patient survival and for preserving lung function. Several key manifold-learning approaches efficiently reveal intrinsic low-dimensional structures latent in high-dimensional data spaces. This study was performed to investigate whether the dimensionality reduction can identify embedded structures from the CT histogram feature of non-small-cell lung cancer (NSCLC) space to improve the performance in predicting the likelihood of RFS for patients with NSCLC.
Probabilistic Dimensionality Reduction via Structure Learning
Wang, Li
2016-01-01
We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth skeleton of embedding points in a low-dimensional space from high-dimensional noisy data. The formulation of the new model can be equivalently interpreted as two coupled learning problem, i.e., structure learning and the learning of projection matrix. This int...
Dimensionality Reduction Algorithms on High Dimensional Datasets
Directory of Open Access Journals (Sweden)
Iwan Syarif
2014-12-01
Full Text Available Classification problem especially for high dimensional datasets have attracted many researchers in order to find efficient approaches to address them. However, the classification problem has become very complicatedespecially when the number of possible different combinations of variables is so high. In this research, we evaluate the performance of Genetic Algorithm (GA and Particle Swarm Optimization (PSO as feature selection algorithms when applied to high dimensional datasets.Our experiments show that in terms of dimensionality reduction, PSO is much better than GA. PSO has successfully reduced the number of attributes of 8 datasets to 13.47% on average while GA is only 31.36% on average. In terms of classification performance, GA is slightly better than PSO. GA‐ reduced datasets have better performance than their original ones on 5 of 8 datasets while PSO is only 3 of 8 datasets. Keywords: feature selection, dimensionality reduction, Genetic Algorithm (GA, Particle Swarm Optmization (PSO.
Algorithmic dimensionality reduction for molecular structure analysis.
Brown, W Michael; Martin, Shawn; Pollock, Sara N; Coutsias, Evangelos A; Watson, Jean-Paul
2008-08-14
Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation--a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation.
Algorithmic dimensionality reduction for molecular structure analysis
Brown, W. Michael; Martin, Shawn; Pollock, Sara N.; Coutsias, Evangelos A.; Watson, Jean-Paul
2008-01-01
Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation—a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation. PMID:18715062
Algorithmic dimensionality reduction for molecular structure analysis
Brown, W. Michael; Martin, Shawn; Pollock, Sara N.; Coutsias, Evangelos A.; Watson, Jean-Paul
2008-08-01
Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation-a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation.
Nonlinear dimensionality reduction in climate data
Directory of Open Access Journals (Sweden)
A. J. Gámez
2004-01-01
Full Text Available Linear methods of dimensionality reduction are useful tools for handling and interpreting high dimensional data. However, the cumulative variance explained by each of the subspaces in which the data space is decomposed may show a slow convergence that makes the selection of a proper minimum number of subspaces for successfully representing the variability of the process ambiguous. The use of nonlinear methods can improve the embedding of multivariate data into lower dimensional manifolds. In this article, a nonlinear method for dimensionality reduction, Isomap, is applied to the sea surface temperature and thermocline data in the tropical Pacific Ocean, where the El Niño-Southern Oscillation (ENSO phenomenon and the annual cycle phenomena interact. Isomap gives a more accurate description of the manifold dimensionality of the physical system. The knowledge of the minimum number of dimensions is expected to improve the development of low dimensional models for understanding and predicting ENSO.
Nonlinear dimensionality reduction in climate data
Gámez, A. J.; Zhou, C. S.; Timmermann, A.; Kurths, J.
2004-09-01
Linear methods of dimensionality reduction are useful tools for handling and interpreting high dimensional data. However, the cumulative variance explained by each of the subspaces in which the data space is decomposed may show a slow convergence that makes the selection of a proper minimum number of subspaces for successfully representing the variability of the process ambiguous. The use of nonlinear methods can improve the embedding of multivariate data into lower dimensional manifolds. In this article, a nonlinear method for dimensionality reduction, Isomap, is applied to the sea surface temperature and thermocline data in the tropical Pacific Ocean, where the El Niño-Southern Oscillation (ENSO) phenomenon and the annual cycle phenomena interact. Isomap gives a more accurate description of the manifold dimensionality of the physical system. The knowledge of the minimum number of dimensions is expected to improve the development of low dimensional models for understanding and predicting ENSO.
Dimensionality reduction for dimension-specific search
Huang, Zi; Hengtao, Shen; Zhou, Xiaofang; Song, Dawei; Rüger, Stefan
2007-01-01
Dimensionality reduction plays an important role in efficient similarity search, which is often based on k-nearest neighbor (k-NN) queries over a high-dimensional feature space. In this paper, we introduce a novel type of k-NN query, namely conditional k-NN (ck-NN), which considers dimension-specific constraint in addition to the inter-point distances. However, existing dimensionality reduction methods are not applicable to this new type of queries. We propose a novel Mean-Std (standard devia...
Nonlinear Dimensionality Reduction on Graphs
Shen, Yanning; Traganitis, Panagiotis A.; Giannakis, Georgios B.
2018-01-01
In this era of data deluge, many signal processing and machine learning tasks are faced with high-dimensional datasets, including images, videos, as well as time series generated from social, commercial and brain network interactions. Their efficient processing calls for dimensionality reduction techniques capable of properly compressing the data while preserving task-related characteristics, going beyond pairwise data correlations. The present paper puts forth a nonlinear dimensionality redu...
Dimensional reduction for conformal blocks
Hogervorst, Matthijs
2016-09-01
We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO( d + 1 , 1) up into multiplets of SO( d, 1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d - 1 dimensions. In particular, we obtain a formula for 3 d conformal blocks as an infinite sum over 2 F 1 hypergeometric functions with closed-form coefficients.
Reduction of Dimensionality for Classification
Cuevas-Covarrubias, Carlos; Riccomagno, Eva
2017-01-01
We present an algorithm for the reduction of dimensionality useful in statistical classification problems where observations from two multivariate normal distributions are discriminated. It is based on Principal Components Analysis and consists of a simultaneous diagonalization of two covariance matrices. The criterion for reduction of dimensionality is given by the contribution of each principal component to the area under the ROC curve of a discriminant function. Linear and quadratic scores...
Dimensional Reduction Near the Horizon
Haba, Z.
2008-11-01
In the Euclidean formulation of functional integration we discuss a dimensional reduction of quantum field theory near the horizon in terms of Green functions. We show that a massless scalar quantum field in D dimensions can be approximated near the bifurcate Killing horizon by a massless two-dimensional conformal field.
ANDRomeda: adaptive nonlinear dimensionality reduction
Marchette, David J.; Priebe, Carey E.
2000-03-01
Standard approaches for the classification of high dimensional data require the selection of features, the projection of the features to a lower dimensional space, and the construction of the classifier in the lower dimensional space. Two fundamental issues arise in determining an appropriate projection to a lower dimensional space: the target dimensionality for the projection must be determined, and a particular projection must be selected from a specified family. We present an algorithm which is designed specifically for classification task and addresses both these issues. The family of nonlinear projections considered is based on interpoint distances - in particular, we consider point-to-subset distances. Our algorithm selects both the number of subsets to use and the subsets themselves. The methodology is applied to an artificial nose odorant classification task.
Koudsi, Badia; Refai, Hakki H.; Sluss, James J., Jr.
2010-10-01
An ongoing public-private research partnership has demonstrated a three-dimensional (3D) volumetric display system that incorporates a static image space. The 3D display system uses micro-electro-mechanical systems (MEMS) based mirror arrays to direct infrared light beams into an image space that exhibits two-step, twofrequency upconversion. A number of candidate image space materials have been evaluated, with 2%Er: NYF4 appearing to be most promising at this stage of the research. In this paper, the authors build upon prior work by investigating the response time of 2%Er:NYF4. In addition, a new technique for reducing flicker in the 3D images is described. The technique includes interlacing the 3D image slices in a way similar to the interlacing that occurs in the generation of television images. Adopting this technique has the potential to reduce the flicker that is presently evident, thereby improving the overall 3D image quality.
Robust linear dimensionality reduction.
Koren, Yehuda; Carmel, Liran
2004-01-01
We present a novel family of data-driven linear transformations, aimed at finding low-dimensional embeddings of multivariate data, in a way that optimally preserves the structure of the data. The well-studied PCA and Fisher's LDA are shown to be special members in this family of transformations, and we demonstrate how to generalize these two methods such as to enhance their performance. Furthermore, our technique is the only one, to the best of our knowledge, that reflects in the resulting embedding both the data coordinates and pairwise relationships between the data elements. Even more so, when information on the clustering (labeling) decomposition of the data is known, this information can also be integrated in the linear transformation, resulting in embeddings that clearly show the separation between the clusters, as well as their internal structure. All of this makes our technique very flexible and powerful, and lets us cope with kinds of data that other techniques fail to describe properly.
Nonlinear dimensionality reduction in climate data
Gámez, A. J.; Zhou, C. S.; Timmermann, A.; Kurths, J.
2004-01-01
International audience; Linear methods of dimensionality reduction are useful tools for handling and interpreting high dimensional data. However, the cumulative variance explained by each of the subspaces in which the data space is decomposed may show a slow convergence that makes the selection of a proper minimum number of subspaces for successfully representing the variability of the process ambiguous. The use of nonlinear methods can improve the embedding of multivariate data into lower di...
Central subspace dimensionality reduction using covariance operators.
Kim, Minyoung; Pavlovic, Vladimir
2011-04-01
We consider the task of dimensionality reduction informed by real-valued multivariate labels. The problem is often treated as Dimensionality Reduction for Regression (DRR), whose goal is to find a low-dimensional representation, the central subspace, of the input data that preserves the statistical correlation with the targets. A class of DRR methods exploits the notion of inverse regression (IR) to discover central subspaces. Whereas most existing IR techniques rely on explicit output space slicing, we propose a novel method called the Covariance Operator Inverse Regression (COIR) that generalizes IR to nonlinear input/output spaces without explicit target slicing. COIR's unique properties make DRR applicable to problem domains with high-dimensional output data corrupted by potentially significant amounts of noise. Unlike recent kernel dimensionality reduction methods that employ iterative nonconvex optimization, COIR yields a closed-form solution. We also establish the link between COIR, other DRR techniques, and popular supervised dimensionality reduction methods, including canonical correlation analysis and linear discriminant analysis. We then extend COIR to semi-supervised settings where many of the input points lack their labels. We demonstrate the benefits of COIR on several important regression problems in both fully supervised and semi-supervised settings.
Weakly infinite-dimensional spaces
International Nuclear Information System (INIS)
Fedorchuk, Vitalii V
2007-01-01
In this survey article two new classes of spaces are considered: m-C-spaces and w-m-C-spaces, m=2,3,...,∞. They are intermediate between the class of weakly infinite-dimensional spaces in the Alexandroff sense and the class of C-spaces. The classes of 2-C-spaces and w-2-C-spaces coincide with the class of weakly infinite-dimensional spaces, while the compact ∞-C-spaces are exactly the C-compact spaces of Haver. The main results of the theory of weakly infinite-dimensional spaces, including classification via transfinite Lebesgue dimensions and Luzin-Sierpinsky indices, extend to these new classes of spaces. Weak m-C-spaces are characterised by means of essential maps to Henderson's m-compacta. The existence of hereditarily m-strongly infinite-dimensional spaces is proved.
Dimensional-reduction anomaly in spherically symmetric spacetimes
Sutton, P.
2000-08-01
In D-dimensional spacetimes which can be foliated by n-dimensional homogeneous subspaces, a quantum field can be decomposed in terms of modes on the subspaces, reducing the system to a collection of (D-n)-dimensional fields. This allows one to write bare D-dimensional field quantities like the Green function and the effective action as sums of their (D-n)-dimensional counterparts in the dimensionally reduced theory. It has been shown, however, that renormalization breaks this relationship between the original and dimensionally reduced theories, an effect called the dimensional-reduction anomaly. We examine the dimensional-reduction anomaly for the important case of spherically symmetric spaces.
Analyzing Protein Dynamics Using Dimensionality Reduction
Eryol, Atahan
2015-01-01
This thesis investigates dimensionality reduction for analyzing the dynamics ofprotein simulations, particularly disordered proteins which do not fold into a xedshape but are thought to perform their functions through their movements. Ratherthan analyze the movement of the proteins in 3D space, we use dimensionalityreduction to project the molecular structure of the proteins into a target space inwhich each structure is represented as a point. All that is needed to do this arethe pairwise dis...
Significance of Dimensionality Reduction in Image Processing
Shereena V. B; Julie M. David
2015-01-01
The aim of this paper is to present a comparative study of two linear dimension reduction methods namely PCA (Principal Component Analysis) and LDA (Linear Discriminant Analysis). The main idea of PCA is to transform the high dimensional input space onto the feature space where the maximal variance is displayed. The feature selection in traditional LDA is obtained by maximizing the difference between classes and minimizing the distance within classes. PCA finds the axes with maximum variance ...
Dimensional reduction at a quantum critical point
Sebastian, S. E.; Harrison, N.; Batista, C. D.; Balicas, L.; Jaime, M.; Sharma, P. A.; Kawashima, N.; Fisher, I. R.
2006-06-01
Competition between electronic ground states near a quantum critical point (QCP)-the location of a zero-temperature phase transition driven solely by quantum-mechanical fluctuations-is expected to lead to unconventional behaviour in low-dimensional systems. New electronic phases of matter have been predicted to occur in the vicinity of a QCP by two-dimensional theories, and explanations based on these ideas have been proposed for significant unsolved problems in condensed-matter physics, such as non-Fermi-liquid behaviour and high-temperature superconductivity. But the real materials to which these ideas have been applied are usually rendered three-dimensional by a finite electronic coupling between their component layers; a two-dimensional QCP has not been experimentally observed in any bulk three-dimensional system, and mechanisms for dimensional reduction have remained the subject of theoretical conjecture. Here we show evidence that the Bose-Einstein condensate of spin triplets in the three-dimensional Mott insulator BaCuSi2O6 (refs 12-16) provides an experimentally verifiable example of dimensional reduction at a QCP. The interplay of correlations on a geometrically frustrated lattice causes the individual two-dimensional layers of spin-½ Cu2+ pairs (spin dimers) to become decoupled at the QCP, giving rise to a two-dimensional QCP characterized by linear power law scaling distinctly different from that of its three-dimensional counterpart. Thus the very notion of dimensionality can be said to acquire an `emergent' nature: although the individual particles move on a three-dimensional lattice, their collective behaviour occurs in lower-dimensional space.
Three dimensional reductions of four-dimensional quasilinear systems
Pavlov, Maxim V.; Stoilov, Nikola M.
2017-11-01
In this paper, we show that four-dimensional quasilinear systems of first order integrable by the method of two-dimensional hydrodynamic reductions possess infinitely many three-dimensional hydrodynamic reductions, which are also integrable systems. These three-dimensional multi-component integrable systems are irreducible to two-dimensional hydrodynamic reductions in a generic case.
Dimensional reduction for a SIR type model
Cahyono, Edi; Soeharyadi, Yudi; Mukhsar
2018-03-01
Epidemic phenomena are often modeled in the form of dynamical systems. Such model has also been used to model spread of rumor, spread of extreme ideology, and dissemination of knowledge. Among the simplest is SIR (susceptible, infected and recovered) model, a model that consists of three compartments, and hence three variables. The variables are functions of time which represent the number of subpopulations, namely suspect, infected and recovery. The sum of the three is assumed to be constant. Hence, the model is actually two dimensional which sits in three-dimensional ambient space. This paper deals with the reduction of a SIR type model into two variables in two-dimensional ambient space to understand the geometry and dynamics better. The dynamics is studied, and the phase portrait is presented. The two dimensional model preserves the equilibrium and the stability. The model has been applied for knowledge dissemination, which has been the interest of knowledge management.
Remarks on Dimensional Reduction of Multidimensional Cosmological Models
Günther, Uwe; Zhuk, Alexander
2006-02-01
Multidimensional cosmological models with factorizable geometry and their dimensional reduction to effective four-dimensional theories are analyzed on sensitivity to different scalings. It is shown that a non-correct gauging of the effective four-dimensional gravitational constant within the dimensional reduction results in a non-correct rescaling of the cosmological constant and the gravexciton/radion masses. The relationship between the effective gravitational constants of theories with different dimensions is discussed for setups where the lower dimensional theory results via dimensional reduction from the higher dimensional one and where the compactified space components vary dynamically.
Dimensionality reduction with image data
Peña, Daniel; Benito, Mónica
2004-01-01
A common objective in image analysis is dimensionality reduction. The most common often used data-exploratory technique with this objective is principal component analysis. We propose a new method based on the projection of the images as matrices after a Procrustes rotation and show that it leads to a better reconstruction of images.
Dimensional Reduction and Hadronic Processes
Signer, Adrian; Stöckinger, Dominik
2008-11-01
We consider the application of regularization by dimensional reduction to NLO corrections of hadronic processes. The general collinear singularity structure is discussed, the origin of the regularization-scheme dependence is identified and transition rules to other regularization schemes are derived.
Dimensionality reduction in epidemic spreading models
Frasca, M.; Rizzo, A.; Gallo, L.; Fortuna, L.; Porfiri, M.
2015-09-01
Complex dynamical systems often exhibit collective dynamics that are well described by a reduced set of key variables in a low-dimensional space. Such a low-dimensional description offers a privileged perspective to understand the system behavior across temporal and spatial scales. In this work, we propose a data-driven approach to establish low-dimensional representations of large epidemic datasets by using a dimensionality reduction algorithm based on isometric features mapping (ISOMAP). We demonstrate our approach on synthetic data for epidemic spreading in a population of mobile individuals. We find that ISOMAP is successful in embedding high-dimensional data into a low-dimensional manifold, whose topological features are associated with the epidemic outbreak. Across a range of simulation parameters and model instances, we observe that epidemic outbreaks are embedded into a family of closed curves in a three-dimensional space, in which neighboring points pertain to instants that are close in time. The orientation of each curve is unique to a specific outbreak, and the coordinates correlate with the number of infected individuals. A low-dimensional description of epidemic spreading is expected to improve our understanding of the role of individual response on the outbreak dynamics, inform the selection of meaningful global observables, and, possibly, aid in the design of control and quarantine procedures.
Pyragas, K.; Lange, F.; Letz, T.; Parisi, J.; Kittel, A.
2001-01-01
We suggest a quantitatively correct procedure for reducing the spatial degrees of freedom of the space-dependent rate equations of a multimode laser that describe the dynamics of the population inversion of the active medium and the mode intensities of the standing waves in the laser cavity. The key idea of that reduction is to take advantage of the small value of the parameter that defines the ratio between the population inversion decay rate and the cavity decay rate. We generalize the reduction procedure for the case of an intracavity frequency doubled laser. Frequency conversion performed by an optically nonlinear crystal placed inside the laser cavity may cause a pronounced instability in the laser performance, leading to chaotic oscillations of the output intensity. Based on the reduced equations, we analyze the dynamical properties of the system as well as the problem of stabilizing the steady state. The numerical analysis is performed considering the specific system of a Nd:YAG (neodymium-doped yttrium aluminum garnet) laser with an intracavity KTP (potassium titanyl phosphate) crystal.
Inoue, Yuuji; Yoneyama, Masami; Nakamura, Masanobu; Takemura, Atsushi
2018-03-07
The two-dimensional Cartesian turbo spin-echo (TSE) sequence is widely used in routine clinical studies, but it is sensitive to respiratory motion. We investigated the k-space orders in Cartesian TSE that can effectively reduce motion artifacts. The purpose of this study was to demonstrate the relationship between k-space order and degree of motion artifacts using a moving phantom. We compared the degree of motion artifacts between linear and asymmetric k-space orders. The actual spacing of ghost artifacts in the asymmetric order was doubled compared with that in the linear order in the free-breathing situation. The asymmetric order clearly showed less sensitivity to incomplete breath-hold at the latter half of the imaging period. Because of the actual number of partitions of the k-space and the temporal filling order, the asymmetric k-space order of Cartesian TSE was superior to the linear k-space order for reduction of ghosting motion artifacts.
Yan, Zhen-Ya
2001-11-01
In this paper,eight types of (1+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators by using the direct method.In addition,the cnoidal wave solution and dromion-like solution are also derived by using the reduced nonlinear ordinary differential equations.The (1+1) dromion obtained by Lou [J.Phys.A28 (1995) 7227] and Zhang [Chin.Phys.9 (2000) 1] is only a special case of our results.Moreover,some properties of the dromion-like solutions are analyzed. The project supported by National Natural Science Foundation of China under Grant No. 10072013, the National Key Basic Research Development Project Program of China under Grant No. G1998030600 and Doctoral Foundation of China under Grant No. 98014119
Dimensional reduction in anomaly mediation
Boyda, Ed; Murayama, Hitoshi; Pierce, Aaron
2002-04-01
We offer a guide to dimensional reduction in theories with anomaly-mediated supersymmetry breaking. Evanescent operators proportional to ɛ arise in the bare Lagrangian when it is reduced from d=4 to d=4-2ɛ dimensions. In the course of a detailed diagrammatic calculation, we show that inclusion of these operators is crucial. The evanescent operators conspire to drive the supersymmetry-breaking parameters along anomaly-mediation trajectories across heavy particle thresholds, guaranteeing the ultraviolet insensitivity.
Generalized Hebbian Algorithm for Dimensionality Reduction in Natural Language Processing
Gorrell, Genevieve
2006-01-01
The current surge of interest in search and comparison tasks in natural language processing has brought with it a focus on vector space approaches and vector space dimensionality reduction techniques. Presenting data as points in hyperspace provides opportunities to use a variety of welldeveloped tools pertinent to this representation. Dimensionality reduction allows data to be compressed and generalised. Eigen decomposition and related algorithms are one category of approaches to dimensional...
Random Projections for Dimensionality Reduction in ICA
Sabrina Gaito; Andrea Greppi; Giuliano Grossi
2008-01-01
In this paper we present a technique to speed up ICA based on the idea of reducing the dimensionality of the data set preserving the quality of the results. In particular we refer to FastICA algorithm which uses the Kurtosis as statistical property to be maximized. By performing a particular Johnson-Lindenstrauss like projection of the data set, we find the minimum dimensionality reduction rate ¤ü, defined as the ratio between the size k of the reduced space and the origi...
Unsupervised Dimensionality Reduction for Transfer Learning
Blöbaum, Patrick; Schulz, Alexander; Hammer, Barbara; Verleysen, Michel
2015-01-01
We investigate the suitability of unsupervised dimensionality reduction (DR) for transfer learning in the context of different representations of the source and target domain. Essentially, unsupervised DR establishes a link of source and target domain by representing the data in a common latent space. We consider two settings: a linear DR of source and target data which establishes correspondences of the data and an according transfer, and its combination with a non-linear D...
Dimensionality reduction in hyperspectral imagery
Gillis, David; Bowles, Jeffrey H.; Winter, Michael E.
2003-09-01
In this paper we examine how the projection of hyperspectral data into smaller dimensional subspaces can effect the propagation of error. In particular, we show that the nonorthogonality of endmembers in the linear mixing model can cause small changes in band space (as, for example, from the addition of noise) to lead to relatively large changes in the estimated abundance coefficients. We also show that increasing the number of endmembers can actually lead to an increase in the amount of possible error.
Topics on dimensional reduction: Solutions and consistency
Martinez Acosta, Rene R.
2000-10-01
We exploit the dimensional reduction ideas to interpret several solutions of low dimensional effective theories from the viewpoint of string theory and M-theory. We report that a rectangular three-dimensional lattice of intersecting domain walls in D = 4 dimensions, with arbitrary spacing, emerges naturally as a classical solution of M-theory. We also construct the non-linear Kaluza-Klein ansätze describing the embeddings of the U(1)3, U(1)4 and U(1)2 truncations of D = 5, D = 4 and D = 7 gauged supergravities into the type IIB string and M-theory. We use these general ansätze to embed and interpret the charged AdS5, AdS4 and AdS7 black hole solutions in ten and eleven dimensions. We then elaborate on the consistent truncation of Kaluza- Klein theories to their massless sector while retaining the full gauge symmetry associated with the isometry group G of the internal manifold M. We derive and test in general a consistency condition for Kaluza-Klein dimensional reduction for any Einstein space M̂ that is constructed as a U(1) bundle over a product of complex projective base spaces.
A complexity-regularized quantization approach to nonlinear dimensionality reduction
Raginsky, Maxim
2005-01-01
We consider the problem of nonlinear dimensionality reduction: given a training set of high-dimensional data whose ``intrinsic'' low dimension is assumed known, find a feature extraction map to low-dimensional space, a reconstruction map back to high-dimensional space, and a geometric description of the dimension-reduced data as a smooth manifold. We introduce a complexity-regularized quantization approach for fitting a Gaussian mixture model to the training set via a Lloyd algorithm. Complex...
Nearly-Kaehler dimensional reduction of the heterotic string
Chatzistavrakidis, Athanasios
2010-01-01
The effective action in four dimensions resulting from the ten-dimensional N=1 heterotic supergravity coupled to N=1 supersymmetric Yang-Mills upon dimensional reduction over nearly-Kaehler manifolds is discussed. Nearly-Kaehler manifolds are an interesting class of manifolds admitting an SU(3)-structure and in six dimensions all homogeneous nearly-Kaehler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold. Therefore it is natural to apply the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds in order to determine the four-dimensional theory.
Nearly-Kähler dimensional reduction of the heterotic string
Chatzistavrakidis, A.; Zoupanos, G.
2010-07-01
The effective action in four dimensions resulting from the ten-dimensional N=1 heterotic supergravity coupled to N=1 supersymmetric Yang-Mills upon dimensional reduction over nearly-Kaehler manifolds is discussed. Nearly-Kaehler manifolds are an interesting class of manifolds admitting an SU(3)-structure and in six dimensions all homogeneous nearly-Kaehler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold. Therefore it is natural to apply the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds in order to determine the four-dimensional theory.
Dimensionality Reduction by Weighted Connections between Neighborhoods
Xie, Fuding; Fan, Yutao; Zhou, Ming
2014-01-01
Dimensionality reduction is the transformation of high-dimensional data into a meaningful representation of reduced dimensionality. This paper introduces a dimensionality reduction technique by weighted connections between neighborhoods to improve $K$ -Isomap method, attempting to preserve perfectly the relationships between neighborhoods in the process of dimensionality reduction. The validity of the proposal is tested by three typical examples which are widely employed in the algorithms bas...
Confining membranes and dimensional reduction
Antonov, Dmitri
2001-11-01
The dual theory describing the 4D Coulomb gas of point-like magnetically charged objects, which confines closed electric strings, is considered. The respective generalization of the theory of confining strings to confining membranes is further constructed. The same is done for the analogous SU(3)-inspired model. We then consider a combined model which confines both electric charges and closed strings. Such a model is a mixture of the above-mentioned Coulomb gas with the condensate of the dual Higgs field, where the latter one is described by the dual abelian Higgs model. It is demonstrated that in a certain limit of this dual abelian Higgs model, the system under study undergoes naively the dimensional reduction and becomes described by the (completely integrable) 2D sine-Gordon theory. In particular, at finite temperature, this fact leads to the phase transition of the Berezinskii-Kosterlitz-Thouless type with the respective critical temperature expressed in terms of the parameters of the dual abelian Higgs model. However, it is finally discussed that the dimensional reduction is rigorously valid only in the strong coupling limit of the original 4D Coulomb gas. In such a limit, this reduction transforms the combined model into the 2D free bosonic theory.
Dimensional reduction and moment maps
Nagatomo, Yasuyuki
2002-03-01
We give a unified viewpoint of moment maps in the case of symplectic, hyper-Kähler, quaternion-Kähler and holomorphic contact manifolds. The Higgs field can be regarded as a moment map under some additional conditions in each case. Using dimensional reductions and moment maps, we reduce the standard 1 instanton on HP 1≅S 4 to an SO(3) instanton on CP 1× CP 1 and the standard 1 instanton on HP n to the standard 1 instanton on Gr 2( Cn+1) .
Multiple kernel learning for dimensionality reduction.
Lin, Yen-Yu; Liu, Tyng-Luh; Fuh, Chiou-Shann
2011-06-01
In solving complex visual learning tasks, adopting multiple descriptors to more precisely characterize the data has been a feasible way for improving performance. The resulting data representations are typically high-dimensional and assume diverse forms. Hence, finding a way of transforming them into a unified space of lower dimension generally facilitates the underlying tasks such as object recognition or clustering. To this end, the proposed approach (termed MKL-DR) generalizes the framework of multiple kernel learning for dimensionality reduction, and distinguishes itself with the following three main contributions: first, our method provides the convenience of using diverse image descriptors to describe useful characteristics of various aspects about the underlying data. Second, it extends a broad set of existing dimensionality reduction techniques to consider multiple kernel learning, and consequently improves their effectiveness. Third, by focusing on the techniques pertaining to dimensionality reduction, the formulation introduces a new class of applications with the multiple kernel learning framework to address not only the supervised learning problems but also the unsupervised and semi-supervised ones.
Dimensionality reduction via locally reconstructive patch alignment
Chen, Yi; Yin, Jun; Zhu, Jie; Jin, Zhong
2012-07-01
Based on the local patch concept, we proposed locally reconstructive patch alignment (LRPA) for dimensionality reduction. For each patch, LRPA aims to find the low-dimensional subspace in which the reconstruction error of the within-class nearest neighbors is minimized and the reconstruction error of the between-class nearest neighbors is maximized. LRPA preserves the local structure hidden in the high-dimensional space. More importantly, LRPA has natural connections with linear regression classification (LRC). While LRC uses reconstruction errors as the classification rule, a sample can be classified correctly when the within-class reconstruction error is minimal. The goal of LRPA makes it cooperate well with LRC. The experimental results on the extended Yale B (YALE-B), AR, PolyU finger knuckle print, and the palm print databases demonstrate LRPA plus LRC is an effective and robust pattern-recognition system.
Dimensionality reduction with unsupervised nearest neighbors
Kramer, Oliver
2013-01-01
This book is devoted to a novel approach for dimensionality reduction based on the famous nearest neighbor method that is a powerful classification and regression approach. It starts with an introduction to machine learning concepts and a real-world application from the energy domain. Then, unsupervised nearest neighbors (UNN) is introduced as efficient iterative method for dimensionality reduction. Various UNN models are developed step by step, reaching from a simple iterative strategy for discrete latent spaces to a stochastic kernel-based algorithm for learning submanifolds with independent parameterizations. Extensions that allow the embedding of incomplete and noisy patterns are introduced. Various optimization approaches are compared, from evolutionary to swarm-based heuristics. Experimental comparisons to related methodologies taking into account artificial test data sets and also real-world data demonstrate the behavior of UNN in practical scenarios. The book contains numerous color figures to illustr...
Dynamics of coset dimensional reduction
Karthauser, Josef L. P.; Saffin, P. M.
2006-04-01
The evolution of multiple scalar fields in cosmology has been much studied, particularly when the potential is formed from a series of exponentials. For a certain subclass of such systems it is possible to get “assisted“ behavior, where the presence of multiple terms in the potential effectively makes it shallower than the individual terms indicate. It is also known that when compactifying on coset spaces one can achieve a consistent truncation to an effective theory which contains many exponential terms; however, if there are too many exponentials then exact scaling solutions do not exist. In this paper we study the potentials arising from such compactifications of eleven-dimensional supergravity and analyze the regions of parameter space which could lead to scaling behavior.
Maximal Linear Embedding for Dimensionality Reduction.
Wang, Ruiping; Shan, Shiguang; Chen, Xilin; Chen, Jie; Gao, Wen
2011-09-01
Over the past few decades, dimensionality reduction has been widely exploited in computer vision and pattern analysis. This paper proposes a simple but effective nonlinear dimensionality reduction algorithm, named Maximal Linear Embedding (MLE). MLE learns a parametric mapping to recover a single global low-dimensional coordinate space and yields an isometric embedding for the manifold. Inspired by geometric intuition, we introduce a reasonable definition of locally linear patch, Maximal Linear Patch (MLP), which seeks to maximize the local neighborhood in which linearity holds. The input data are first decomposed into a collection of local linear models, each depicting an MLP. These local models are then aligned into a global coordinate space, which is achieved by applying MDS to some randomly selected landmarks. The proposed alignment method, called Landmarks-based Global Alignment (LGA), can efficiently produce a closed-form solution with no risk of local optima. It just involves some small-scale eigenvalue problems, while most previous aligning techniques employ time-consuming iterative optimization. Compared with traditional methods such as ISOMAP and LLE, our MLE yields an explicit modeling of the intrinsic variation modes of the observation data. Extensive experiments on both synthetic and real data indicate the effectivity and efficiency of the proposed algorithm.
Stochastic inflation and dimensional reduction
Kühnel, Florian; Schwarz, Dominik J.
2008-11-01
We adopt methods that are well-known in statistical physics to the problem of stochastic inflation. The effective power spectrum for the classical, stochastic long-wavelength fluctuations is calculated for free scalar fields in a de Sitter background. For a smooth separation into long and short wavelengths, we identify an infrared divergence of the effective power spectrum, which has its correspondence in statistical physics in the phenomenon of dimensional reduction. The inflationary dynamics pushes the affected scales exponentially fast to large superhorizon scales, and establishes scale-invariant behavior for smaller scales (for massless fields). In the limit of a sharp separation of wavelengths, the scale of the infrared divergence is pushed to infinity.
Limitations on quantum dimensionality reduction
Harrow, Aram W.; Montanaro, Ashley; Short, Anthony J.
2015-06-01
The Johnson-Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(logn) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over quantum channels that significantly reduces the dimension of quantum states while preserving the 2-norm distance with high probability. We discuss two tasks for which the 2-norm distance is indeed the correct figure of merit. In the case of the trace norm, we show that the dimension of low-rank mixed states can be reduced by up to a square root, but that essentially no dimensionality reduction is possible for highly mixed states.
Robust Nonnegative Patch Alignment for Dimensionality Reduction.
You, Xinge; Ou, Weihua; Chen, Chun Lung Philip; Li, Qiang; Zhu, Ziqi; Tang, Yuanyan
2015-11-01
Dimensionality reduction is an important method to analyze high-dimensional data and has many applications in pattern recognition and computer vision. In this paper, we propose a robust nonnegative patch alignment for dimensionality reduction, which includes a reconstruction error term and a whole alignment term. We use correntropy-induced metric to measure the reconstruction error, in which the weight is learned adaptively for each entry. For the whole alignment, we propose locality-preserving robust nonnegative patch alignment (LP-RNA) and sparsity-preserviing robust nonnegative patch alignment (SP-RNA), which are unsupervised and supervised, respectively. In the LP-RNA, we propose a locally sparse graph to encode the local geometric structure of the manifold embedded in high-dimensional space. In particular, we select large p -nearest neighbors for each sample, then obtain the sparse representation with respect to these neighbors. The sparse representation is used to build a graph, which simultaneously enjoys locality, sparseness, and robustness. In the SP-RNA, we simultaneously use local geometric structure and discriminative information, in which the sparse reconstruction coefficient is used to characterize the local geometric structure and weighted distance is used to measure the separability of different classes. For the induced nonconvex objective function, we formulate it into a weighted nonnegative matrix factorization based on half-quadratic optimization. We propose a multiplicative update rule to solve this function and show that the objective function converges to a local optimum. Several experimental results on synthetic and real data sets demonstrate that the learned representation is more discriminative and robust than most existing dimensionality reduction methods.
Dimensionality Reduction Algorithms on High Dimensional Datasets
Iwan Syarif
2014-01-01
Classification problem especially for high dimensional datasets have attracted many researchers in order to find efficient approaches to address them. However, the classification problem has become very complicatedespecially when the number of possible different combinations of variables is so high. In this research, we evaluate the performance of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) as feature selection algorithms when applied to high dimensional datasets.Our experime...
Sampling, Metric Entropy and Dimensionality Reduction
Batenkov, D.; Friedland, O.; Yomdin, Y.
2013-01-01
Let $Q$ be a relatively compact subset in a Hilbert space $V$. For a given $\\e>0$ let $N(\\e,Q)$ be the minimal number of linear measurements, sufficient to reconstruct any $x \\in Q$ with the accuracy $\\e$. We call $N(\\e,Q)$ a sampling $\\e$-entropy of $Q$. Using Dimensionality Reduction, as provided by the Johnson-Lindenstrauss lemma, we show that, in an appropriate probabilistic setting, $N(\\e,Q)$ is bounded from above by the Kolmogorov's $\\e$-entropy $H(\\e,Q)$, defined as $H(\\e,Q)=\\log M(\\e,...
Unsupervised 2D Dimensionality Reduction with Adaptive Structure Learning.
Zhao, Xiaowei; Nie, Feiping; Wang, Sen; Guo, Jun; Xu, Pengfei; Chen, Xiaojiang
2017-05-01
In recent years, unsupervised two-dimensional (2D) dimensionality reduction methods for unlabeled large-scale data have made progress. However, performance of these degrades when the learning of similarity matrix is at the beginning of the dimensionality reduction process. A similarity matrix is used to reveal the underlying geometry structure of data in unsupervised dimensionality reduction methods. Because of noise data, it is difficult to learn the optimal similarity matrix. In this letter, we propose a new dimensionality reduction model for 2D image matrices: unsupervised 2D dimensionality reduction with adaptive structure learning (DRASL). Instead of using a predetermined similarity matrix to characterize the underlying geometry structure of the original 2D image space, our proposed approach involves the learning of a similarity matrix in the procedure of dimensionality reduction. To realize a desirable neighbors assignment after dimensionality reduction, we add a constraint to our model such that there are exact [Formula: see text] connected components in the final subspace. To accomplish these goals, we propose a unified objective function to integrate dimensionality reduction, the learning of the similarity matrix, and the adaptive learning of neighbors assignment into it. An iterative optimization algorithm is proposed to solve the objective function. We compare the proposed method with several 2D unsupervised dimensionality methods. K-means is used to evaluate the clustering performance. We conduct extensive experiments on Coil20, AT&T, FERET, USPS, and Yale data sets to verify the effectiveness of our proposed method.
Dimensionality reduction of collective motion by principal manifolds
Gajamannage, Kelum; Butail, Sachit; Porfiri, Maurizio; Bollt, Erik M.
2015-01-01
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods is not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the mapping from the original high-dimensional space to the embedding space. Here, we propose an alternative approach that demands a two-dimensional embedding which topologically summarizes the high-dimensional data. In this sense, our approach is closely related to the construction of one-dimensional principal curves that minimize orthogonal error to data points subject to smoothness constraints. Specifically, we construct a two-dimensional principal manifold directly in the high-dimensional space using cubic smoothing splines, and define the embedding coordinates in terms of geodesic distances. Thus, the mapping from the high-dimensional data to the manifold is defined in terms of local coordinates. Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets. The principal manifold finding algorithm is applied to configurations obtained from a dynamical system of multiple agents simulating a complex maneuver called predator mobbing, and the resulting two-dimensional embedding is compared with that of a well-established nonlinear dimensionality reduction method.
Rotational Linear Discriminant Analysis Using Bayes Rule for Dimensionality Reduction
Alok Sharma; Kuldip K. Paliwal
2006-01-01
Linear discriminant analysis (LDA) finds an orientation that projects high dimensional feature vectors to reduced dimensional feature space in such a way that the overlapping between the classes in this feature space is minimum. This overlapping is usually finite and produces finite classification error which is further minimized by rotational LDA technique. This rotational LDA technique rotates the classes individually in the original feature space in a manner that enables further reduction ...
Visualizing the quality of dimensionality reduction
Mokbel, Bassam; Lueks, Wouter; Gisbrecht, Andrej; Hammer, Barbara
2013-01-01
The growing number of dimensionality reduction methods available for data visualization has recently inspired the development of formal measures to evaluate the resulting low-dimensional representation independently from the methods' inherent criteria. Many evaluation measures can be summarized
Efficient Kernelization of Discriminative Dimensionality Reduction
Schulz, Alexander; Brinkrolf, Johannes; Hammer, Barbara
2017-01-01
Modern nonlinear dimensionality reduction (DR) techniques project high dimensional data to low dimensions for their visual inspection. Provided the intrinsic data dimensionality is larger than two, DR nec- essarily faces information loss and the problem becomes ill-posed. Dis- criminative dimensionality reduction (DiDi) offers one intuitive way to reduce this ambiguity: it allows a practitioner to identify what is relevant and what should be regarded as noise by means of int...
Dimensionality Reduction by Local Discriminative Gaussians
Parrish, Nathan; Gupta, Maya
2012-01-01
We present local discriminative Gaussian (LDG) dimensionality reduction, a supervised dimensionality reduction technique for classification. The LDG objective function is an approximation to the leave-one-out training error of a local quadratic discriminant analysis classifier, and thus acts locally to each training point in order to find a mapping where similar data can be discriminated from dissimilar data. While other state-of-the-art linear dimensionality reduction methods require gradien...
Ensembles of Classifiers based on Dimensionality Reduction
Schclar, Alon; Rokach, Lior; Amit, Amir
2013-01-01
We present a novel approach for the construction of ensemble classifiers based on dimensionality reduction. Dimensionality reduction methods represent datasets using a small number of attributes while preserving the information conveyed by the original dataset. The ensemble members are trained based on dimension-reduced versions of the training set. These versions are obtained by applying dimensionality reduction to the original training set using different values of the input parameters. Thi...
Universal features of dimensional reduction schemes from general covariance breaking
Maraner, Paolo; Pachos, Jiannis K.
2008-08-01
Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors, that successfully generalize to one side Kaluza-Klein gauge fields and to the other side extrinsic curvature and torsion of embedded spaces, thus fully characterizing the geometry of dimensional reduction. We obtain general formulas for the reduction of the main tensors and operators of Riemannian geometry. In particular, we provide what is probably the maximal possible generalization of Gauss, Codazzi and Ricci equations and various other standard formulas in Kaluza-Klein and embedded spacetimes theories. After general covariance breaking, part of the residual covariance is perceived by effective lower dimensional observers as an infinite dimensional gauge group. This reduces to finite dimensions in Kaluza-Klein and other few remarkable backgrounds, all characterized by the vanishing of appropriate lower dimensional tensors.
Using dimensional reduction for hadronic collisions
Signer, Adrian; Stöckinger, Dominik
2009-02-01
We discuss how to apply regularization by dimensional reduction for computing hadronic cross sections at next-to-leading order. We analyze the infrared singularity structure, demonstrate that there are no problems with factorization, and show how to use dimensional reduction in conjunction with standard parton distribution functions. We clarify that different versions of dimensional reduction with different infrared and factorization behaviour have been used in the literature. Finally, we give transition rules for translating the various parts of next-to-leading order cross sections from dimensional reduction to other regularization schemes.
Equivariant dimensional reduction and quiver gauge theories
Dolan, Brian P.; Szabo, Richard J.
2011-09-01
We review recent applications of equivariant dimensional reduction techniques to the construction of Yang-Mills-Higgs-Dirac theories with dynamical mass generation and exactly massless chiral fermions.
Nonequilibrium quantum meson gas: Dimensional reduction
Alvarez-Estrada, R. F.
2009-07-01
A nonequilibrium quantum gas of interacting relativistic effective mesons, ressembling qualitatively those produced in a heavy-ion collision, is described by a scalar φ^{{4}}_{} quantum field in (1 + 3) -dimensional Minkowski space. For high temperature and large temporal and spatial scales, we justify that classical statistical mechanics including quantum renormalization effects describe approximately the gas: nonequilibrium dimensional reduction (NEDR). As a source of hints, we treat the gas at equilibrium in real-time formalism and obtain simplifications for high temperature and large spatial scales, thereby extending a useful equilibrium dimensional reduction known for the imaginary-time formalism. By assumption, the nonequilibrium initial state of the gas, not far from thermal equilibrium, includes interactions and inhomogeneities. We use nonequilibrium real-time generating functionals and correlators at nonzero temperature. In the NEDR regime, our arguments yield: 1) renormalized correlators simplify, 2) the perturbative series for those simplified correlators can be resummed into a new nonequilibrium generating functional, Z’ r, dr , which is super-renormalizable and includes renormalization effects (large position-dependent thermal self-energies and effective couplings). Z’ r, dr could enable to study nonperturbatively changes in the phase structures of the field, by proceeding from the nonequilibrium quantum regime to the NEDR one.
Dimensionality reduction in Bayesian estimation algorithms
Directory of Open Access Journals (Sweden)
G. W. Petty
2013-09-01
Full Text Available An idealized synthetic database loosely resembling 3-channel passive microwave observations of precipitation against a variable background is employed to examine the performance of a conventional Bayesian retrieval algorithm. For this dataset, algorithm performance is found to be poor owing to an irreconcilable conflict between the need to find matches in the dependent database versus the need to exclude inappropriate matches. It is argued that the likelihood of such conflicts increases sharply with the dimensionality of the observation space of real satellite sensors, which may utilize 9 to 13 channels to retrieve precipitation, for example. An objective method is described for distilling the relevant information content from N real channels into a much smaller number (M of pseudochannels while also regularizing the background (geophysical plus instrument noise component. The pseudochannels are linear combinations of the original N channels obtained via a two-stage principal component analysis of the dependent dataset. Bayesian retrievals based on a single pseudochannel applied to the independent dataset yield striking improvements in overall performance. The differences between the conventional Bayesian retrieval and reduced-dimensional Bayesian retrieval suggest that a major potential problem with conventional multichannel retrievals – whether Bayesian or not – lies in the common but often inappropriate assumption of diagonal error covariance. The dimensional reduction technique described herein avoids this problem by, in effect, recasting the retrieval problem in a coordinate system in which the desired covariance is lower-dimensional, diagonal, and unit magnitude.
Dimensionality reduction in Bayesian estimation algorithms
Petty, G. W.
2013-09-01
An idealized synthetic database loosely resembling 3-channel passive microwave observations of precipitation against a variable background is employed to examine the performance of a conventional Bayesian retrieval algorithm. For this dataset, algorithm performance is found to be poor owing to an irreconcilable conflict between the need to find matches in the dependent database versus the need to exclude inappropriate matches. It is argued that the likelihood of such conflicts increases sharply with the dimensionality of the observation space of real satellite sensors, which may utilize 9 to 13 channels to retrieve precipitation, for example. An objective method is described for distilling the relevant information content from N real channels into a much smaller number (M) of pseudochannels while also regularizing the background (geophysical plus instrument) noise component. The pseudochannels are linear combinations of the original N channels obtained via a two-stage principal component analysis of the dependent dataset. Bayesian retrievals based on a single pseudochannel applied to the independent dataset yield striking improvements in overall performance. The differences between the conventional Bayesian retrieval and reduced-dimensional Bayesian retrieval suggest that a major potential problem with conventional multichannel retrievals - whether Bayesian or not - lies in the common but often inappropriate assumption of diagonal error covariance. The dimensional reduction technique described herein avoids this problem by, in effect, recasting the retrieval problem in a coordinate system in which the desired covariance is lower-dimensional, diagonal, and unit magnitude.
Dimensionality reduction for registration of high-dimensional data sets.
Xu, Min; Chen, Hao; Varshney, Pramod K
2013-08-01
Registration of two high-dimensional data sets often involves dimensionality reduction to yield a single-band image from each data set followed by pairwise image registration. We develop a new application-specific algorithm for dimensionality reduction of high-dimensional data sets such that the weighted harmonic mean of Cramér-Rao lower bounds for the estimation of the transformation parameters for registration is minimized. The performance of the proposed dimensionality reduction algorithm is evaluated using three remotes sensing data sets. The experimental results using mutual information-based pairwise registration technique demonstrate that our proposed dimensionality reduction algorithm combines the original data sets to obtain the image pair with more texture, resulting in improved image registration.
Holography, Dimensional Reduction and the Bekenstein Bound
Bak, Dongsu; Yee, Ho-Ung
2004-04-01
We consider dimensional reduction of the lightlike holography of the covariant entropy bound from D+1 dimensional geometry of M × S1 to the D dimensional geometry M. With a warping factor, the local Bekenstein bound in D+1 dimensions leads to a more refined form of the bound from the D dimensional view point. With this new local Bekenstein bound, it is quite possible to saturate the lightlike holography even with nonvanishing expansion rate. With a Kaluza-Klein gauge field, the dimensional reduction implies a stronger bound where the energy momentum tensor contribution is replaced by the energy momentum tensor with the electromagnetic contribution subtracted.
Dimensional Reduction for Generalized Continuum Polymers
Helmuth, Tyler
2016-10-01
The Brydges-Imbrie dimensional reduction formula relates the pressure of a d-dimensional gas of hard spheres to a model of (d+2)-dimensional branched polymers. Brydges and Imbrie's proof was non-constructive and relied on a supersymmetric localization lemma. The main result of this article is a constructive proof of a more general dimensional reduction formula that contains the Brydges-Imbrie formula as a special case. Central to the proof are invariance lemmas, which were first introduced by Kenyon and Winkler for branched polymers. The new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to Mészáros and Postnikov. Several applications are presented, notably dimensional reduction formulas for (i) non-spherical bodies and (ii) for corrections to the pressure due to symmetry effects.
Algorithmic dimensionality reduction for molecular structure analysis
Brown, W. Michael; Martin, Shawn; Pollock, Sara N.; Coutsias, Evangelos A.; Watson, Jean-Paul
2008-01-01
Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated alg...
DROP: Dimensionality Reduction Optimization for Time Series
Suri, Sahaana; Bailis, Peter
2017-01-01
Dimensionality reduction is critical in analyzing increasingly high-volume, high-dimensional time series. In this paper, we revisit a now-classic study of time series dimensionality reduction operators and find that for a given quality constraint, Principal Component Analysis (PCA) uncovers representations that are over 2x smaller than those obtained via alternative techniques favored in the literature. However, as classically implemented via Singular Value Decomposition (SVD), PCA is incredi...
Dimensionality reduction methods for molecular simulations
Doerr, Stefan; Ariz-Extreme, Igor; Harvey, Matthew J.; De Fabritiis, Gianni
2017-01-01
Molecular simulations produce very high-dimensional data-sets with millions of data points. As analysis methods are often unable to cope with so many dimensions, it is common to use dimensionality reduction and clustering methods to reach a reduced representation of the data. Yet these methods often fail to capture the most important features necessary for the construction of a Markov model. Here we demonstrate the results of various dimensionality reduction methods on two simulation data-set...
Dimensional regularization in configuration space
International Nuclear Information System (INIS)
Bollini, C.G.; Giambiagi, J.J.
1995-09-01
Dimensional regularization is introduced in configuration space by Fourier transforming in D-dimensions the perturbative momentum space Green functions. For this transformation, Bochner theorem is used, no extra parameters, such as those of Feynman or Bogoliubov-Shirkov are needed for convolutions. The regularized causal functions in x-space have ν-dependent moderated singularities at the origin. They can be multiplied together and Fourier transformed (Bochner) without divergence problems. The usual ultraviolet divergences appear as poles of the resultant functions of ν. Several example are discussed. (author). 9 refs
Understanding protein flexibility through dimensionality reduction.
Teodoro, Miguel L; Phillips, George N; Kavraki, Lydia E
2003-01-01
This work shows how to decrease the complexity of modeling flexibility in proteins by reducing the number of dimensions necessary to model important macromolecular motions such as the induced-fit process. Induced fit occurs during the binding of a protein to other proteins, nucleic acids, or small molecules (ligands) and is a critical part of protein function. It is now widely accepted that conformational changes of proteins can affect their ability to bind other molecules and that any progress in modeling protein motion and flexibility will contribute to the understanding of key biological functions. However, modeling protein flexibility has proven a very difficult task. Experimental laboratory methods, such as x-ray crystallography, produce rather limited information, while computational methods such as molecular dynamics are too slow for routine use with large systems. In this work, we show how to use the principal component analysis method, a dimensionality reduction technique, to transform the original high-dimensional representation of protein motion into a lower dimensional representation that captures the dominant modes of motions of proteins. For a medium-sized protein, this corresponds to reducing a problem with a few thousand degrees of freedom to one with less than fifty. Although there is inevitably some loss in accuracy, we show that we can obtain conformations that have been observed in laboratory experiments, starting from different initial conformations and working in a drastically reduced search space.
Cascade Support Vector Machines with Dimensionality Reduction
Directory of Open Access Journals (Sweden)
Oliver Kramer
2015-01-01
Full Text Available Cascade support vector machines have been introduced as extension of classic support vector machines that allow a fast training on large data sets. In this work, we combine cascade support vector machines with dimensionality reduction based preprocessing. The cascade principle allows fast learning based on the division of the training set into subsets and the union of cascade learning results based on support vectors in each cascade level. The combination with dimensionality reduction as preprocessing results in a significant speedup, often without loss of classifier accuracies, while considering the high-dimensional pendants of the low-dimensional support vectors in each new cascade level. We analyze and compare various instantiations of dimensionality reduction preprocessing and cascade SVMs with principal component analysis, locally linear embedding, and isometric mapping. The experimental analysis on various artificial and real-world benchmark problems includes various cascade specific parameters like intermediate training set sizes and dimensionalities.
Dimensionality Reduction Library v 0.2
Energy Technology Data Exchange (ETDEWEB)
2009-06-12
Dimensionality Reduction Library is a C++ library for dimensionality reduction. In the context of this library, dimensionality reduction is considered to consist of 1)estimation of the intrinsic dimensionality using sampled data, 2) Finding maps that reduce the diemsionality of data (forward map) or increase the dimensionality of data (reverse map) and 3) mapping arbitray coordiantes to high and low dimensionalities. The library is intended toprovide a consistent interface to multiple dimensionality reduction algorithms with an efficient C++ interface that runs efficiently on multicore architectures. A few routines have been optimized with an option for GPU acceleration or distributed computation. Currently the library offers intrinsic dimensionality estimation using point-PCA reconstruction error and/ residual variance. The following dimensionality reduction methods have been implemented: Principal Component Analysis Multidimensional Scaling Locally Linear Embedding IsoMap Autoencoder Neutral Networks An executable is also supplied that can be built to allow for command-line access to the library routines. A description for an applciation of the library for molecular structure analysis has been published.
Dimensional reduction for D3-brane moduli
Energy Technology Data Exchange (ETDEWEB)
Cownden, Brad [Department of Physics & Astronomy, University of Manitoba,Winnipeg, Manitoba R3T 2N2 (Canada); Frey, Andrew R. [Department of Physics & Astronomy, University of Manitoba,Winnipeg, Manitoba R3T 2N2 (Canada); Department of Physics, University of Winnipeg,Winnipeg, Manitoba R3B 2E9 (Canada); Marsh, M.C. David [Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Cambridge, CB3 0WA (United Kingdom); Underwood, Bret [Department of Physics, Pacific Lutheran University,Tacoma, WA 98447 (United States)
2016-12-28
Warped string compactifications are central to many attempts to stabilize moduli and connect string theory with cosmology and particle phenomenology. We present a first-principles derivation of the low-energy 4D effective theory from dimensional reduction of a D3-brane in a warped Calabi-Yau compactification of type IIB string theory with imaginary self-dual 3-form flux, including effects of D3-brane motion beyond the probe approximation, and find the metric on the moduli space of brane positions, the universal volume modulus, and axions descending from the 4-form potential. As D3-branes may be considered as carrying either electric or magnetic charges for the self-dual 5-form field strength, we present calculations in both duality frames. Our results are consistent with, but extend significantly, earlier results on the low-energy effective theory arising from D3-branes in string compactifications.
Dimensional reduction for D3-brane moduli
Cownden, Brad; Frey, Andrew R.; Marsh, M. C. David; Underwood, Bret
2016-12-01
Warped string compactifications are central to many attempts to stabilize moduli and connect string theory with cosmology and particle phenomenology. We present a first-principles derivation of the low-energy 4D effective theory from dimensional reduction of a D3-brane in a warped Calabi-Yau compactification of type IIB string theory with imaginary self-dual 3-form flux, including effects of D3-brane motion beyond the probe approximation, and find the metric on the moduli space of brane positions, the universal volume modulus, and axions descending from the 4-form potential. As D3-branes may be considered as carrying either electric or magnetic charges for the self-dual 5-form field strength, we present calculations in both duality frames. Our results are consistent with, but extend significantly, earlier results on the low-energy effective theory arising from D3-branes in string compactifications.
Multiple Kernel Spectral Regression for Dimensionality Reduction
Directory of Open Access Journals (Sweden)
Bing Liu
2013-01-01
Full Text Available Traditional manifold learning algorithms, such as locally linear embedding, Isomap, and Laplacian eigenmap, only provide the embedding results of the training samples. To solve the out-of-sample extension problem, spectral regression (SR solves the problem of learning an embedding function by establishing a regression framework, which can avoid eigen-decomposition of dense matrices. Motivated by the effectiveness of SR, we incorporate multiple kernel learning (MKL into SR for dimensionality reduction. The proposed approach (termed MKL-SR seeks an embedding function in the Reproducing Kernel Hilbert Space (RKHS induced by the multiple base kernels. An MKL-SR algorithm is proposed to improve the performance of kernel-based SR (KSR further. Furthermore, the proposed MKL-SR algorithm can be performed in the supervised, unsupervised, and semi-supervised situation. Experimental results on supervised classification and semi-supervised classification demonstrate the effectiveness and efficiency of our algorithm.
Dimensional reductions of BKP and CKP hierarchies
Loris, Ignace
2001-04-01
A discussion of dimensional reductions, which are not classical symmetry reductions, is made for the BKP and CKP hierarchies of integrable evolution equations. A novel direct method for testing Pfaffian solutions to bilinear identities is presented and applied to these reductions.
Reduction of infinite dimensional equations
Directory of Open Access Journals (Sweden)
Zhongding Li
2006-02-01
Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.
A Difference Criterion for Dimensionality Reduction
Aved, A. J.; Blasch, E.; Peng, J.
2015-12-01
A dynamic data-driven geoscience application includes hyperspectral scene classification which has shown promising potential in many remote-sensing applications. A hyperspectral image of a scene spectral radiance is typically measured by hundreds of contiguous spectral bands or features, ranging from visible/near-infrared (VNIR) to shortwave infrared (SWIR). Spectral-reflectance measurements provide rich information for object detection and classification. On the other hand, they generate a large number of features, resulting in a high dimensional measurement space. However, a large number of features often poses challenges and can result in poor classification performance. This is due to the curse of dimensionality which requires model reduction, uncertainty quantification and optimization for real-world applications. In such situations, feature extraction or selection methods play an important role by significantly reducing the number of features for building classifiers. In this work, we focus on efficient feature extraction using the dynamic data-driven applications systems (DDDAS) paradigm. Many dimension reduction techniques have been proposed in the literature. A well-known technique is Fisher's linear discriminant analysis (LDA). LDA finds the projection matrix that simultaneously maximizes a within class scatter matrix and minimizes a between class scatter matrix. However, LDA requires matrix inverse which can be a major issue when the within matrix is singular. We propose a difference criterion for dimension reduction that does not require a matrix inverse for software implementation. We show how to solve the optimization problem with semi-definite programming. In addition, we establish an error bound for the proposed algorithm. We demonstrate the connection between relief feature selection and a two class formulation of multi-class problems, thereby providing a sound basis for observed benefits associated with this formulation. Finally, we provide
Fukunaga-Koontz transform based dimensionality reduction for hyperspectral imagery
Ochilov, S.; Alam, M. S.; Bal, A.
2006-05-01
Fukunaga-Koontz Transform based technique offers some attractive properties for desired class oriented dimensionality reduction in hyperspectral imagery. In FKT, feature selection is performed by transforming into a new space where feature classes have complimentary eigenvectors. Dimensionality reduction technique based on these complimentary eigenvector analysis can be described under two classes, desired class and background clutter, such that each basis function best represent one class while carrying the least amount of information from the second class. By selecting a few eigenvectors which are most relevant to desired class, one can reduce the dimension of hyperspectral cube. Since the FKT based technique reduces data size, it provides significant advantages for near real time detection applications in hyperspectral imagery. Furthermore, the eigenvector selection approach significantly reduces computation burden via the dimensionality reduction processes. The performance of the proposed dimensionality reduction algorithm has been tested using real-world hyperspectral dataset.
Nonlinear dimensionality reduction by locally linear embedding.
Roweis, S T; Saul, L K
2000-12-22
Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text.
Nonlinear Dimensionality Reduction by Locally Linear Embedding
Roweis, Sam T.; Saul, Lawrence K.
2000-12-01
Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text.
Exploring Dimensionality Reduction for Text Mining
National Research Council Canada - National Science Library
Underhill, David G
2007-01-01
.... Both of these challenges can be addressed with "dimensionality reduction" (DR). DR is the process of transforming a large amount of data into a much smaller, less noisy representation that preserves...
High Temperature Dimensional Reduction and Parity Violation
Kajantie, Keijo; Rummukainen, K; Shaposhnikov, Mikhail E
1998-01-01
The effective super-renormalizable 3-dimensional Lagrangian, describing the high temperature limit of chiral gauge theories, has more symmetry than the original 4d Lagrangian: parity violation is absent. Parity violation appears in the 3d theory only through higher-dimensional operators. We compute the coefficients of dominant P-odd operators in the Standard Electroweak theory and discuss their implications. We also clarify the parametric accuracy obtained with dimensional reduction.
Sufficient Dimensionality Reduction with Irrelevant Statistics
Globerson, Amir; Chechik, Gal; Tishby, Naftali
2012-01-01
The problem of finding a reduced dimensionality representation of categorical variables while preserving their most relevant characteristics is fundamental for the analysis of complex data. Specifically, given a co-occurrence matrix of two variables, one often seeks a compact representation of one variable which preserves information about the other variable. We have recently introduced ``Sufficient Dimensionality Reduction' [GT-2003], a method that extracts continuous reduced dimensional fea...
On nonlinear dimensionality reduction for face recognition
Huang, Weilin; Yin, Hujun
2012-01-01
The curse of dimensionality has prompted intensive research in effective methods of mapping high dimensional data. Dimensionality reduction and subspace learning have been studied extensively and widely applied to feature extraction and pattern representation in image and vision applications. Although PCA has long been regarded as a simple, efficient linear subspace technique, many nonlinear methods such as kernel PCA, local linear embedding, and self-organizing networks have been proposed re...
Incomplete Pivoted QR-based Dimensionality Reduction
Bermanis, Amit; Rotbart, Aviv; Salhov, Moshe; Averbuch, Amir
2016-01-01
High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of dimensionality' phenomenon. Therefore, dimensionality reduction methods are applied to the data prior to its analysis. Many of these methods are based on principal components analysis, which is statistically driven, namely they map the data into a low-dimension subsp...
Outlier preservation by dimensionality reduction techniques
Onderwater, Martijn
2015-01-01
htmlabstractSensors are increasingly part of our daily lives: motion detection, lighting control, and energy consumption all rely on sensors. Combining this information into, for instance, simple and comprehensive graphs can be quite challenging. Dimensionality reduction is often used to address this problem, by decreasing the number of variables in the data and looking for shorter representations. However, dimensionality reduction is often aimed at normal daily data, and applying it to event...
Dimension and dimensional reduction in quantum gravity
Carlip, S.
2017-10-01
A number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of ‘dimension’ and concluding with some speculative ideas of what dimensional reduction might mean for physics.
Dimensional reduction for generalized Poisson brackets
Acatrinei, Ciprian Sorin
2008-02-01
We discuss dimensional reduction for Hamiltonian systems which possess nonconstant Poisson brackets between pairs of coordinates and between pairs of momenta. The associated Jacobi identities imply that the dimensionally reduced brackets are always constant. Some examples are given alongside the general theory.
Spontaneous dimensional reduction in quantum gravity
Carlip, S.
2016-07-01
Hints from a number of different approaches to quantum gravity point to a phenomenon of “spontaneous dimensional reduction” to two spacetime dimensions near the Planck scale. I examine the physical meaning of the term “dimension” in this context, summarize the evidence for dimensional reduction, and discuss possible physical explanations.
Local coordinates alignment with global preservation for dimensionality reduction.
Chen, Jing; Ma, Zhengming; Liu, Yang
2013-01-01
Dimensionality reduction is vital in many fields, and alignment-based methods for nonlinear dimensionality reduction have become popular recently because they can map the high-dimensional data into a low-dimensional subspace with the property of local isometry. However, the relationships between patches in original high-dimensional space cannot be ensured to be fully preserved during the alignment process. In this paper, we propose a novel method for nonlinear dimensionality reduction called local coordinates alignment with global preservation. We first introduce a reasonable definition of topology-preserving landmarks (TPLs), which not only contribute to preserving the global structure of datasets and constructing a collection of overlapping linear patches, but they also ensure that the right landmark is allocated to the new test point. Then, an existing method for dimensionality reduction that has good performance in preserving the global structure is used to derive the low-dimensional coordinates of TPLs. Local coordinates of each patch are derived using tangent space of the manifold at the corresponding landmark, and then these local coordinates are aligned into a global coordinate space with the set of landmarks in low-dimensional space as reference points. The proposed alignment method, called landmarks-based alignment, can produce a closed-form solution without any constraints, while most previous alignment-based methods impose the unit covariance constraint, which will result in the deficiency of global metrics and undesired rescaling of the manifold. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed algorithm.
Dictionary Learning Based Dimensionality Reduction for Classification
Schnass, Karin; Vandergheynst, Pierre
2008-01-01
In this article we present a signal model for classification based on a low dimensional dictionary embedded into the high dimensional signal space. We develop an alternate projection algorithm to find the embedding and the dictionary and finally test the classification performance of our scheme in comparison to Fisher’s LDA.
Dimensionality Reduction Through Classifier Ensembles
Oza, Nikunj C.; Tumer, Kagan; Norwig, Peter (Technical Monitor)
1999-01-01
In data mining, one often needs to analyze datasets with a very large number of attributes. Performing machine learning directly on such data sets is often impractical because of extensive run times, excessive complexity of the fitted model (often leading to overfitting), and the well-known "curse of dimensionality." In practice, to avoid such problems, feature selection and/or extraction are often used to reduce data dimensionality prior to the learning step. However, existing feature selection/extraction algorithms either evaluate features by their effectiveness across the entire data set or simply disregard class information altogether (e.g., principal component analysis). Furthermore, feature extraction algorithms such as principal components analysis create new features that are often meaningless to human users. In this article, we present input decimation, a method that provides "feature subsets" that are selected for their ability to discriminate among the classes. These features are subsequently used in ensembles of classifiers, yielding results superior to single classifiers, ensembles that use the full set of features, and ensembles based on principal component analysis on both real and synthetic datasets.
Mixed projection pursuit-based dimensionality reduction
Safavi, Haleh; Chang, Chein-I.
2009-05-01
Projection Pursuit (PP) is a component transform technique which seeks a component whose projection vector points to a direction of interestingness in data space which can be specified by a Projection Index (PI). Two most popular component analysis-based techniques, Principal Components Analysis (PCA), Independent Component Analysis (ICA) can be considered as special cases with their PIs specified by data variance and statistical independency respectively. Despite the fact that various component analysis-based techniques have been used for Dimensionality Reduction (DR) the components are generally generated by a specific technique. Even in the case of PP, the same PI has been used to generate project components. This paper explores the utility of PP in DR where various projection indexes are used for DR in context of PP. It further lays out a general setting for PP-based DR and develops algorithms to perform one dimension reduction at a time by using different PIs. In order to substantiate our findings, experiments are conducted to demonstrate advantages of the PP with mixed PIs-based DR over traditional PCA-based, ICA-based and PP-based DR techniques.
Adaptive Sampling for Nonlinear Dimensionality Reduction Based on Manifold Learning
DEFF Research Database (Denmark)
Franz, Thomas; Zimmermann, Ralf; Goertz, Stefan
2017-01-01
We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space that is approxi......We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space...... that is approximately isometric to the manifold that is assumed to be formed by the high-fidelity Navier-Stokes flow solutions under smooth variations of the inflow conditions. The focus of the work at hand is the adaptive construction and refinement of the Isomap emulator: We exploit the non-Euclidean Isomap metric...
A Fourier dimensionality reduction model for big data interferometric imaging
Vijay Kartik, S.; Carrillo, Rafael E.; Thiran, Jean-Philippe; Wiaux, Yves
2017-06-01
Data dimensionality reduction in radio interferometry can provide savings of computational resources for image reconstruction through reduced memory footprints and lighter computations per iteration, which is important for the scalability of imaging methods to the big data setting of the next-generation telescopes. This article sheds new light on dimensionality reduction from the perspective of the compressed sensing theory and studies its interplay with imaging algorithms designed in the context of convex optimization. We propose a post-gridding linear data embedding to the space spanned by the left singular vectors of the measurement operator, providing a dimensionality reduction below image size. This embedding preserves the null space of the measurement operator and hence its sampling properties are also preserved in light of the compressed sensing theory. We show that this can be approximated by first computing the dirty image and then applying a weighted subsampled discrete Fourier transform to obtain the final reduced data vector. This Fourier dimensionality reduction model ensures a fast implementation of the full measurement operator, essential for any iterative image reconstruction method. The proposed reduction also preserves the independent and identically distributed Gaussian properties of the original measurement noise. For convex optimization-based imaging algorithms, this is key to justify the use of the standard ℓ2-norm as the data fidelity term. Our simulations confirm that this dimensionality reduction approach can be leveraged by convex optimization algorithms with no loss in imaging quality relative to reconstructing the image from the complete visibility data set. Reconstruction results in simulation settings with no direction dependent effects or calibration errors show promising performance of the proposed dimensionality reduction. Further tests on real data are planned as an extension of the current work. matlab code implementing the
Four Dimensional Trace Space Measurement
Energy Technology Data Exchange (ETDEWEB)
Hernandez, M.
2005-02-10
Future high energy colliders and FELs (Free Electron Lasers) such as the proposed LCLS (Linac Coherent Light Source) at SLAC require high brightness electron beams. In general a high brightness electron beam will contain a large number of electrons that occupy a short longitudinal duration, can be focused to a small transverse area while having small transverse divergences. Therefore the beam must have a high peak current and occupy small areas in transverse phase space and so have small transverse emittances. Additionally the beam should propagate at high energy and have a low energy spread to reduce chromatic effects. The requirements of the LCLS for example are pulses which contain 10{sup 10} electrons in a temporal duration of 10 ps FWHM with projected normalized transverse emittances of 1{pi} mm mrad[1]. Currently the most promising method of producing such a beam is the RF photoinjector. The GTF (Gun Test Facility) at SLAC was constructed to produce and characterize laser and electron beams which fulfill the LCLS requirements. Emittance measurements of the electron beam at the GTF contain evidence of strong coupling between the transverse dimensions of the beam. This thesis explores the effects of this coupling on the determination of the projected emittances of the electron beam. In the presence of such a coupling the projected normalized emittance is no longer a conserved quantity. The conserved quantity is the normalized full four dimensional phase space occupied by the beam. A method to determine the presence and evaluate the strength of the coupling in emittance measurements made in the laboratory is developed. A method to calculate the four dimensional volume the beam occupies in phase space using quantities available in the laboratory environment is also developed. Results of measurements made of the electron beam at the GTF that demonstrate these concepts are presented and discussed.
High dimensional feature reduction via projection pursuit
Jimenez, Luis; Landgrebe, David
1994-01-01
The recent development of more sophisticated remote sensing systems enables the measurement of radiation in many more spectral intervals than previously possible. An example of that technology is the AVIRIS system, which collects image data in 220 bands. As a result of this, new algorithms must be developed in order to analyze the more complex data effectively. Data in a high dimensional space presents a substantial challenge, since intuitive concepts valid in a 2-3 dimensional space to not necessarily apply in higher dimensional spaces. For example, high dimensional space is mostly empty. This results from the concentration of data in the corners of hypercubes. Other examples may be cited. Such observations suggest the need to project data to a subspace of a much lower dimension on a problem specific basis in such a manner that information is not lost. Projection Pursuit is a technique that will accomplish such a goal. Since it processes data in lower dimensions, it should avoid many of the difficulties of high dimensional spaces. In this paper, we begin the investigation of some of the properties of Projection Pursuit for this purpose.
High Temperature QCD and Dimensional Reduction
Petersson, Bengt
2001-04-01
In this talk I will first give a short discussion of some lattice results for QCD at finite temperature. I will then describe in some detail the technique of dimensional reduction, which in principle is a powerful technique to obtain results on the long distance properties of the quark-gluon plasma. Finally I will describe some new results, which test the technique in a simpler model, namely three dimensional gauge theory.
Quantization as a dimensional reduction phenomenon
Gozzi, E.; Mauro, D.
2006-06-01
Classical mechanics, in the operatorial formulation of Koopman and von Neumann, can be written also in a functional form. In this form two Grassmann partners of time make their natural appearance extending in this manner time to a three dimensional supermanifold. Quantization is then achieved by a process of dimensional reduction of this supermanifold. We prove that this procedure is equivalent to the well-known method of geometric quantization.
Dimensionality reduction when data are density functions
Delicado Useros, Pedro Francisco
2011-01-01
Functional Data Analysis deals with samples where a whole function is observed for each individual. A relevant case of FDA is when the observed functions are density functions. Among the particular characteristics of density functions, the most of the fact that they are an example of infinite dimensional compositional data (parts of some whole which only carry relative information) is made. Several dimensionality reduction methods for this particular type of data are compared: fun...
Joint Dimensionality Reduction for Two Feature Vectors
Li, Yanjun; Bresler, Yoram
2016-01-01
Many machine learning problems, especially multi-modal learning problems, have two sets of distinct features (e.g., image and text features in news story classification, or neuroimaging data and neurocognitive data in cognitive science research). This paper addresses the joint dimensionality reduction of two feature vectors in supervised learning problems. In particular, we assume a discriminative model where low-dimensional linear embeddings of the two feature vectors are sufficient statisti...
Discriminative dimensionality reduction: variations, applications, interpretations
Schulz, Alexander
2017-01-01
The amount of digital data increases rapidly as a result of advances in information and sensor technology. Because the data sets grow with respect to their size, complexity and dimensionality, they are no longer easily accessible to a human user. The framework of dimensionality reduction addresses this problem by aiming to visualize complex data sets in two dimensions while preserving the relevant structure. While these methods can provide significant insights, the problem formulation of str...
Dimensional reduction, truncations, constraints and the issue of consistency
Pons, J. M.
2007-05-01
A brief overview of dimensional reductions for diffeomorphism invariant theories is given. The distinction between the physical idea of compactification and the mathematical problem of a consistent truncation is discussed, and the typical ingredients of the latter-reduction of spacetime dimensions and the introduction of constraints-are examined. The consistency in the case of of group manifold reductions, when the structure constants satisfy the unimodularity condition, is shown together with the associated reduction of the gauge group. The problem of consistent truncations on coset spaces is also discussed and we comment on examples of some remarkable consistent truncations that have been found in this context.
Factorization and regularization by dimensional reduction
Signer, Adrian; Stöckinger, Dominik
2005-10-01
Since an old observation by Beenakker et al., the evaluation of QCD processes in dimensional reduction has repeatedly led to terms that seem to violate the QCD factorization theorem. We reconsider the example of the process gg → ttbar and show that the factorization problem can be completely resolved. A natural interpretation of the seemingly non-factorizing terms is found, and they are rewritten in a systematic and factorized form. The key to the solution is that the D- and (4 - D)-dimensional parts of the 4-dimensional gluon have to be regarded as independent partons.
On dimensional reduction of magical supergravity theories
Kan, Naoto; Mizoguchi, Shun'ya
2016-11-01
We prove, by a direct dimensional reduction and an explicit construction of the group manifold, that the nonlinear sigma model of the dimensionally reduced three-dimensional A = R magical supergravity is F 4 (+ 4) / (USp (6) × SU (2)). This serves as a basis for the solution generating technique in this supergravity as well as allows to give the Lie algebraic characterizations to some of the parameters and functions in the original D = 5 Lagrangian. Generalizations to other magical supergravities are also discussed.
Improving dimensionality reduction with spectral gradient descent.
Memisevic, Roland; Hinton, Geoffrey
2005-01-01
We introduce spectral gradient descent, a way of improving iterative dimensionality reduction techniques. The method uses information contained in the leading eigenvalues of a data affinity matrix to modify the steps taken during a gradient-based optimization procedure. We show that the approach is able to speed up the optimization and to help dimensionality reduction methods find better local minima of their objective functions. We also provide an interpretation of our approach in terms of the power method for finding the leading eigenvalues of a symmetric matrix and verify the usefulness of the approach in some simple experiments.
Dimensional reduction of a generalized flux problem
International Nuclear Information System (INIS)
Moroz, A.
1992-01-01
In this paper, a generalized flux problem with Abelian and non-Abelian fluxes is considered. In the Abelian case we shall show that the generalized flux problem for tight-binding models of noninteracting electrons on either 2n- or (2n + 1)-dimensional lattice can always be reduced to an n-dimensional hopping problem. A residual freedom in this reduction enables one to identify equivalence classes of hopping Hamiltonians which have the same spectrum. In the non-Abelian case, the reduction is not possible in general unless the flux tensor factorizes into an Abelian one times are element of the corresponding algebra
Manousselis, Pantelis; Zoupanos, George
2004-11-01
A ten-dimensional supersymmetric gauge theory is written in terms of Script N = 1, D = 4 superfields. The theory is dimensionally reduced over six-dimensional coset spaces. We find that the resulting four-dimensional theory is either a softly broken Script N = 1 supersymmetric gauge theory or a non-supersymmetric gauge theory depending on whether the coset spaces used in the reduction are non-symmetric or symmetric. In both cases examples susceptible to yield realistic models are presented.
A sparse grid based method for generative dimensionality reduction of high-dimensional data
Bohn, Bastian; Garcke, Jochen; Griebel, Michael
2016-03-01
Generative dimensionality reduction methods play an important role in machine learning applications because they construct an explicit mapping from a low-dimensional space to the high-dimensional data space. We discuss a general framework to describe generative dimensionality reduction methods, where the main focus lies on a regularized principal manifold learning variant. Since most generative dimensionality reduction algorithms exploit the representer theorem for reproducing kernel Hilbert spaces, their computational costs grow at least quadratically in the number n of data. Instead, we introduce a grid-based discretization approach which automatically scales just linearly in n. To circumvent the curse of dimensionality of full tensor product grids, we use the concept of sparse grids. Furthermore, in real-world applications, some embedding directions are usually more important than others and it is reasonable to refine the underlying discretization space only in these directions. To this end, we employ a dimension-adaptive algorithm which is based on the ANOVA (analysis of variance) decomposition of a function. In particular, the reconstruction error is used to measure the quality of an embedding. As an application, the study of large simulation data from an engineering application in the automotive industry (car crash simulation) is performed.
Dimensionality Reduction via Regression in Hyperspectral Imagery
Laparra, Valero; Malo, Jesus; Camps-Valls, Gustau
2015-09-01
This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate regression to ensure the reduction in redundancy between he PCA coefficients, the reduction of the variance of the scores, and the reduction in the reconstruction error. More importantly, unlike other nonlinear dimensionality reduction methods, the invertibility, volume-preservation, and straightforward out-of-sample extension, makes DRR interpretable and easy to apply. The properties of DRR enable learning a more broader class of data manifolds than the recently proposed Non-linear Principal Components Analysis (NLPCA) and Principal Polynomial Analysis (PPA). We illustrate the performance of the representation in reducing the dimensionality of remote sensing data. In particular, we tackle two common problems: processing very high dimensional spectral information such as in hyperspectral image sounding data, and dealing with spatial-spectral image patches of multispectral images. Both settings pose collinearity and ill-determination problems. Evaluation of the expressive power of the features is assessed in terms of truncation error, estimating atmospheric variables, and surface land cover classification error. Results show that DRR outperforms linear PCA and recently proposed invertible extensions based on neural networks (NLPCA) and univariate regressions (PPA).
Dimensional Reduction of Nonlinear Gauge Theories
Ikeda, Noriaki; Izawa, K.-I.
2004-09-01
We extend 2D nonlinear gauge theory from the Poisson sigma model based on Lie algebroid to a model with additional two-form gauge fields. Dimensional reduction of 3D nonlinear gauge theory yields an example of such a model, which provides a realization of Courant algebroid by 2D nonlinear gauge theory. We see that the reduction of the base structure generically results in a modification of the target (algebroid) structure.
Fringe pattern denoising by image dimensionality reduction
Vargas, J.; Sorzano, C. O. S.; Antonio Quiroga, J.; Estrada, J. C.; Carazo, J. M.
2013-07-01
Noise is a key problem in fringe pattern processing, especially in single frame demodulation of interferograms. In this work, we propose to filter the pattern noise using a straightforward, fast and easy to implement denoising method, which is based on a dimensionality reduction approach, in the sense of image rank reduction. The proposed technique has been applied to simulated and experimental ESPI interferograms obtaining satisfactory results.
Teleportation schemes in infinite dimensional Hilbert spaces
International Nuclear Information System (INIS)
Fichtner, Karl-Heinz; Freudenberg, Wolfgang; Ohya, Masanori
2005-01-01
The success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. We describe the teleportation process in an infinite dimensional Hilbert space by giving simple examples
Denoising and dimensionality reduction of genomic data
Capobianco, Enrico
2005-05-01
Genomics represents a challenging research field for many quantitative scientists, and recently a vast variety of statistical techniques and machine learning algorithms have been proposed and inspired by cross-disciplinary work with computational and systems biologists. In genomic applications, the researcher deals with noisy and complex high-dimensional feature spaces; a wealth of genes whose expression levels are experimentally measured, can often be observed for just a few time points, thus limiting the available samples. This unbalanced combination suggests that it might be hard for standard statistical inference techniques to come up with good general solutions, likewise for machine learning algorithms to avoid heavy computational work. Thus, one naturally turns to two major aspects of the problem: sparsity and intrinsic dimensionality. These two aspects are studied in this paper, where for both denoising and dimensionality reduction, a very efficient technique, i.e., Independent Component Analysis, is used. The numerical results are very promising, and lead to a very good quality of gene feature selection, due to the signal separation power enabled by the decomposition technique. We investigate how the use of replicates can improve these results, and deal with noise through a stabilization strategy which combines the estimated components and extracts the most informative biological information from them. Exploiting the inherent level of sparsity is a key issue in genetic regulatory networks, where the connectivity matrix needs to account for the real links among genes and discard many redundancies. Most experimental evidence suggests that real gene-gene connections represent indeed a subset of what is usually mapped onto either a huge gene vector or a typically dense and highly structured network. Inferring gene network connectivity from the expression levels represents a challenging inverse problem that is at present stimulating key research in biomedical
Dimensional reduction to hypersurface of foliation
Park, I. Y.
2014-11-01
When the bulk spacetime has a foliation structure, the collective dynamics of the hypersurfaces should reveal certain aspects of the bulk physics. The procedure of reducing the bulk to a hypersurface, called ADM reduction, was implemented in \\cite{Park:2013iqa} where the 4D Einstein-Hilbert action was reduced along the radial reduction. In this work, reduction along the angular directions is considered {with a main goal to firmly establish the method of dimensional reduction to a hypersurface of foliation.} We obtain a theory on a 2D plane (the $(t,r)$-plane) and observe that novel and elaborate boundary effects are crucial for the consistency of the reduction. The reduction leads to a 2D interacting quantum field theory. We comment on its application to black hole information physics.
Multichannel transfer function with dimensionality reduction
Kim, Han Suk; Schulze, Jürgen P.; Cone, Angela C.; Sosinsky, Gina E.; Martone, Maryann E.
2010-01-01
The design of transfer functions for volume rendering is a difficult task. This is particularly true for multichannel data sets, where multiple data values exist for each voxel. In this paper, we propose a new method for transfer function design. Our new method provides a framework to combine multiple approaches and pushes the boundary of gradient-based transfer functions to multiple channels, while still keeping the dimensionality of transfer functions to a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. The high-dimensional data of the domain is reduced by applying recently developed nonlinear dimensionality reduction algorithms. In this paper, we used Isomap as well as a traditional algorithm, Principle Component Analysis (PCA). Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. In this publication we report on the impact of the dimensionality reduction algorithms on transfer function design for confocal microscopy data.
Self-completeness and spontaneous dimensional reduction
Mureika, Jonas; Nicolini, Piero
2013-07-01
A viable quantum theory of gravity is one of the biggest challenges physicists are facing. We discuss the confluence of two highly expected features which might be instrumental in the quest of a finite and renormalizable quantum gravity —spontaneous dimensional reduction and self-completeness. The former suggests the spacetime background at the Planck scale may be effectively two-dimensional, while the latter implies a condition of maximal compression of matter by the formation of an event horizon for Planckian scattering. We generalize such a result to an arbitrary number of dimensions, and show that gravity in higher than four dimensions remains self-complete, but in lower dimensions it does not. In such a way we established an "exclusive disjunction" or "exclusive or" (XOR) between the occurrence of self-completeness and dimensional reduction, with the goal of actually reducing the unknowns for the scenario of the physics at the Planck scale. Potential phenomenological implications of this result are considered by studying the case of a two-dimensional dilaton gravity model resulting from dimensional reduction of the Einstein gravity.
Multichannel transfer function with dimensionality reduction
Kim, Han Suk
2010-01-17
The design of transfer functions for volume rendering is a difficult task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel. In this paper, we propose a new method for transfer function design. Our new method provides a framework to combine multiple approaches and pushes the boundary of gradient-based transfer functions to multiple channels, while still keeping the dimensionality of transfer functions to a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. The high-dimensional data of the domain is reduced by applying recently developed nonlinear dimensionality reduction algorithms. In this paper, we used Isomap as well as a traditional algorithm, Principle Component Analysis (PCA). Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. In this publication we report on the impact of the dimensionality reduction algorithms on transfer function design for confocal microscopy data.
Outlier preservation by dimensionality reduction Techniques
M. Onderwater (Martijn)
2015-01-01
textabstractSensors are increasingly part of our daily lives: motion detection, lighting control, and energy consumption all rely on sensors. Combining this information into, for instance, simple and comprehensive graphs can be quite challenging. Dimensionality reduction is often used to address
Outlier preservation by dimensionality reduction techniques
M. Onderwater (Martijn)
2015-01-01
htmlabstractSensors are increasingly part of our daily lives: motion detection, lighting control, and energy consumption all rely on sensors. Combining this information into, for instance, simple and comprehensive graphs can be quite challenging. Dimensionality reduction is often used to address
Generalized Elitzur's theorem and dimensional reductions
Batista, C. D.; Nussinov, Zohar
2005-07-01
We extend Elitzur’s theorem to systems with symmetries intermediate between global and local. In general, our theorem formalizes the idea of dimensional reduction. We apply the results of this generalization to many systems that are of current interest. These include liquid crystalline phases of quantum Hall systems, orbital systems, geometrically frustrated spin lattices, Bose metals, and models of superconducting arrays.
Pole masses of quarks in dimensional reduction
International Nuclear Information System (INIS)
Avdeev, L.V.; Kalmykov, M.Yu.
1997-01-01
Pole masses of quarks in quantum chromodynamics are calculated to the two-loop order in the framework of the regularization by dimensional reduction. For the diagram with a light quark loop, the non-Euclidean asymptotic expansion is constructed with the external momentum on the mass shell of a heavy quark
Directory of Open Access Journals (Sweden)
Dmitri A. Viattchenin
2009-06-01
Full Text Available This paper describes a modification of a possibilistic clustering method based on the concept of allotment among fuzzy clusters. Basic ideas of the method are considered and the concept of a principal allotment among fuzzy clusters is introduced. The paper provides the description of the plan of the algorithm for detection principal allotment. An analysis of experimental results of the proposed algorithm’s application to the Tamura’s portrait data in comparison with the basic version of the algorithm and with the NERFCM-algorithm is carried out. A methodology of the algorithm’s application to the dimensionality reduction problem is outlined and the application of the methodology is illustrated on the example of Anderson’s Iris data in comparison with the result of principal component analysis. Preliminary conclusions are formulated also.
Time-like reductions of five-dimensional supergravity
Cortés, V.; Dempster, P.; Mohaupt, T.
2014-04-01
In this paper we study the scalar geometries occurring in the dimensional reduction of minimal five-dimensional supergravity to three Euclidean dimensions, and find that these depend on whether one first reduces over space or over time. In both cases the scalar manifold of the reduced theory is described as an eight-dimensional Lie group L (the Iwasawa subgroup of G 2(2)) with a left-invariant para-quaternionic-Kähler structure. We show that depending on whether one reduces first over space or over time, the group L is mapped to two different open L-orbits on the pseudo-Riemannian symmetric space G 2(2) /(SL(2) · SL(2)). These two orbits are inequivalent in the sense that they are distinguished by the existence of integrable L-invariant complex or para-complex structures.
Effective Image Database Search via Dimensionality Reduction
DEFF Research Database (Denmark)
Dahl, Anders Bjorholm; Aanæs, Henrik
2008-01-01
of the visual vocabulary is typically done using k-means. We investigate a clustering algorithm based on the leader follower principle (LF-clustering), in which the number of clusters is not fixed. The adaptive nature of LF-clustering is shown to improve the quality of the visual vocabulary using this...... results compared to the traditional bag-of-words approach based on 128 dimensional SIFT feature and k-means clustering........ In the query step, features from the query image are assigned to the visual vocabulary. The dimensionality reduction enables us to do exact feature labeling using kD-tree, instead of approximate approaches normally used. Despite the dimensionality reduction to between 6 and 15 dimensions we obtain improved...
Multiloop integrand reduction for dimensionally regulated amplitudes
Mastrolia, Pierpaolo; Mirabella, Edoardo; Ossola, Giovanni; Peraro, Tiziano
2013-12-01
We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to integrands with denominators appearing with arbitrary powers.
Incremental nonlinear dimensionality reduction by manifold learning.
Law, Martin H C; Jain, Anil K
2006-03-01
Understanding the structure of multidimensional patterns, especially in unsupervised cases, is of fundamental importance in data mining, pattern recognition, and machine learning. Several algorithms have been proposed to analyze the structure of high-dimensional data based on the notion of manifold learning. These algorithms have been used to extract the intrinsic characteristics of different types of high-dimensional data by performing nonlinear dimensionality reduction. Most of these algorithms operate in a "batch" mode and cannot be efficiently applied when data are collected sequentially. In this paper, we describe an incremental version of ISOMAP, one of the key manifold learning algorithms. Our experiments on synthetic data as well as real world images demonstrate that our modified algorithm can maintain an accurate low-dimensional representation of the data in an efficient manner.
Data dimensionality reduction in anthropometrical investigations.
Kordecki, Henryk; Knapik-Kordecka, Maria; Karmowski, Mikołaj; Gworys, Bohdan; Karmowski, Andrzei
2012-01-01
Very often it is necessary to make a decision or to establish a diagnosis on the basis of great amounts of different kinds of data. In this paper the principal component analysis procedure was applied to anthropometrical data analysis. The aim was to simplify the process of decision making by data dimensionality reduction. A second aim was to check how the reduction affected an analysis of the pubertal growth process. A group of 400 boys was investigated. Three main components were calculated and interpreted. In order to investigate growth changes, the variability of each component was approximated by fourth order polynomials. It was shown that the loss of information resulting from data dimensionality reduction is about 25%, so the three calculated principal components contained 75% of the entire information. It seems possible to make an appropriate decision on the basis of that amount of information. The results obtained fully supported using the approach presented for data analysis in the case under consideration.
Construction of N=8 supergravity theories by dimensional reduction
International Nuclear Information System (INIS)
Boucher, W.
1985-01-01
In this paper I ask which N=8 supergravity theories in four dimensions can be obtained by dimensional reduction of the N=1 supergravity theory in eleven dimensions. Several years ago Scherk and Schwarz produced a particular class of N = 8 theories by giving a dimensional reduction scheme on the restricted class of coset spaces, G/H, with dim H=0 (and therefore dim G=7). I generalize their considerations by looking at arbitrary (seven-dimensional) coset spaces. Also, instead of giving a particular ansatz which happens to work, I set about the distinctly more difficult task of determining all ansatzes which produce N=8 theories. The basic ingredient of my dimensional reduction scheme is the demand that certain symmetries, including supersymmetry, be truncated consistently. I find the surprising result that the only N=8 theories obtainable within the contexts of my scheme are those theories already written down by Scherk and Schwarz. In particular dim H=0 and dim G=7. Independently of these considerations, I prove that any dimensional reduction scheme which consistently truncates supersymmetry must also be consistent with the equations of motion. I discuss Lorentz-invariant solutions of the theories of Scherk and Schwarz, pointing out that since the ansatz of Scherk and Schwarz consistently truncates supersymmetry, any solution of these theories is also a solution of the N=1 supergravity theory in eleven dimensions and, hence, in particular that there is a Freund-Rubin-type ansatz for these theories. However I demonstrate that for most gauge groups the ansatz must be trivial which implies that for these theories the cosmological constant of any Lorentz-invariant solution must be zero (classically). Finally, I make some comparisons with work by Manton on dimensional reduction. (orig.)
Can fermions save large N dimensional reduction?
Bedaque, Paulo F.; Buchoff, Michael I.; Cherman, Aleksey; Springer, Roxanne P.
2009-10-01
This paper explores whether Eguchi-Kawai reduction for gauge theories with adjoint fermions is valid. The Eguchi-Kawai reduction relates gauge theories in different numbers of dimensions in the large N limit provided that certain conditions are met. In principle, this relation opens up the possibility of learning about the dynamics of 4D gauge theories through techniques only available in lower dimensions. Dimensional reduction can be understood as a special case of large N equivalence between theories related by an orbifold projection. In this work, we focus on the simplest case of dimensional reduction, relating a 4D gauge theory to a 3D gauge theory via an orbifold projection. A necessary condition for the large N equivalence between the 4D and 3D theories to hold is that certain discrete symmetries in the two theories must not be broken spontaneously. In pure 4D Yang-Mills theory, these symmetries break spontaneously as the size of one of the spacetime dimensions shrinks. An analysis of the effect of adjoint fermions on the relevant symmetries of the 4D theory shows that the fermions help stabilize the symmetries. We consider the same problem from the point of view of the lower dimensional 3D theory and find that, surprisingly, adjoint fermions are not generally enough to stabilize the necessary symmetries of the 3D theory. In fact, a rich phase diagram arises, with a complicated pattern of symmetry breaking. We discuss the possible causes and consequences of this finding.
Universal spaces for almost n-dimensionality
Abry, M.; Dijkstra, J.J.
2007-01-01
We find universal functions for the class of lower semi-continuous (LSC) functions with at most n-dimensional domain. In an earlier paper we proved that a space is almost n-dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most n-dimensional domain. We conclude
TPSLVM: a dimensionality reduction algorithm based on thin plate splines.
Jiang, Xinwei; Gao, Junbin; Wang, Tianjiang; Shi, Daming
2014-10-01
Dimensionality reduction (DR) has been considered as one of the most significant tools for data analysis. One type of DR algorithms is based on latent variable models (LVM). LVM-based models can handle the preimage problem easily. In this paper we propose a new LVM-based DR model, named thin plate spline latent variable model (TPSLVM). Compared to the well-known Gaussian process latent variable model (GPLVM), our proposed TPSLVM is more powerful especially when the dimensionality of the latent space is low. Also, TPSLVM is robust to shift and rotation. This paper investigates two extensions of TPSLVM, i.e., the back-constrained TPSLVM (BC-TPSLVM) and TPSLVM with dynamics (TPSLVM-DM) as well as their combination BC-TPSLVM-DM. Experimental results show that TPSLVM and its extensions provide better data visualization and more efficient dimensionality reduction compared to PCA, GPLVM, ISOMAP, etc.
Supervised Gaussian process latent variable model for dimensionality reduction.
Gao, Xinbo; Wang, Xiumei; Tao, Dacheng; Li, Xuelong
2011-04-01
The Gaussian process latent variable model (GP-LVM) has been identified to be an effective probabilistic approach for dimensionality reduction because it can obtain a low-dimensional manifold of a data set in an unsupervised fashion. Consequently, the GP-LVM is insufficient for supervised learning tasks (e.g., classification and regression) because it ignores the class label information for dimensionality reduction. In this paper, a supervised GP-LVM is developed for supervised learning tasks, and the maximum a posteriori algorithm is introduced to estimate positions of all samples in the latent variable space. We present experimental evidences suggesting that the supervised GP-LVM is able to use the class label information effectively, and thus, it outperforms the GP-LVM and the discriminative extension of the GP-LVM consistently. The comparison with some supervised classification methods, such as Gaussian process classification and support vector machines, is also given to illustrate the advantage of the proposed method.
Dimensional reduction of the generalized DBI
Ho, Jun-Kai; Ma, Chen-Te
2015-08-01
We study the generalized Dirac-Born-Infeld (DBI) action, which describes a q-brane ending on a p-brane with a (q + 1)-form background. This action has the equivalent descriptions in commutative and non-commutative settings, which can be shown from the generalized metric and Nambu-Sigma model. We mainly discuss the dimensional reduction of the generalized DBI at the massless level on the flat spacetime and constant antisymmetric background in the case of flat spacetime, constant antisymmetric background and the gauge potential vanishes for all time-like components. In the case of q = 2, we can do the dimensional reduction to get the DBI theory. We also try to extend this theory by including a one-form gauge potential.
Kantowski-Sachs multidimensional cosmological models and dynamical dimensional reduction
International Nuclear Information System (INIS)
Demianski, M.; Rome Univ.; Golda, Z.A.; Heller, M.; Szydlowski, M.
1988-01-01
Einstein's field equations are solved for a multidimensional spacetime (KS) x Tsup(m), where (KS) is a four-dimensional Kantowski-Sachs spacetime and Tsup(m) is an m-dimensional torus. Among all possible vacuum solutions there is a large class of spacetimes in which the macroscopic space expands and the microscopic space contracts to a finite volume. We also consider a non-vacuum case and we explicitly solve the field equations for the matter satisfying the Zel'dovich equation of state. In non-vacuum models, with matter satisfying an equation of state p = γρ, O ≤ γ < 1, at a sufficiently late stage of evolution the microspace always expands and the dynamical dimensional reduction does not occur. (author)
Control point selection for dimensionality reduction by radial basis function
Directory of Open Access Journals (Sweden)
Kotryna Paulauskienė
2016-02-01
Full Text Available This research deals with dimensionality reduction technique which is based on radial basis function (RBF theory. The technique uses RBF for mapping multidimensional data points into a low-dimensional space by interpolating the previously calculated position of so-called control points. This paper analyses various ways of selection of control points (regularized orthogonal least squares method, random and stratified selections. The experiments have been carried out with 8 real and artificial data sets. Positions of the control points in a low-dimensional space are found by principal component analysis. We demonstrate that random and stratified selections of control points are efficient and acceptable in terms of balance between projection error (stress and time-consumption.DOI: 10.15181/csat.v4i1.1095
Epistasis analysis using multifactor dimensionality reduction.
Moore, Jason H; Andrews, Peter C
2015-01-01
Here we introduce the multifactor dimensionality reduction (MDR) methodology and software package for detecting and characterizing epistasis in genetic association studies. We provide a general overview of the method and then highlight some of the key functions of the open-source MDR software package that is freely distributed. We end with a few examples of published studies of complex human diseases that have used MDR.
Non-Redundant Spectral Dimensionality Reduction
Blau, Yochai; Michaeli, Tomer
2016-01-01
Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known as the "repeated Eigen-directions" phenomenon. That is, many of the embedding coordinates they produce typically capture the same direction along the data manifold. This leads to redundant and inefficient representations that do not reveal the true intrinsi...
Dimensionality reduction of clustered data sets.
Sanguinetti, Guido
2008-03-01
We present a novel probabilistic latent variable model to perform linear dimensionality reduction on data sets which contain clusters. We prove that the maximum likelihood solution of the model is an unsupervised generalisation of linear discriminant analysis. This provides a completely new approach to one of the most established and widely used classification algorithms. The performance of the model is then demonstrated on a number of real and artificial data sets.
Semantic coding by supervised dimensionality reduction
Kokiopoulou, Effrosyni; Frossard, Pascal
2008-01-01
This paper addresses the problem of representing multimedia information under a compressed form that permits efficient classification. The semantic coding problem starts from a subspace method where dimensionality reduction is formulated as a matrix factorization problem. Data samples are jointly represented in a common subspace extracted from a redundant dictionary of basis functions. We first build on greedy pursuit algorithms for simultaneous sparse approximations to...
Robust inversion via semistochastic dimensionality reduction
Aravkin, Aleksandr; Friedlander, Michael P.; van Leeuwen, Tristan
2011-01-01
We consider a class of inverse problems where it is possible to aggregate the results of multiple experiments. This class includes problems where the forward model is the solution operator to linear ODEs or PDEs. The tremendous size of such problems motivates dimensionality reduction techniques based on randomly mixing experiments. These techniques break down, however, when robust data-fitting formulations are used, which are essential in cases of missing data, unusually large errors, and sys...
Visualizing dimensionality reduction of systems biology data
Lehrmann, Andreas; Huber, Michael; Polatkan, Aydin C.; Pritzkau, Albert; Nieselt, Kay
2012-01-01
One of the challenges in analyzing high-dimensional expression data is the detection of important biological signals. A common approach is to apply a dimension reduction method, such as principal component analysis. Typically, after application of such a method the data is projected and visualized in the new coordinate system, using scatter plots or profile plots. These methods provide good results if the data have certain properties which become visible in the new coordinate system and which...
Dimensionality reduction for time series data
Vidaurre, Diego; Rezek, Iead; Harrison, Samuel L.; Smith, Stephen S.; Woolrich, Mark
2014-01-01
Despite the fact that they do not consider the temporal nature of data, classic dimensionality reduction techniques, such as PCA, are widely applied to time series data. In this paper, we introduce a factor decomposition specific for time series that builds upon the Bayesian multivariate autoregressive model and hence evades the assumption that data points are mutually independent. The key is to find a low-rank estimation of the autoregressive matrices. As in the probabilistic version of othe...
Dimensionality Reduction with Subspace Structure Preservation
Arpit, Devansh; Nwogu, Ifeoma; Govindaraju, Venu
2014-01-01
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have not been well studied. Our key contribution is to show that $2K$ projection vectors are sufficient for the independence preservation of any $K$ class data sampled from a union of independent subspaces. It is this non-trivial observation that we use for desi...
Foundations of Coupled Nonlinear Dimensionality Reduction
Mohri, Mehryar; Rostamizadeh, Afshin; Storcheus, Dmitry
2015-01-01
In this paper we introduce and analyze the learning scenario of \\emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we introduce an algorithm that follows from these bounds. The generalization error bound is based on a careful analysis of the empirical Rademacher complexity of the relevant hypothes...
Using Discriminative Dimensionality Reduction to Visualize Classifiers
Schulz, Alexander; Gisbrecht, Andrej; Hammer, Barbara
2015-01-01
Albeit automated classifiers offer a standard tool in many application areas, there exists hardly a generic possibility to directly inspect their behavior, which goes beyond the mere classification of (sets of) data points. In this contribution, we propose a general framework how to visualize a given classifier and its behavior as concerns a given data set in two dimensions. More specifically, we use modern nonlinear dimensionality reduction (DR) techniques to project a given set of data poin...
Sequential Dimensionality Reduction for Extracting Localized Features
Casalino, Gabriella; Gillis, Nicolas
2015-01-01
Linear dimensionality reduction techniques are powerful tools for image analysis as they allow the identification of important features in a data set. In particular, nonnegative matrix factorization (NMF) has become very popular as it is able to extract sparse, localized and easily interpretable features by imposing an additive combination of nonnegative basis elements. Nonnegative matrix underapproximation (NMU) is a closely related technique that has the advantage to identify features seque...
Dimensionality Reduction using Similarity-induced Embeddings
Passalis, Nikolaos; Tefas, Anastasios
2017-01-01
The vast majority of Dimensionality Reduction (DR) techniques rely on second-order statistics to define their optimization objective. Even though this provides adequate results in most cases, it comes with several shortcomings. The methods require carefully designed regularizers and they are usually prone to outliers. In this work, a new DR framework, that can directly model the target distribution using the notion of similarity instead of distance, is introduced. The proposed framework, call...
Recursive support vector machines for dimensionality reduction.
Tao, Qing; Chu, Dejun; Wang, Jue
2008-01-01
The usual dimensionality reduction technique in supervised learning is mainly based on linear discriminant analysis (LDA), but it suffers from singularity or undersampled problems. On the other hand, a regular support vector machine (SVM) separates the data only in terms of one single direction of maximum margin, and the classification accuracy may be not good enough. In this letter, a recursive SVM (RSVM) is presented, in which several orthogonal directions that best separate the data with the maximum margin are obtained. Theoretical analysis shows that a completely orthogonal basis can be derived in feature subspace spanned by the training samples and the margin is decreasing along the recursive components in linearly separable cases. As a result, a new dimensionality reduction technique based on multilevel maximum margin components and then a classifier with high accuracy are achieved. Experiments in synthetic and several real data sets show that RSVM using multilevel maximum margin features can do efficient dimensionality reduction and outperform regular SVM in binary classification problems.
Improved outlier identification in hyperspectral imaging via nonlinear dimensionality reduction
Olson, C. C.; Nichols, J. M.; Michalowicz, J. V.; Bucholtz, F.
2010-04-01
We use a nonlinear dimensionality reduction technique to improve anomaly detection in a hyperspectral imaging application. A nonlinear transformation, diffusion map, is used to map pixels from the high-dimensional spectral space to a (possibly) lower-dimensional manifold. The transformation is designed to retain a measure of distance between the selected pixels. This lower-dimensional manifold represents the background of the scene with high probability and selecting a subset of points reduces the computational overhead associated with diffusion map. The remaining pixels are mapped to the manifold by means of a Nyströom extension. A distance measure is computed for each new pixel and those that do not reside near the background manifold, as determined by a threshold, are identified as anomalous. We compare our results with the RX and subspace RX methods of anomaly detection.
Coset space dimension reduction of gauge theories
International Nuclear Information System (INIS)
Farakos, K.; Kapetanakis, D.; Koutsoumbas, G.; Zoupanos, G.
1989-01-01
A very interesting approach in the attempts to unify all the interactions is to consider that a unification takes place in higher than four dimensions. The most ambitious program based on the old Kaluza-Klein idea is not able to reproduce the low energy chiral nature of the weak interactions. A suggested way out was the introduction of Yang-Mills fields in the higher dimensional theory. From the particle physics point of view the most important question is how such a theory behaves in four dimensions and in particular in low energies. Therefore most of our efforts concern studies of the properties of an attractive scheme, the Coset-Space-Dimensional-Reduction (C.S.D.R.) scheme, which permits the study of the effective four dimensional theory coming from a gauge theory defined in higher dimensions. Here we summarize the C.S.D.R. procedure the main the rems which are obeyed and to present a realistic model which is the result of the model building efforts that take into account all the C.S.D.R. properties. (orig./HSI)
Dimensionality Reduction via Euclidean Distance Embeddings
Šarić, Marin; Ek, Carl Henrik; Kragić, Danica
2011-01-01
This report provides a mathematically thorough review and investigation of Metric Multidimensional scaling (MDS) through the analysis of Euclidean distances in input and output spaces. By combining a geometric approach with modern linear algebra and multivariate analysis, Metric MDS is viewed as a Euclidean distance embedding transformation that converts between coordinate and coordinate-free representations of data. In this work we link Mercer kernel functions, data in infinite-dimensional H...
N-Dimensional LLL Reduction Algorithm with Pivoted Reflection
Directory of Open Access Journals (Sweden)
Zhongliang Deng
2018-01-01
Full Text Available The Lenstra-Lenstra-Lovász (LLL lattice reduction algorithm and many of its variants have been widely used by cryptography, multiple-input-multiple-output (MIMO communication systems and carrier phase positioning in global navigation satellite system (GNSS to solve the integer least squares (ILS problem. In this paper, we propose an n-dimensional LLL reduction algorithm (n-LLL, expanding the Lovász condition in LLL algorithm to n-dimensional space in order to obtain a further reduced basis. We also introduce pivoted Householder reflection into the algorithm to optimize the reduction time. For an m-order positive definite matrix, analysis shows that the n-LLL reduction algorithm will converge within finite steps and always produce better results than the original LLL reduction algorithm with n > 2. The simulations clearly prove that n-LLL is better than the original LLL in reducing the condition number of an ill-conditioned input matrix with 39% improvement on average for typical cases, which can significantly reduce the searching space for solving ILS problem. The simulation results also show that the pivoted reflection has significantly declined the number of swaps in the algorithm by 57%, making n-LLL a more practical reduction algorithm.
Discriminative Dimensionality Reduction for Multi-Dimensional Sequences.
Su, Bing; Ding, Xiaoqing; Wang, Hao; Wu, Ying
2018-01-01
Since the observables at particular time instants in a temporal sequence exhibit dependencies, they are not independent samples. Thus, it is not plausible to apply i.i.d. assumption-based dimensionality reduction methods to sequence data. This paper presents a novel supervised dimensionality reduction approach for sequence data, called Linear Sequence Discriminant Analysis (LSDA). It learns a linear discriminative projection of the feature vectors in sequences to a lower-dimensional subspace by maximizing the separability of the sequence classes such that the entire sequences are holistically discriminated. The sequence class separability is constructed based on the sequence statistics, and the use of different statistics produces different LSDA methods. This paper presents and compares two novel LSDA methods, namely M-LSDA and D-LSDA. M-LSDA extracts model-based statistics by exploiting the dynamical structure of the sequence classes, and D-LSDA extracts the distance-based statistics by computing the pairwise similarity of samples from the same sequence class. Extensive experiments on several different tasks have demonstrated the effectiveness and the general applicability of the proposed methods.
Nonlinear dimensionality reduction methods for synthetic biology biobricks' visualization.
Yang, Jiaoyun; Wang, Haipeng; Ding, Huitong; An, Ning; Alterovitz, Gil
2017-01-19
Visualizing data by dimensionality reduction is an important strategy in Bioinformatics, which could help to discover hidden data properties and detect data quality issues, e.g. data noise, inappropriately labeled data, etc. As crowdsourcing-based synthetic biology databases face similar data quality issues, we propose to visualize biobricks to tackle them. However, existing dimensionality reduction methods could not be directly applied on biobricks datasets. Hereby, we use normalized edit distance to enhance dimensionality reduction methods, including Isomap and Laplacian Eigenmaps. By extracting biobricks from synthetic biology database Registry of Standard Biological Parts, six combinations of various types of biobricks are tested. The visualization graphs illustrate discriminated biobricks and inappropriately labeled biobricks. Clustering algorithm K-means is adopted to quantify the reduction results. The average clustering accuracy for Isomap and Laplacian Eigenmaps are 0.857 and 0.844, respectively. Besides, Laplacian Eigenmaps is 5 times faster than Isomap, and its visualization graph is more concentrated to discriminate biobricks. By combining normalized edit distance with Isomap and Laplacian Eigenmaps, synthetic biology biobircks are successfully visualized in two dimensional space. Various types of biobricks could be discriminated and inappropriately labeled biobricks could be determined, which could help to assess crowdsourcing-based synthetic biology databases' quality, and make biobricks selection.
Liu, Wenyuan; Wang, Chunlei; Wang, Baowen; Wang, Changwu
2014-02-01
Cancer gene expression data have the characteristics of high dimensionalities and small samples so it is necessary to perform dimensionality reduction of the data. Traditional linear dimensionality reduction approaches can not find the nonlinear relationship between the data points. In addition, they have bad dimensionality reduction results. Therefore a multiple weights locally linear embedding (LLE) algorithm with improved distance is introduced to perform dimensionality reduction in this study. We adopted an improved distance to calculate the neighbor of each data point in this algorithm, and then we introduced multiple sets of linearly independent local weight vectors for each neighbor, and obtained the embedding results in the low-dimensional space of the high-dimensional data by minimizing the reconstruction error. Experimental result showed that the multiple weights LLE algorithm with improved distance had good dimensionality reduction functions of the cancer gene expression data.
Dimensionally continued infinite reduction of couplings
Anselmi, Damiano; Halat, Milenko
2006-01-01
The infinite reduction of couplings is a tool to consistently renormalize a wide class of non-renormalizable theories with a reduced, eventually finite, set of independent couplings, and classify the non-renormalizable interactions. Several properties of the reduction of couplings, both in renormalizable and non-renormalizable theories, can be better appreciated working at the regularized level, using the dimensional-regularization technique. We show that, when suitable invertibility conditions are fulfilled, the reduction follows uniquely from the requirement that both the bare and renormalized reduction relations be analytic in ɛ = D-d, where D and d are the physical and continued spacetime dimensions, respectively. In practice, physically independent interactions are distinguished by relatively non-integer powers of ɛ. We discuss the main physical and mathematical properties of this criterion for the reduction and compare it with other equivalent criteria. The leading-log approximation is solved explicitly and contains sufficient information for the existence and uniqueness of the reduction to all orders.
Dimensional Reduction via Noncommutative Spacetime: Bootstrap and Holography
Li, Miao
2002-05-01
Unlike noncommutative space, when space and time are noncommutative, it seems necessary to modify the usual scheme of quantum mechanics. We propose in this paper a simple generalization of the time evolution equation in quantum mechanics to incorporate the feature of a noncommutative spacetime. This equation is much more constraining than the usual Schrödinger equation in that the spatial dimension noncommuting with time is effectively reduced to a point in low energy. We thus call the new evolution equation the spacetime bootstrap equation, the dimensional reduction called for by this evolution seems close to what is required by the holographic principle. We will discuss several examples to demonstrate this point.
Radial dimensional reduction: (anti) de Sitter theories from flat
Biswas, Tirthabir; Siegel, Warren
2002-07-01
We propose a new form of dimensional reduction that constrains dilatation instead of a component of momentum. It corresponds to replacing toroidal compactification in a cartesian coordinate with that in the logarithm of the radius. Massive theories in de Sitter or anti de Sitter space are thus produced from massless (scale invariant) theories in one higher space or time dimension. As an example, we derive free massive actions for arbitrary representations of the (anti) de Sitter group in arbitrary dimensions. (Previous general results were restricted to symmetric tensors.) We also discuss generalizations to interacting theories.
Nonlinear dimensionality reduction by locally linear inlaying.
Hou, Yuexian; Zhang, Peng; Xu, Xingxing; Zhang, Xiaowei; Li, Wenjie
2009-02-01
High-dimensional data is involved in many fields of information processing. However, sometimes, the intrinsic structures of these data can be described by a few degrees of freedom. To discover these degrees of freedom or the low-dimensional nonlinear manifold underlying a high-dimensional space, many manifold learning algorithms have been proposed. Here we describe a novel algorithm, locally linear inlaying (LLI), which combines simple geometric intuitions and rigorously established optimality to compute the global embedding of a nonlinear manifold. Using a divide-and-conquer strategy, LLI gains some advantages in itself. First, its time complexity is linear in the number of data points, and hence LLI can be implemented efficiently. Second, LLI overcomes problems caused by the nonuniform sample distribution. Third, unlike existing algorithms such as isometric feature mapping (Isomap), local tangent space alignment (LTSA), and locally linear coordination (LLC), LLI is robust to noise. In addition, to evaluate the embedding results quantitatively, two criteria based on information theory and Kolmogorov complexity theory, respectively, are proposed. Furthermore, we demonstrated the efficiency and effectiveness of our proposal by synthetic and real-world data sets.
Dimensional Reduction in Quantum Dipolar Antiferromagnets
Babkevich, P.; Jeong, M.; Matsumoto, Y.; Kovacevic, I.; Finco, A.; Toft-Petersen, R.; Ritter, C.; Mânsson, M.; Nakatsuji, S.; Rønnow, H. M.
2016-05-01
We report ac susceptibility, specific heat, and neutron scattering measurements on a dipolar-coupled antiferromagnet LiYbF4 . For the thermal transition, the order-parameter critical exponent is found to be 0.20(1) and the specific-heat critical exponent -0.25 (1 ) . The exponents agree with the 2D X Y /h4 universality class despite the lack of apparent two-dimensionality in the structure. The order-parameter exponent for the quantum phase transitions is found to be 0.35(1) corresponding to (2 +1 )D . These results are in line with those found for LiErF4 which has the same crystal structure, but largely different TN, crystal field environment and hyperfine interactions. Our results therefore experimentally establish that the dimensional reduction is universal to quantum dipolar antiferromagnets on a distorted diamond lattice.
Fermions Obstruct Dimensional Reduction in Hot QCD
Gavai, R. V.; Gupta, Sourendu
2000-09-01
We have studied, for the first time, screening masses obtained from glueball-like correlators in quantum chromodynamics with four light dynamical flavors of quarks in the temperature range 1.5Tc<=T<=3Tc, where Tc is the temperature at which the chiral transition occurs. We have also studied pionlike and sigmalike screening masses and found that they are degenerate in the entire range of T. These obstruct perturbative dimensional reduction since the lowest glueball screening mass is heavier than them. Extrapolation of our results suggests that this obstruction may affect the entire range of temperature expected to be reached even at the Large Hadron Collider.
Epistasis, complexity, and multifactor dimensionality reduction.
Pan, Qinxin; Hu, Ting; Moore, Jason H
2013-01-01
Genome-wide association studies (GWASs) and other high-throughput initiatives have led to an information explosion in human genetics and genetic epidemiology. Conversion of this wealth of new information about genomic variation to knowledge about public health and human biology will depend critically on the complexity of the genotype to phenotype mapping relationship. We review here computational approaches to genetic analysis that embrace, rather than ignore, the complexity of human health. We focus on multifactor dimensionality reduction (MDR) as an approach for modeling one of these complexities: epistasis or gene-gene interaction.
Alternative dimensional reduction via the density matrix
de Carvalho, C. A.; Cornwall, J. M.; da Silva, A. J.
2001-07-01
We give graphical rules, based on earlier work for the functional Schrödinger equation, for constructing the density matrix for scalar and gauge fields in equilibrium at finite temperature T. More useful is a dimensionally reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expansions; it encodes all thermal matrix elements. We term the DREA procedure alternative dimensional reduction, to distinguish it from the conventional dimensionally reduced field theory (DRFT) which applies at infinite T. The DREA is useful because it gives a dimensionally reduced theory usable at any T including infinity, where it yields the DRFT, and because it does not and cannot have certain spurious infinities which sometimes occur in the density matrix itself or the conventional DRFT; these come from ln T factors at infinite temperature. The DREA can be constructed to all orders (in principle) and the only regularizations needed are those which control the ultraviolet behavior of the zero-T theory. An example of spurious divergences in the DRFT occurs in d=2+1φ4 theory dimensionally reduced to d=2. We study this theory and show that the rules for the DREA replace these ``wrong'' divergences in physical parameters by calculable powers of ln T; we also compute the phase transition temperature of this φ4 theory in one-loop order. Our density-matrix construction is equivalent to a construction of the Landau-Ginzburg ``coarse-grained free energy'' from a microscopic Hamiltonian.
Fourierdimredn: Fourier dimensionality reduction model for interferometric imaging
Kartik, S. Vijay; Carrillo, Rafael; Thiran, Jean-Philippe; Wiaux, Yves
2016-10-01
Fourierdimredn (Fourier dimensionality reduction) implements Fourier-based dimensionality reduction of interferometric data. Written in Matlab, it derives the theoretically optimal dimensionality reduction operator from a singular value decomposition perspective of the measurement operator. Fourierdimredn ensures a fast implementation of the full measurement operator and also preserves the i.i.d. Gaussian properties of the original measurement noise.
Comparative efficiency of dimensionality reduction schemes in global optimization
Grishagin, Vladimir; Israfilov, Ruslan; Sergeyev, Yaroslav
2016-10-01
This work presents results of a comparative efficiency for global optimization methods based on ideas of reducing the dimensionality of the multiextremal optimization problems. Two approaches to the dimensionality reduction are considered. One of them applies Peano-type space filling curves for reducing the multidimensional problem to an equivalent univariate one. The second approach is based on the nested optimization scheme that transforms the multidimensional problem to a family of one-dimensional subproblems connected recursively. In the frameworks of both approaches, the so-called characteristical algorithms are used for executing the univariate optimization. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator being at present a classical tool for testing global optimization algorithms. Results for problems of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme used in combination with the information-statistical univariate algorithm over its rivals.
Cortical spatiotemporal dimensionality reduction for visual grouping.
Cocci, Giacomo; Barbieri, Davide; Citti, Giovanna; Sarti, Alessandro
2015-06-01
The visual systems of many mammals, including humans, are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at the single cell level and geometric processing by means of cell connectivity. We present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low-level object segmentation. The main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher-dimensional spaces. Neural plausibility of the proposed arguments will be discussed.
Dimensionality Reduction in Multiple Ordinal Regression.
Zeng, Jiabei; Liu, Yang; Leng, Biao; Xiong, Zhang; Cheung, Yiu-Ming
2017-10-10
Supervised dimensionality reduction (DR) plays an important role in learning systems with high-dimensional data. It projects the data into a low-dimensional subspace and keeps the projected data distinguishable in different classes. In addition to preserving the discriminant information for binary or multiple classes, some real-world applications also require keeping the preference degrees of assigning the data to multiple aspects, e.g., to keep the different intensities for co-occurring facial expressions or the product ratings in different aspects. To address this issue, we propose a novel supervised DR method for DR in multiple ordinal regression (DRMOR), whose projected subspace preserves all the ordinal information in multiple aspects or labels. We formulate this problem as a joint optimization framework to simultaneously perform DR and ordinal regression. In contrast to most existing DR methods, which are conducted independently of the subsequent classification or ordinal regression, the proposed framework fully benefits from both of the procedures. We experimentally demonstrate that the proposed DRMOR method (DRMOR-M) well preserves the ordinal information from all the aspects or labels in the learned subspace. Moreover, DRMOR-M exhibits advantages compared with representative DR or ordinal regression algorithms on three standard data sets.
Classical aspects of lightlike dimensional reduction
Minguzzi, E.
2006-12-01
Some aspects of lightlike dimensional reduction in flat spacetime are studied with emphasis to classical applications. Among them the Galilean transformation of shadows induced by inertial frame changes is studied in detail by proving that (i) the shadow of an object has the same shape in every orthogonal-to-light screen, (ii) if two shadows are simultaneous in an orthogonal-to-light screen then they are simultaneous in any such screen. In particular, the Galilean group in 2 + 1 dimensions is recognized as an exact symmetry of nature which acts on the shadows of the events instead that on the events themselves. The group theoretical approach to lightlike dimensional reduction is used to solve the reconstruction problem of a trajectory starting from its acceleration history or from its projected (shadow) trajectory. The possibility of obtaining Galilean projected physics starting from Poincaré invariant physics is stressed through the example of relativistic collisions. In particular, it is shown that the projection of a relativistic collision between massless particles gives a non-relativistic collision in which the kinetic energy is conserved.
Projection pursuit-based dimensionality reduction
Safavi, Haleh; Chang, Chein-I.
2008-04-01
Dimensionality Reduction (DR) has found many applications in hyperspectral image processing, e.g., data compression, endmember extraction. This paper investigates Projection Pursuit (PP)-based data dimensionality reduction where three approaches are developed. One is to use a Projection Index (PI) to produce projection vectors that can be used to generate Projection Index Components (PICs). It is a common practice that PP generally uses random initial conditions to produce PICs. As a result, when the same PP is performed in different times or different users at the same time, the resulting PICs are generally not the same. In order to resolve this issue, two approaches are proposed. One is referred to as PI-based PRioritized PP (PI-PRPP) which uses a PI as a criterion to prioritize PICs that are produced by any component analysis, for example, Principal Components Analysis (PCA) or Independent Component Analysis. The other approach is called Initialization-Driven PP (ID-PP) which specifies an appropriate set of initial conditions that allows PP to not only produce PICs in the same order but also the same PICs regardless of how many times PP is run or who runs the PP.
Supporting regenerative medicine by integrative dimensionality reduction.
Mulas, F; Zagar, L; Zupan, B; Bellazzi, R
2012-01-01
The assessment of the developmental potential of stem cells is a crucial step towards their clinical application in regenerative medicine. It has been demonstrated that genome-wide expression profiles can predict the cellular differentiation stage by means of dimensionality reduction methods. Here we show that these techniques can be further strengthened to support decision making with i) a novel strategy for gene selection; ii) methods for combining the evidence from multiple data sets. We propose to exploit dimensionality reduction methods for the selection of genes specifically activated in different stages of differentiation. To obtain an integrated predictive model, the expression values of the selected genes from multiple data sets are combined. We investigated distinct approaches that either aggregate data sets or use learning ensembles. We analyzed the performance of the proposed methods on six publicly available data sets. The selection procedure identified a reduced subset of genes whose expression values gave rise to an accurate stage prediction. The assessment of predictive accuracy demonstrated a high quality of predictions for most of the data integration methods presented. The experimental results highlighted the main potentials of proposed approaches. These include the ability to predict the true staging by combining multiple training data sets when this could not be inferred from a single data source, and to focus the analysis on a reduced list of genes of similar predictive performance.
Dimensional reduction, gauged /D=5 supergravity and brane solutions
Chamseddine, A. H.; Sabra, W. A.
2000-06-01
The /U(1) gauged version of the Strominger-Vafa five dimensional /N=2 supergravity with one vector multiplet is obtained via dimensional reduction from the /N=1 ten dimensional supergravity. Using such explicit relation between the gauged supergravity theory and ten dimensional supergravity, all known solutions of the five dimensional theory can be lifted up to ten-dimensions. The eleven dimensional solutions can also obtained by lifting the ten-dimensional solutions.
Theory of Space Charge Limited Current in Fractional Dimensional Space
Zubair, Muhammad; Ang, L. K.
The concept of fractional dimensional space has been effectively applied in many areas of physics to describe the fractional effects on the physical systems. We will present some recent developments of space charge limited (SCL) current in free space and solid in the framework of fractional dimensional space which may account for the effect of imperfectness or roughness of the electrode surface. For SCL current in free space, the governing law is known as the Child-Langmuir (CL) law. Its analogy in a trap-free solid (or dielectric) is known as Mott-Gurney (MG) law. This work extends the one-dimensional CL Law and MG Law for the case of a D-dimensional fractional space with 0 < D <= 1 where parameter D defines the degree of roughness of the electrode surface. Such a fractional dimensional space generalization of SCL current theory can be used to characterize the charge injection by the imperfectness or roughness of the surface in applications related to high current cathode (CL law), and organic electronics (MG law). In terms of operating regime, the model has included the quantum effects when the spacing between the electrodes is small.
Classification improvement by optimal dimensionality reduction when training sets are of small size
Starks, S. A.; Defigueiredo, R. J. P.; Vanrooy, D. L.
1976-01-01
A computer simulation was performed to test the conjecture that, when the sizes of the training sets are small, classification in a subspace of the original data space may give rise to a smaller probability of error than the classification in the data space itself; this is because the gain in the accuracy of estimation of the likelihood functions used in classification in the lower dimensional space (subspace) offsets the loss of information associated with dimensionality reduction (feature extraction). A number of pseudo-random training and data vectors were generated from two four-dimensional Gaussian classes. A special algorithm was used to create an optimal one-dimensional feature space on which to project the data. When the sizes of the training sets are small, classification of the data in the optimal one-dimensional space is found to yield lower error rates than the one in the original four-dimensional space.
Dimensional reduction of gravity and relation between static states, cosmologies, and waves
de Alfaro, V.; Filippov, A. T.
2007-12-01
We introduce generalized dimensional reductions of an integrable (1+1)-dimensional dilaton gravity coupled to matter down to one-dimensional static states (black holes in particular), cosmological models, and waves. An unusual feature of these reductions is that the wave solutions depend on two variables: space and time. They are obtained here both by reducing the moduli space (available because of complete integrability) and by a generalized separation of variables (also applicable to nonintegrable models and to higher-dimensional theories). Among these new wavelike solutions, we find a class of solutions for which the matter fields are finite everywhere in space-time, including infinity. These considerations clearly demonstrate that a deep connection exists between static states, cosmologies, and waves. We argue that it should also exist in realistic higher-dimensional theories. Among other things, we also briefly outline the relations existing between the low-dimensional models that we discuss here and the realistic higher-dimensional ones.
Dimensional reduction, monopoles and dynamical symmetry breaking
Dolan, Brian P.; Szabo, Richard J.
2009-03-01
We consider SU(2)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M × Bbb CP1, with emphasis on the effects of non-trivial magnetic flux on Bbb CP1. The reduction of Yang-Mills fields gives a chain of coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking, as a consequence of the monopole fields. The reduction of SU(2)-symmetric fermions gives massless Dirac fermions on M transforming under the low-energy gauge group with Yukawa couplings, again as a result of the internal U(1) fluxes. The tower of massive fermionic Kaluza-Klein states also has Yukawa interactions and admits a natural SU(2)-equivariant truncation by replacing Bbb CP1 with a fuzzy sphere. In this approach it is possible to obtain exactly massless chiral fermions in the effective field theory with Yukawa interactions, without any further requirements. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples.
Stochastic simulation of patterns using ISOMAP for dimensionality reduction of training images
Zhang, Ting; Du, Yi; Huang, Tao; Yang, Jiaqing; Li, Xue
2015-06-01
Most data in the real world are normally nonlinear or difficult to determine whether they are linear or not beforehand. Some linear dimensionality reduction algorithms, e.g., principal component analysis (PCA) and multi-dimensional scaling (MDS) are only suitable for linear dimensionality reduction of spatial data. The patterns extracted from training images (TIs) used in MPS simulation mostly are probably nonlinear, so for some MPS simulation methods based on dimensionality reduction, e.g., FILTERSIM using some filters created via the idea of PCA and DisPAT using MDS as a tool of dimensionality reduction, those linear methods for dimensionality reduction are not appropriate when realizing the dimensionality reduction of nonlinear data of patterns. Therefore, isometric mapping (ISOMAP) working as a nonlinear dimensionality reduction method used in manifold learning is introduced to map those patterns, regardless of being linear or nonlinear, into low-dimensional space. However, because the original ISOMAP has some disadvantages in computing speed and accuracy, landmark points of patterns are selected to improve the speed and neighborhoods of patterns are set to guarantee the quality of dimensionality reduction. Next, the sequential simulation similar to FILTERSIM is performed after low-dimensional data of patterns are classified by a density-based clustering algorithm. The comparisons with FILTERSIM and DisPAT show the improvement of pattern reproductivity and computing speed of our method for both continuous and categorical variables.
International Nuclear Information System (INIS)
Barrow, J.D.
1983-01-01
The role played by the dimensions of space and space-time in determining the form of various physical laws and constants of Nature is examined. Low dimensional manifolds are also seen to possess special mathematical properties. The concept of fractal dimension is introduced and the recent renaissance of Kaluza-Klein theories obtained by dimensional reduction from higher dimensional gravity or supergravity theories is discussed. A formulation of the anthropic principle is suggested. (author)
Quantum discriminant analysis for dimensionality reduction and classification
Cong, Iris; Duan, Luming
2016-07-01
We present quantum algorithms to efficiently perform discriminant analysis for dimensionality reduction and classification over an exponentially large input data set. Compared with the best-known classical algorithms, the quantum algorithms show an exponential speedup in both the number of training vectors M and the feature space dimension N. We generalize the previous quantum algorithm for solving systems of linear equations (2009 Phys. Rev. Lett. 103 150502) to efficiently implement a Hermitian chain product of k trace-normalized N ×N Hermitian positive-semidefinite matrices with time complexity of O({log}(N)). Using this result, we perform linear as well as nonlinear Fisher discriminant analysis for dimensionality reduction over M vectors, each in an N-dimensional feature space, in time O(p {polylog}({MN})/{ε }3), where ɛ denotes the tolerance error, and p is the number of principal projection directions desired. We also present a quantum discriminant analysis algorithm for data classification with time complexity O({log}({MN})/{ε }3).
Dimensionality reduction of hyperspectral data: band selection using curve fitting
Pal, Mahendra K.; Porwal, Alok
2016-04-01
Hyperspectral sensors offer narrow spectral bandwidth facilitating better discrimination of various ground materials. However, high spectral resolutions of these sensors result in larger data volumes, and thus pose computation challenges. The increased computational complexity limit the use of hyperspectral data, where applications demands moderate accuracies but economy of processing and execution time. Also the high dimensionality of the feature space adversely affect classification accuracies when the number of training samples is limited - a consequence of Hughes' effect. A reduction in the number of dimensions lead to the Hughes effect, thus improving classification accuracies. Dimensionality reduction can be accomplished by: (i) feature selection, that is, selection of sub-optimal subset of the original set of features and (ii) feature extraction, that is, projection of the original feature space into a lower dimensional subspace that preserves most of Information. In this contribution, we propose a novel method of feature section by identifying and selecting the optimal bands based on spectral decorrelation using a local curve fitting technique. The technique is implemented on the Hyperion data of a study area from Western India. The results shows that the proposed algorithm is efficient and effective in preserving the useful original information for better classification with reduced data size and dimension.
Relativistic phase space: dimensional recurrences
International Nuclear Information System (INIS)
Delbourgo, R; Roberts, M L
2003-01-01
We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius R and taking the limit as R→∞. These relations take the form of mass integrals, associated with extraneous momenta (relative to the lower dimension), and produce the result in the higher dimension
A roadmap to multifactor dimensionality reduction methods
Gola, Damian; Mahachie John, Jestinah M.; van Steen, Kristel
2016-01-01
Complex diseases are defined to be determined by multiple genetic and environmental factors alone as well as in interactions. To analyze interactions in genetic data, many statistical methods have been suggested, with most of them relying on statistical regression models. Given the known limitations of classical methods, approaches from the machine-learning community have also become attractive. From this latter family, a fast-growing collection of methods emerged that are based on the Multifactor Dimensionality Reduction (MDR) approach. Since its first introduction, MDR has enjoyed great popularity in applications and has been extended and modified multiple times. Based on a literature search, we here provide a systematic and comprehensive overview of these suggested methods. The methods are described in detail, and the availability of implementations is listed. Most recent approaches offer to deal with large-scale data sets and rare variants, which is why we expect these methods to even gain in popularity. PMID:26108231
Progressive dimensionality reduction for hyperspectral imagery
Safavi, Haleh; Liu, Keng-Hao; Chang, Chein-I.
2009-08-01
This paper develops to a new concept, called Progressive Dimensionality Reduction (PDR) which can perform data dimensionality progressive in terms of information preservation. Two procedures can be designed to perform PDR in a forward or backward manner, referred to forward PDR (FPDR) or backward PDR (BPDR) respectively where FPDR starts with a minimum number of spectral-transformed dimensions and increases the spectral-transformed dimension progressively as opposed to BPDR begins with a maximum number of spectral-transformed dimensions and decreases the spectral-transformed dimension progressively. Both procedures are terminated when a stopping rule is satisfied. In order to carry out DR in a progressive manner, DR must be prioritized in accordance with significance of information so that the information after DR can be either increased progressively by FPDR or decreased progressively by BPDR. To accomplish this task, Projection Pursuit (PP)-based DR techniques are further developed where the Projection Index (PI) designed to find a direction of interestingness is used to prioritize directions of Projection Index Components (PICs) so that the DR can be performed by retaining PICs with high priorities via FPDR or BPDR. In the context of PDR, two well-known component analysis techniques, Principal Components Analysis (PCA) and Independent Component Analysis (ICA) can be considered as its special cases when they are used for DR.
Curvilinear component analysis for nonlinear dimensionality reduction of hyperspectral images
Lennon, Marc; Mercier, Gregoire; Mouchot, Marie-Catherine; Hubert-Moy, Laurence
2002-01-01
This paper presents a multidimensional data nonlinear projection method applied to the dimensionality reduction of hyperspectral images. The method, called Curvilinear Component Analysis (CCA) consists in reproducing at best the topology of the joint distribution of the data in a projection subspace whose dimension is lower than the dimension of the initial space, thus preserving a maximum amount of information. The Curvilinear Distance Analysis (CDA) is an improvement of the CCA that allows data including high nonlinearities to be projected. Its interest for reducing the dimension of hyperspectral images is shown. The results are presented on real hyperspectral images and compared with usual linear projection methods.
Three-dimensional group manifold reductions of gravity
Linares, Román
2005-04-01
We review the three-dimensional group manifold reductions of pure Einstein gravity and we exhibit a new consistent group manifold reduction of gravity when the compactification group manifold is S3. The new reduction leads to a lower dimensional theory whose gauge group is SU(2).
Efficient and Robust Classification of Seismic Data through Dimensionality Reduction
Srinivasan, G.; Hickmann, K. S.; Hyman, J.
2016-12-01
Classification of data using supervised learning algorithms, such as the Support Vector Machine (SVM) is popular in many fields including remote sensing, medical imaging, and geophysics. However, the suitability of such methods for classifying seismic data isseverely hampered by assumptions of linearity (linear SVM) or computational limitations with increases in data dimension (nonlinear SVM). We propose an approach to classification using nonlinear SVM in a reduced dimensional space through the application of kernel Principal Component Analysis (kPCA). We demonstrate the robustness and scalability of our approach to classification of nonlinearly correlated seismic data using synthetically generated seismograms. The utility of the method is demonstrated by identifying the number of reflective layers (one or two) buried within the subsurface. We find that by training SVM in a subspace of a high dimensional feature space determined using kPCA, more accrate and efficient classification is achieved on an independenttest set when compared to training SVM using the entire feature space, which is equivalent to performing standard nonlinear SVM. In most test cases considered, optimal SVM performance occurs when a subspace that makes up at most 10% of the entire feature space is used. We also recorded 6 times speedup in computational time for the test cases considered. The results indicate that performing kPCA dimension reduction prior to SVM classification can significantly increase performance, reliability, and robustness of an SVM classifier in seismic problems.
Testing Dimensional Reduction in SU(2) Gauge Theory
Kratochvila, S; Kratochvila, Slavo; Forcrand, Philippe de
2002-01-01
At high temperature, every $(d+1)$-dimensional theory can be reformulated as an effective theory in $d$ dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from $T=4 T_c$ downwards, which may however be due to somewhat different implementations of dimensional reduction in the two cases.
Does Dimensionality Reduction improve the Quality of Motion Interpolation?
Bitzer, Sebastian; Klanke, Stefan; Vijayakumar, Sethu
2009-01-01
In recent years nonlinear dimensionality reduction has frequently been suggested for the modelling of high-dimensional motion data. While it is intuitively plausible to use dimensionality reduction to recover low dimensional manifolds which compactly represent a given set of movements, there is a lack of critical investigation into the quality of resulting representations, in particular with respect to generalisability. Furthermore it is unclear how consistently particular m...
Dimensionality Reduction Using Similarity-Induced Embeddings.
Passalis, Nikolaos; Tefas, Anastasios
2017-08-08
The vast majority of dimensionality reduction (DR) techniques rely on the second-order statistics to define their optimization objective. Even though this provides adequate results in most cases, it comes with several shortcomings. The methods require carefully designed regularizers and they are usually prone to outliers. In this paper, a new DR framework that can directly model the target distribution using the notion of similarity instead of distance is introduced. The proposed framework, called similarity embedding framework (SEF), can overcome the aforementioned limitations and provides a conceptually simpler way to express optimization targets similar to existing DR techniques. Deriving a new DR technique using the SEF becomes simply a matter of choosing an appropriate target similarity matrix. A variety of classical tasks, such as performing supervised DR and providing out-of-sample extensions, as well as, new novel techniques, such as providing fast linear embeddings for complex techniques, are demonstrated in this paper using the proposed framework. Six data sets from a diverse range of domains are used to evaluate the proposed method and it is demonstrated that it can outperform many existing DR techniques.
A roadmap to multifactor dimensionality reduction methods.
Gola, Damian; Mahachie John, Jestinah M; van Steen, Kristel; König, Inke R
2016-03-01
Complex diseases are defined to be determined by multiple genetic and environmental factors alone as well as in interactions. To analyze interactions in genetic data, many statistical methods have been suggested, with most of them relying on statistical regression models. Given the known limitations of classical methods, approaches from the machine-learning community have also become attractive. From this latter family, a fast-growing collection of methods emerged that are based on the Multifactor Dimensionality Reduction (MDR) approach. Since its first introduction, MDR has enjoyed great popularity in applications and has been extended and modified multiple times. Based on a literature search, we here provide a systematic and comprehensive overview of these suggested methods. The methods are described in detail, and the availability of implementations is listed. Most recent approaches offer to deal with large-scale data sets and rare variants, which is why we expect these methods to even gain in popularity. © The Author 2015. Published by Oxford University Press.
Approximate Orthogonal Sparse Embedding for Dimensionality Reduction.
Lai, Zhihui; Wong, Wai Keung; Xu, Yong; Yang, Jian; Zhang, David
2016-04-01
Locally linear embedding (LLE) is one of the most well-known manifold learning methods. As the representative linear extension of LLE, orthogonal neighborhood preserving projection (ONPP) has attracted widespread attention in the field of dimensionality reduction. In this paper, a unified sparse learning framework is proposed by introducing the sparsity or L1-norm learning, which further extends the LLE-based methods to sparse cases. Theoretical connections between the ONPP and the proposed sparse linear embedding are discovered. The optimal sparse embeddings derived from the proposed framework can be computed by iterating the modified elastic net and singular value decomposition. We also show that the proposed model can be viewed as a general model for sparse linear and nonlinear (kernel) subspace learning. Based on this general model, sparse kernel embedding is also proposed for nonlinear sparse feature extraction. Extensive experiments on five databases demonstrate that the proposed sparse learning framework performs better than the existing subspace learning algorithm, particularly in the cases of small sample sizes.
Dimensional reduction of web traffic data
Nikulin, Vladimir
2006-04-01
Dimensional reduction may be effective in order to compress data without loss of essential information. Also, it may be useful in order to smooth data and reduce random noise. The model presented in this paper was motivated by the structure of the msweb web-traffic dataset from the UCI archive. It is proposed to reduce dimension (number of the used web-areas or vroots) as a result of the unsupervised learning process maximizing specially defined average log-likelihood divergence. Two different web-areas will be merged in the case if these areas appear together frequently during the same sessions. Essentially, roles of the web-areas are not symmetrical in the merging process. The web-area or cluster with bigger weight will act as an attractor and will stimulate merging. In difference, the smaller cluster will try to keep independence. In both cases the powers of attraction or resistance will depend on the weights of the corresponding clusters. Above strategy will prevent creation of one super-big cluster, and will help to reduce number of non-significant clusters. The proposed method was illustrated using two synthetic examples. The first example is based on an ideal vlink matrix which characterizes weights of the vroots and relations between them. The vlink matrix for the second example was generated using specially designed web-traffic simulator.
Dimensionality reduction methods in virtual metrology
Zeng, Dekong; Tan, Yajing; Spanos, Costas J.
2008-03-01
The objective of this work is the creation of predictive models that can forecast the electrical or physical parameters of wafers using data collected from the relevant processing tools. In this way, direct measurements from the wafer can be minimized or eliminated altogether, hence the term "virtual" metrology. Challenges include the selection of the appropriate process step to monitor, the pre-treatment of the raw data, and the deployment of a Virtual Metrology Model (VMM) that can track a manufacturing process as it ages. A key step in any VM application is dimensionality reduction, i.e. ensuring that the proper subset of predictors is included in the model. In this paper, a software tool developed with MATLAB is demonstrated for interactive data prescreening and selection. This is combined with a variety of automated statistical techniques. These include step-wise regression and genetic selection in conjunction with linear modeling such as Principal Component Regression (PCR) and Partial Least Squares (PLS). Modeling results based on industrial datasets are used to demonstrate the effectiveness of these methods.
Teleparallel gravity and dimensional reductions of noncommutative gauge theory
Langmann, Edwin; Szabo, Richard J.
2001-11-01
We study dimensional reductions of noncommutative electrodynamics on flat space, which lead to gauge theories of gravitation. For a general class of such reductions, we show that the noncommutative gauge fields naturally yield a Weitzenböck geometry on spacetime and that the induced diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity which macroscopically describes general relativity. The Planck length is determined in this setting by the Yang-Mills coupling constant and the noncommutativity scale. The effective field theory can also contain higher curvature and non-local terms which are characteristic of string theory. Some applications to D-brane dynamics and generalizations to include the coupling of ordinary Yang-Mills theory to gravity are also described.
A local approach to dimensional reduction. I. General formalism
Nikolov, Petko A.; Petrov, Nikola P.
2003-01-01
We present a formalism for dimensional reduction based on the local properties of invariant cross-sections ("fields") and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a "non-canonical" reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.
Spinors in Four-Dimensional Spaces
Torres del Castillo, Gerardo F
2010-01-01
Without using the customary Clifford algebras frequently studied in connection with the representations of orthogonal groups, this book gives an elementary introduction to the two-component spinor formalism for four-dimensional spaces with any signature. Some of the useful applications of four-dimensional spinors, such as Yang–Mills theory, are derived in detail using illustrative examples. Key topics and features: • Uniform treatment of the spinor formalism for four-dimensional spaces of any signature, not only the usual signature (+ + + −) employed in relativity • Examples taken from Riemannian geometry and special or general relativity are discussed in detail, emphasizing the usefulness of the two-component spinor formalism • Exercises in each chapter • The relationship of Clifford algebras and Dirac four-component spinors is established • Applications of the two-component formalism, focusing mainly on general relativity, are presented in the context of actual computations Spinors in Four-Dim...
Optimal band selection for dimensionality reduction of hyperspectral imagery
Stearns, Stephen D.; Wilson, Bruce E.; Peterson, James R.
1993-01-01
Hyperspectral images have many bands requiring significant computational power for machine interpretation. During image pre-processing, regions of interest that warrant full examination need to be identified quickly. One technique for speeding up the processing is to use only a small subset of bands to determine the 'interesting' regions. The problem addressed here is how to determine the fewest bands required to achieve a specified performance goal for pixel classification. The band selection problem has been addressed previously Chen et al., Ghassemian et al., Henderson et al., and Kim et al.. Some popular techniques for reducing the dimensionality of a feature space, such as principal components analysis, reduce dimensionality by computing new features that are linear combinations of the original features. However, such approaches require measuring and processing all the available bands before the dimensionality is reduced. Our approach, adapted from previous multidimensional signal analysis research, is simpler and achieves dimensionality reduction by selecting bands. Feature selection algorithms are used to determine which combination of bands has the lowest probability of pixel misclassification. Two elements required by this approach are a choice of objective function and a choice of search strategy.
Dimensional reduction and vacuum structure of quiver gauge theory
Dolan, Brian P.; Szabo, Richard J.
2009-08-01
We describe the structure of the vacuum states of quiver gauge theories obtained via dimensional reduction over homogeneous spaces, in the explicit example of SU(3)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M × Bbb CP2. We pay particular attention to the role of topology of background gauge fields on the internal coset spaces, in this case U(1) magnetic monopoles and SU(2) instantons on Bbb CP2. The reduction of Yang-Mills theory induces a quiver gauge theory involving coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking. The criterion for a ground state of the Higgs potential can be written as the vanishing of a non-abelian Yang-Mills flux on the quiver diagram, regarded as a lattice with group elements attached to the links. The reduction of SU(3)-symmetric fermions yields Dirac fermions on M transforming under the low-energy gauge group with Yukawa couplings. The fermionic zero modes on Bbb CP2 yield exactly massless chiral fermions on M, though there is a unique choice of spinc structure on Bbb CP2 for which some of the zero modes can acquire masses through Yukawa interactions. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples, some of which possess quantum number assignments qualitatively analogous to the manner in which vector bosons, quarks and leptons acquire masses in the standard model.
Graph embedded nonparametric mutual information for supervised dimensionality reduction.
Bouzas, Dimitrios; Arvanitopoulos, Nikolaos; Tefas, Anastasios
2015-05-01
In this paper, we propose a novel algorithm for dimensionality reduction that uses as a criterion the mutual information (MI) between the transformed data and their corresponding class labels. The MI is a powerful criterion that can be used as a proxy to the Bayes error rate. Furthermore, recent quadratic nonparametric implementations of MI are computationally efficient and do not require any prior assumptions about the class densities. We show that the quadratic nonparametric MI can be formulated as a kernel objective in the graph embedding framework. Moreover, we propose its linear equivalent as a novel linear dimensionality reduction algorithm. The derived methods are compared against the state-of-the-art dimensionality reduction algorithms with various classifiers and on various benchmark and real-life datasets. The experimental results show that nonparametric MI as an optimization objective for dimensionality reduction gives comparable and in most of the cases better results compared with other dimensionality reduction methods.
Regularization by Dimensional Reduction: Consistency, Quantum Action Principle, and Supersymmetry
Stöckinger, Dominik
2005-03-01
It is proven by explicit construction that regularization by dimensional reduction can be formulated in a mathematically consistent way. In this formulation the quantum action principle is shown to hold. This provides an intuitive and elegant relation between the D-dimensional lagrangian and Ward or Slavnov-Taylor identities, and it can be used in particular to study to what extent dimensional reduction preserves supersymmetry. We give several examples of previously unchecked cases.
Nonlinear Dimensionality Reduction Methods in Climate Data Analysis
Ross, Ian
2009-01-01
Linear dimensionality reduction techniques, notably principal component analysis, are widely used in climate data analysis as a means to aid in the interpretation of datasets of high dimensionality. These linear methods may not be appropriate for the analysis of data arising from nonlinear processes occurring in the climate system. Numerous techniques for nonlinear dimensionality reduction have been developed recently that may provide a potentially useful tool for the identification of low-di...
On Dimensionality Reduction for Indexing and Retrieval of Large-Scale Solar Image Data
Banda, J. M.; Angryk, R. A.; Martens, P. C. H.
2013-03-01
This work investigates the applicability of several dimensionality reduction techniques for large-scale solar data analysis. Using a solar benchmark dataset that contains images of multiple types of phenomena, we investigate linear and nonlinear dimensionality reduction methods in order to reduce our storage and processing costs and maintain a good representation of our data in a new vector space. We present a comparative analysis of several dimensionality reduction methods and different numbers of target dimensions by utilizing different classifiers in order to determine the degree of data dimensionality reduction that can be achieved with these methods, and to discover the method that is the most effective for solar images. After determining the optimal number of dimensions, we then present preliminary results on indexing and retrieval of the dimensionally reduced data.
Computation of the string tension in three dimensional Yang-Mills theory using large N reduction
Kiskis, Joe; Narayanan, Rajamani
2008-09-01
We numerically compute the string tension in the large N limit of three dimensional Yang-Mills theory using Wilson loops. Space-time loops are formed as products of smeared space-like links and unsmeared time-like links. We use continuum reduction and both unfolded and folded Wilson loops in the analysis.
Robust Structure Preserving Nonnegative Matrix Factorization for Dimensionality Reduction
Directory of Open Access Journals (Sweden)
Bingfeng Li
2016-01-01
Full Text Available As a linear dimensionality reduction method, nonnegative matrix factorization (NMF has been widely used in many fields, such as machine learning and data mining. However, there are still two major drawbacks for NMF: (a NMF can only perform semantic factorization in Euclidean space, and it fails to discover the intrinsic geometrical structure of high-dimensional data distribution. (b NMF suffers from noisy data, which are commonly encountered in real-world applications. To address these issues, in this paper, we present a new robust structure preserving nonnegative matrix factorization (RSPNMF framework. In RSPNMF, a local affinity graph and a distant repulsion graph are constructed to encode the geometrical information, and noisy data influence is alleviated by characterizing the data reconstruction term of NMF with l2,1-norm instead of l2-norm. With incorporation of the local and distant structure preservation regularization term into the robust NMF framework, our algorithm can discover a low-dimensional embedding subspace with the nature of structure preservation. RSPNMF is formulated as an optimization problem and solved by an effective iterative multiplicative update algorithm. Experimental results on some facial image datasets clustering show significant performance improvement of RSPNMF in comparison with the state-of-the-art algorithms.
Hierarchical discriminant manifold learning for dimensionality reduction and image classification
Chen, Weihai; Zhao, Changchen; Ding, Kai; Wu, Xingming; Chen, Peter C. Y.
2015-09-01
In the field of image classification, it has been a trend that in order to deliver a reliable classification performance, the feature extraction model becomes increasingly more complicated, leading to a high dimensionality of image representations. This, in turn, demands greater computation resources for image classification. Thus, it is desirable to apply dimensionality reduction (DR) methods for image classification. It is necessary to apply DR methods to relieve the computational burden as well as to improve the classification accuracy. However, traditional DR methods are not compatible with modern feature extraction methods. A framework that combines manifold learning based DR and feature extraction in a deeper way for image classification is proposed. A multiscale cell representation is extracted from the spatial pyramid to satisfy the locality constraints for a manifold learning method. A spectral weighted mean filtering is proposed to eliminate noise in the feature space. A hierarchical discriminant manifold learning is proposed which incorporates both category label and image scale information to guide the DR process. Finally, the image representation is generated by concatenating dimensionality reduced cell representations from the same image. Extensive experiments are conducted to test the proposed algorithm on both scene and object recognition datasets in comparison with several well-established and state-of-the-art methods with respect to classification precision and computational time. The results verify the effectiveness of incorporating manifold learning in the feature extraction procedure and imply that the multiscale cell representations may be distributed on a manifold.
Decentralized Dimensionality Reduction for Distributed Tensor Data Across Sensor Networks.
Liang, Junli; Yu, Guoyang; Chen, Badong; Zhao, Minghua
2016-11-01
This paper develops a novel decentralized dimensionality reduction algorithm for the distributed tensor data across sensor networks. The main contributions of this paper are as follows. First, conventional centralized methods, which utilize entire data to simultaneously determine all the vectors of the projection matrix along each tensor mode, are not suitable for the network environment. Here, we relax the simultaneous processing manner into the one-vector-by-one-vector (OVBOV) manner, i.e., determining the projection vectors (PVs) related to each tensor mode one by one. Second, we prove that in the OVBOV manner each PV can be determined without modifying any tensor data, which simplifies corresponding computations. Third, we cast the decentralized PV determination problem as a set of subproblems with consensus constraints, so that it can be solved in the network environment only by local computations and information communications among neighboring nodes. Fourth, we introduce the null space and transform the PV determination problem with complex orthogonality constraints into an equivalent hidden convex one without any orthogonality constraint, which can be solved by the Lagrange multiplier method. Finally, experimental results are given to show that the proposed algorithm is an effective dimensionality reduction scheme for the distributed tensor data across the sensor networks.
Spectral Reduction of Two and Three-Dimensional Turbulence
Bowman, John; Shadwick, B. A.; Morrison, P. J.
2000-10-01
The method of spectral reduction(J. C. Bowman et al. Phys. Rev. Lett.) 83, 5491 (1999) performs a coarse-graining in wavenumber space to greatly reduce the number of modes required to simulate incompressible homogeneous turbulence. It has recently been extended from two to three dimensions. A Liouville theorem for the inviscid dynamics leads to statistical equipartition solutions. However, if the wavenumber bins are of nonuniform size (as is desirable for efficiency), an additional bin-dependent rescaling of time by the bin size must be introduced to obtain the correct equipartition of modal (rather than bin) energies. Unfortunately, a practical numerical method has not yet been developed to solve the rescaled spectrally reduced equations. However, for the two dimensional enstrophy cascade, this rescaling is shown not to be necessary to obtain the correct small-scale nonlinear relaxation. Spectral reduction in its present form may thus be ideally suited to certain two-dimensional plasma turbulence problems (e.g. as a subgrid model).
Stochastic confinement and dimensional reduction. 1
International Nuclear Information System (INIS)
Ambjoern, J.; Olesen, P.; Peterson, C.
1984-03-01
By Monte Carlo calculations on a 16 4 lattice the authors investigate four dimensional SU(2) lattice guage theory with respect to the conjecture that at large distances this theory reduces approximately to two dimensional SU(2) lattice gauge theory. Good numerical evidence is found for this conjecture. As a by-product the SU(2) string tension is also measured and good agreement is found with scaling. The 'adjoint string tension' is also found to have a reasonable scaling behaviour. (Auth.)
Dimensionality reduction in translational noninvariant wave guides
Voo, Khee-Kyun
2008-01-01
A scheme to reduce translational noninvariant quasi-one-dimensional wave guides into singly or multiply connected one-dimensional (1D) lines is proposed. It is meant to simplify the analysis of wave guides, with the low-energy properties of the guides preserved. Guides comprising uniform-cross-sectional sections and discontinuities such as bends and branching junctions are considered. The uniform sections are treated as 1D lines, and the discontinuities are described by equations sets connect...
Stochastic confinement and dimensional reduction. Pt. 1
International Nuclear Information System (INIS)
Ambjoern, J.; Olesen, P.; Peterson, C.
1984-01-01
By Monte Carlo calculations on a 12 4 lattice we investigate four-dimensional SU(2) lattice gauge theory with respect to the conjecture that at large distances this theory reduces approximately to two-dimensional SU(2) lattice gauge theory. We find good numerical evidence for this conjecture. As a by-product we also measure the SU(2) string tension and find reasonable agreement with scaling. The 'adjoint string tension' is also found to have a reasonable scaling behaviour. (orig.)
Parallel Framework for Dimensionality Reduction of Large-Scale Datasets
Directory of Open Access Journals (Sweden)
Sai Kiranmayee Samudrala
2015-01-01
Full Text Available Dimensionality reduction refers to a set of mathematical techniques used to reduce complexity of the original high-dimensional data, while preserving its selected properties. Improvements in simulation strategies and experimental data collection methods are resulting in a deluge of heterogeneous and high-dimensional data, which often makes dimensionality reduction the only viable way to gain qualitative and quantitative understanding of the data. However, existing dimensionality reduction software often does not scale to datasets arising in real-life applications, which may consist of thousands of points with millions of dimensions. In this paper, we propose a parallel framework for dimensionality reduction of large-scale data. We identify key components underlying the spectral dimensionality reduction techniques, and propose their efficient parallel implementation. We show that the resulting framework can be used to process datasets consisting of millions of points when executed on a 16,000-core cluster, which is beyond the reach of currently available methods. To further demonstrate applicability of our framework we perform dimensionality reduction of 75,000 images representing morphology evolution during manufacturing of organic solar cells in order to identify how processing parameters affect morphology evolution.
Minimizing the loss of entanglement under dimensional reduction
Petersen, V.; Madsen, L. B.; Mølmer, K.
2004-05-01
We investigate the possibility of transforming, under local operations and classical communication, a general bipartite quantum state on a d A x d B tensor-product space into a final state in 2 x 2 dimensions, while maintaining as much entanglement as possible. For pure states, we prove that Nielsen’s theorem provides the optimal protocol, and we present quantitative results on the degree of entanglement before and after the dimensional reduction. For mixed states, we identify a protocol that we argue is optimal for isotropic and Werner states. In the literature, it has been conjectured that some Werner states are bound entangled and in support of this conjecture our protocol gives final states without entanglement for this class of states. For all other entangled Werner states and for all entangled isotropic states some degree of free entanglement is maintained. In this sense, our protocol may be used to discriminate between bound and free entanglement.
Gillis, Nicolas; Plemmons, Robert J.
2010-04-01
Nonnegative Matrix Factorization (NMF) and its variants have recently been successfully used as dimensionality reduction techniques for identification of the materials present in hyperspectral images. In this paper, we present a new variant of NMF called Nonnegative Matrix Underapproximation (NMU): it is based on the introduction of underapproximation constraints which enables one to extract features in a recursive way, like PCA, but preserving nonnegativity. Moreover, we explain why these additional constraints make NMU particularly wellsuited to achieve a parts-based and sparse representation of the data, enabling it to recover the constitutive elements in hyperspectral data. We experimentally show the efficiency of this new strategy on hyperspectral images associated with space object material identification, and on HYDICE and related remote sensing images.
Immersive Visualization of the Quality of Dimensionality Reduction
Babaee, M.; Datcu, M.; Rigoll, G.
2013-09-01
Dimensionality reduction is the most widely used approach for extracting the most informative low-dimensional features from highdimensional ones. During the last two decades, different techniques (linear and nonlinear) have been proposed by researchers in various fields. However, the main question is now how well a specific technique does this job. In this paper, we introduce a qualitative method to assess the quality of dimensionality reduction. In contrast to numerical assessment, we focus here on visual assessment. We visualize the Minimum Spanning Tree (MST) of neighborhood graphs of data before and after dimensionality reduction in an immersive 3D virtual environment. We employe a mixture of linear and nonlinear dimension reduction techniques to apply to both synthetic and real datasets. The visualization depicts the quality of each technique in term of preserving distances and neighborhoods. The results show that a specific dimension reduction technique exhibits different performance in dealing with different datasets.
Locally Linear Embedding for dimensionality reduction in QSAR
L'Heureux, P.-J.; Carreau, J.; Bengio, Y.; Delalleau, O.; Yue, S. Y.
2004-07-01
Current practice in Quantitative Structure Activity Relationship (QSAR) methods usually involves generating a great number of chemical descriptors and then cutting them back with variable selection techniques. Variable selection is an effective method to reduce the dimensionality but may discard some valuable information. This paper introduces Locally Linear Embedding (LLE), a local non-linear dimensionality reduction technique, that can statistically discover a low-dimensional representation of the chemical data. LLE is shown to create more stable representations than other non-linear dimensionality reduction algorithms, and to be capable of capturing non-linearity in chemical data.
Continuous symmetry reduction and return maps for high-dimensional flows
Siminos, Evangelos; Cvitanović, Predrag
2011-01-01
We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a five-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.
Dimensional reduction of the ABJM model
Nastase, Horatiu; Papageorgakis, Constantinos
2011-03-01
We dimensionally reduce the ABJM model, obtaining a two-dimensional theory that can be thought of as a `master action'. This encodes information about both T- and S-duality, i.e. describes fundamental (F1) and D-strings (D1) in 9 and 10 dimensions. The Higgsed theory at large VEV, tilde{v} , and large k yields D1-brane actions in 9d and 10d, depending on which auxiliary fields are integrated out. For N = 1thereisamaptoa Green-Schwarz string wrapping a nontrivial circle in {{{{mathbb{C}^4}}} left/ {{{mathbb{Z}_k}}} right.}.
Dimensionality reduction for probabilistic movement primitives
Colome, A.; Neumann, G.; Peters, J.; Torras, C.
2014-01-01
Humans as well as humanoid robots can use a large number of degrees of freedom to solve very complex motor tasks. The high-dimensionality of these motor tasks adds difficulties to the control problem and machine learning algorithms. However, it is well known that the intrinsic dimensionality of many human movements is small in comparison to the number of employed DoFs, and hence, the movements can be represented by a small number of synergies encoding the couplings between DoFs. In this paper...
Flexible Multi-View Dimensionality Co-Reduction.
Zhang, Changqing; Fu, Huazhu; Hu, Qinghua; Zhu, Pengfei; Cao, Xiaochun
2017-02-01
Dimensionality reduction aims to map the high-dimensional inputs onto a low-dimensional subspace, in which the similar points are close to each other and vice versa. In this paper, we focus on unsupervised dimensionality reduction for the data with multiple views, and propose a novel method, called Multi-view Dimensionality co-Reduction. Our method flexibly exploits the complementarity of multiple views during the dimensionality reduction and respects the similarity relationships between data points across these different views. The kernel matching constraint based on Hilbert-Schmidt Independence Criterion enhances the correlations and penalizes the disagreement of different views. Specifically, our method explores the correlations within each view independently, and maximizes the dependence among different views with kernel matching jointly. Thus, the locality within each view and the consistence between different views are guaranteed in the subspaces corresponding to different views. More importantly, benefiting from the kernel matching, our method need not depend on a common low-dimensional subspace, which is critical to reduce the influence of the unbalanced dimensionalities of multiple views. Specifically, our method explicitly produces individual low-dimensional projections for individual views, which could be applied for new coming data in the out-of-sample manner. Experiments on both clustering and recognition tasks demonstrate the advantages of the proposed method over the state-of-the-art approaches.
Effects of dimensionality reduction on the statististical distribution of hyperspectral backgrounds
Rossacci, M.; Manolakis, D.; Cipar, J.; Lockwood, R.; Cooley, T.; Jacobson, J.
2006-08-01
The objective of this paper is to investigate the effects of dimensionality reduction on the statistical distribution of natural hyperspectral backgrounds. The statistical modeling is based on application of the multivariate t-elliptically contoured distribution to background regions which have been shown to exhibit "long-tail" behavior. Hyperspectral backgrounds are commonly represented with reduced dimensionality in order to minimize statistical redundancies in the spectral dimension and to satisfy data processing and storage requirements. In this investigation, we extend the statistical characterization of these backgrounds by modeling their Mahalanobis distance distributions in reduced dimensional space. The dimensionality reduction techniques applied in this paper include Principal Components Analysis (PCA) and spectral band aggregation. The knowledge gained from a better understanding of the effects of dimensionality reduction will be beneficial toward improving threshold selection for target detection applications. These investigations are done using hyperspectral data from the AVIRIS sensor and include spectrally homogeneous regions of interest obtained by visual interactive spatial segmentation.
Dimensional regularization and dimensional reduction in the light cone
Qiu, J.
2008-06-01
We calculate all of the 2 to 2 scattering process in Yang-Mills theory in the light cone gauge, with the dimensional regulator as the UV regulator. The IR is regulated with a cutoff in q+. It supplements our earlier work, where a Lorentz noncovariant regulator was used, and the final results bear some problems in gauge fixing. Supersymmetry relations among various amplitudes are checked by using the light cone superfields.
Dimensional reduction in causal set gravity
Carlip, S.
2015-12-01
Results from a number of different approaches to quantum gravity suggest that the effective dimension of spacetime may drop to d = 2 at small scales. I show that two different dimensional estimators in causal set theory display the same behavior, and argue that a third, the spectral dimension, may exhibit a related phenomenon of ‘asymptotic silence.’
Efficient EMD-based Similarity Search in Multimedia Databases via Flexible Dimensionality Reduction
DEFF Research Database (Denmark)
Wichterich, Marc; Assent, Ira; Philipp, Kranen
2008-01-01
The Earth Mover's Distance (EMD) was developed in computer vision as a flexible similarity model that utilizes similarities in feature space to define a high quality similarity measure in feature representation space. It has been successfully adopted in a multitude of applications with low...... dimensionality reduction techniques for the EMD in a filter-and-refine architecture for efficient lossless retrieval. Thorough experimental evaluation on real world data sets demonstrates a substantial reduction of the number of expensive high-dimensional EMD computations and thus remarkably faster response...
Linear Dimensionality Reduction in Random Motion Planning
Dalibard, Sébastien; Laumond, Jean-Paul
2011-01-01
Accepté pour publication à International Journal of Robotics Research; International audience; The paper presents a method to control probabilistic diffusion in motion planning algorithms. The principle of the method is to use on line the results of a diffusion algorithm to describe the free space in which the planning takes place, by computing a Principal Component Analysis (PCA). This method identifies the locally free directions of the free space. Given that description, our algorithm acce...
Dimensionality reduction in translational noninvariant wave guides
Voo, Khee-Kyun
2009-01-01
A scheme to reduce translational noninvariant quasi-one-dimensional wave guides into singly or multiply connected one-dimensional (1D) lines is proposed. It is meant to simplify the analysis of wave guides, with the low-energy properties of the guides preserved. Guides comprising uniform-cross-sectional sections and discontinuities such as bends and branching junctions are considered. The uniform sections are treated as 1D lines, and the discontinuities are described by equations sets connecting the wave functions on the lines. The procedures to derive the equations and to solve reduced systems are illustrated by examples, and the scheme is found to apply when the discontinuities are distant and the energy is low. When the scheme applies, it may substantially simplify the analysis of a wave guide, and hence the scheme may find uses in the study of related problems, such as quantum wire networks.
Testing dimensional reduction in SU(2) gauge theory
Kratochvila, Slavo; de Forcrand, Philippe
2002-03-01
At high temperature, every ( d + 1)-dimensional theory can be reformulated as an effective theory in d dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from T = 4 T c downwards.
Aspects of dynamical dimensional reduction in multigraph ensembles of CDT
Giasemidis, Georgios; Wheater, John F.; Zohren, Stefan
2013-02-01
We study the continuum limit of a "radially reduced" approximation of Causal Dynamical Triangulations (CDT), so-called multigraph ensembles, and explain why they serve as realistic toy models to study the dimensional reduction observed in numerical simulations of four-dimensional CDT. We present properties of this approximation in two, three and four dimensions comparing them with the numerical simulations and pointing out some common features with 2+1 dimensional Hořava-Lifshitz gravity.
Preserving Global and Local Structures for Supervised Dimensionality Reduction
Directory of Open Access Journals (Sweden)
Song Yinglei
2015-01-01
Full Text Available In this paper, we develop a new approach for dimensionality reduction of labeled data. This approach integrates both global and local structures of data into a new objective, we show that the objective can be optimized by solving an eigenvalue problem. Testing results on benchmark data sets show that this new approach can effectively capture both the crucial global and local structures of data and thus lead to more accurate results for dimensionality reduction than existing approaches.
Exact Dimensional Reduction of Linear Dynamics: Application to Confined Diffusion
Kalinay, Pavol; Percus, Jerome K.
2006-06-01
In their stochastic versions, dynamical systems take the form of the linear dynamics of a probability distribution. We show that exact dimensional reduction of such systems can be carried out, and is physically relevant when the dimensions to be eliminated can be identified with those that represent transient behavior, disappearing under typical coarse graining. Application is made to non-uniform quasi-low dimensional diffusion, resulting in a systematic extension of the "classical" Fick-Jacobs approximate reduction to an exact subdynamics.
Randomized Dimensionality Reduction for k-means Clustering
Boutsidis, Christos; Zouzias, Anastasios; Mahoney, Michael W.; Drineas, Petros
2011-01-01
We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \\emph{feature selection} and \\emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$...
Fourier dimensionality reduction of radio-interferometric data
Kartik, Vijay; Carrillo, Rafael; Thiran, Jean-Philippe; Wiaux, Yves
2017-01-01
Next-generation radio-interferometers face a computing challenge with respect to the imaging techniques that can be applied in the big data setting in which they are designed. Dimensionality reduction can thus provide essential savings of computing resources, allowing imaging methods to scale with data. The work presented here approaches dimensionality reduction from a compressed sensing theory perspective, and links to its role in convex optimization-based imaging algorithms. We describe a n...
Dimensionality reduction of quality of life indicators
Andrea Jindrová; Julie Poláčková
2012-01-01
Selecting indicators for assessing the quality of life at the regional level is not unambigous. Currently, there are no precisely defined indicators that would give comprehensive information about the quality of life on a local level. In this paper we focus on the determination (selection) of groups of indicators that can be interpreted, on the basis of studied literature, as factors characterizing the quality of life. Furthermore, on the application of methods to reduce the dimensionality of...
Multiple Kernel Spectral Regression for Dimensionality Reduction
Liu, Bing; Xia, Shixiong; Zhou, Yong
2013-01-01
Traditional manifold learning algorithms, such as locally linear embedding, Isomap, and Laplacian eigenmap, only provide the embedding results of the training samples. To solve the out-of-sample extension problem, spectral regression (SR) solves the problem of learning an embedding function by establishing a regression framework, which can avoid eigen-decomposition of dense matrices. Motivated by the effectiveness of SR, we incorporate multiple kernel learning (MKL) into SR for dimensionality...
Efficient Supervised Dimensionality Reduction for Image Categorization
Benmokhtar, Rachid; Delhumeau, Jonathan; Gosselin, Philippe-Henri
2013-01-01
International audience; This paper addresses the problem of large scale image repre- sentation for object recognition and classification. Our work deals with the problem of optimizing the classification accu- racy and the dimensionality of the image representation. We propose to iteratively select sets of projections from an ex- ternal dataset, using Bagging and feature selection thanks to SVM normals. Features are selected using weights of SVM normals in orthogonalized sets of projections. T...
Evaluating Information Loss from Phonological Dimensionality Reduction
Macklin-Cordes, Jayden L.; Moran, Steven; Round, Erich R.
2016-01-01
As in many sciences, cross-linguistic data is often complex and multi-dimensional. The PHOIBLE database of phonological inventories (Moran et al. 2014) is one example, containing 2160 distinct segments across 2155 phoneme inventories. Each segment type is defined by a unique vector of (mostly binary) distinctive phonetic and phonological features. This multivariate dataset can be modeled as a set of coordinates, where each variable (e.g. segment, its distinctive featur...
Symmetry Reductions of Two-Dimensional Variable Coefficient Burgers Equation
Zhang, Xiao-Ling; Li, Biao
2005-05-01
By use of a direct method, we discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient (VCBurgers). Five types of symmetry-reducing VCBurgers to (1+1)-dimensional partial differential equation and three types of symmetry reducing VCBurgers to ordinary differential equation are obtained.
Higher-derivative massive actions from dimensional reduction
Joung, Euihun; Mkrtchyan, Karapet
2013-02-01
A procedure to obtain higher-derivative free massive actions is proposed. It consists in dimensional reduction of conventional two-derivative massless actions, where solutions to constraints bring in higher derivatives. We apply this procedure to derive the arbitrary dimensional generalizations of (linearized) New Massive Gravity and New Topologically Massive Gravity.
Setting the Scale of Dimensional Reduction in Causal Dynamical Triangulations
Cooperman, Joshua
2011-04-01
Within the causal dynamical triangulations approach to quantization of gravity, striking evidence has emerged that the effective dimensionality of spacetime dynamically reduces at small scales. Specifically, in the case of topological sphericity, the expectation value of the spectral dimension decreases with the scale being probed from the topological value of four to an apparent value of two. Thus far the physical scale at which this dynamical dimensional reduction occurs has not been ascertained. In this talk I present the first determinations of this scale. By fitting the expectation value of the spacetime geometry to a classical minisuperspace model, I extract the triangulation lattice spacing in units of the Planck length and the effective cosmological constant in units of the inverse Planck length squared. The former value allows me to establish directly the scales probed by the random walk that defines the spectral dimension. The latter value allows me to establish indirectly these scales via the heat trace for the minisuperspace geometry. This work also yields preliminary indications of the flow of the cosmological constant within this model of quantum geometry.
Dimensional reduction of the heterotic string over nearly-Kähler manifolds
Chatzistavrakidis, Athanasios; Zoupanos, George
2009-09-01
Our aim is to derive the effective action in four dimensions resulting by reducing dimensionally the ten-dimensional Script N = 1 heterotic supergravity coupled to Script N = 1 super Yang-Mills over manifolds admitting a nearly-Kähler structure. Given the fact that all homogeneous six-dimensional nearly-Kähler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold, our procedure amounts in applying the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds. In our examination firstly the rules of the reduction of the theory over a general six-dimensional non-symmetric manifold are stated and subsequently a detailed case by case analysis is performed for all the three non-symmetric coset spaces. For each case the four-dimensional scalar potential is derived and the corresponding nearly-Kähler limit is obtained. Finally, we determine the corresponding supergravity description of the four-dimensional theory employing the heterotic Gukov-Vafa-Witten formula and results of the special Kähler geometry.
Fiziev, P. P.; Shirkov, D. V.
2011-05-01
We develop the recent proposal to use dimensional reduction from the four-dimensional space-time (D = 1 + 3) to the variant with a smaller number of space dimensions D = 1 + d, d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation with a few toy examples ("educational toys") of a space-time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: ( 1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). ( 2) The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its "fingerprints." ( 3) The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation.
Obstructions to dimensional reduction in hot QCD
Gupta, Sourendu
2001-03-01
I describe results on screening masses in hot gauge theories. Wilsonian effective long distance theories called dimensionally reduced (DR) theories describe very well the longest screening length in pure gauge theories. In the presence of fermions, meson-like screening lengths dominate the long-distance physics for 3 Tc/2 ≤ T < 3 Tc, and thus obstruct perturbative DR. Extrapolation of our results indicates that a form of this obstruction may remain till temperatures of 10 Tc or higher, and therefore affect the entire range of temperature expected to be reached even at the Large Hadron Collider.
Two-color QCD via dimensional reduction
Zhang, Tian; Brauner, Tomáš; Kurkela, Aleksi; Vuorinen, Aleksi
2012-02-01
We study the thermodynamics of two-color QCD at high temperature and/or density using a dimensionally reduced superrenormalizable effective theory, formulated in terms of a coarse grained Wilson line. In the absence of quarks, the theory is required to respect the Z(2) center symmetry, while the effects of quarks of arbitrary masses and chemical potentials are introduced via soft Z(2) breaking operators. Perturbative matching of the effective theory parameters to the full theory is carried out explicitly, and it is argued how the new theory can be used to explore the phase diagram of two-color QCD.
Representation and Dimensional Reduction of the Universe
Wu, Zhong-Chao
2004-01-01
The external space we live in or the apparent dimension in the Kaluza Klein model can be identified by using the right representation in quantum cosmology. The external dimension of the Freund Rubin model is min(s,n-s), where s is the rank of the antisymmetric field strength in the model.
Wang, Shijun; Yao, Jianhua; Summers, Ronald M.
2008-01-01
Computer-aided detection (CAD) has been shown to be feasible for polyp detection on computed tomography (CT) scans. After initial detection, the dataset of colonic polyp candidates has large-scale and high dimensional characteristics. In this article, we propose a nonlinear dimensionality reduction method based on diffusion map and locally linear embedding (DMLLE) for large-scale datasets. By selecting partial data as landmarks, we first map these points into a low dimensional embedding space using the diffusion map. The embedded landmarks can be viewed as a skeleton of whole data in the low dimensional space. Then by using the locally linear embedding algorithm, nonlandmark samples are mapped into the same low dimensional space according to their nearest landmark samples. The local geometry is preserved in both the original high dimensional space and the embedding space. In addition, DMLLE provides a faithful representation of the original high dimensional data at coarse and fine scales. Thus, it can capture the intrinsic distance relationship between samples and reduce the influence of noisy features, two aspects that are crucial to achieving high classifier performance. We applied the proposed DMLLE method to a colonic polyp dataset of 175 269 polyp candidates with 155 features. Visual inspection shows that true polyps with similar shapes are mapped to close vicinity in the low dimensional space. We compared the performance of a support vector machine (SVM) classifier in the low dimensional embedding space with that in the original high dimensional space, SVM with principal component analysis dimensionality reduction and SVM committee using feature selection technology. Free-response receiver operating characteristic analysis shows that by using our DMLLE dimensionality reduction method, SVM achieves higher sensitivity with a lower false positive rate compared with other methods. For 6–9mm polyps (193 true polyps contained in test set), when the number of
Wang, Shijun; Yao, Jianhua; Summers, Ronald M
2008-04-01
Computer-aided detection (CAD) has been shown to be feasible for polyp detection on computed tomography (CT) scans. After initial detection, the dataset of colonic polyp candidates has large-scale and high dimensional characteristics. In this article, we propose a nonlinear dimensionality reduction method based on diffusion map and locally linear embedding (DMLLE) for large-scale datasets. By selecting partial data as landmarks, we first map these points into a low dimensional embedding space using the diffusion map. The embedded landmarks can be viewed as a skeleton of whole data in the low dimensional space. Then by using the locally linear embedding algorithm, nonlandmark samples are mapped into the same low dimensional space according to their nearest landmark samples. The local geometry is preserved in both the original high dimensional space and the embedding space. In addition, DMLLE provides a faithful representation of the original high dimensional data at coarse and fine scales. Thus, it can capture the intrinsic distance relationship between samples and reduce the influence of noisy features, two aspects that are crucial to achieving high classifier performance. We applied the proposed DMLLE method to a colonic polyp dataset of 175 269 polyp candidates with 155 features. Visual inspection shows that true polyps with similar shapes are mapped to close vicinity in the low dimensional space. We compared the performance of a support vector machine (SVM) classifier in the low dimensional embedding space with that in the original high dimensional space, SVM with principal component analysis dimensionality reduction and SVM committee using feature selection technology. Free-response receiver operating characteristic analysis shows that by using our DMLLE dimensionality reduction method, SVM achieves higher sensitivity with a lower false positive rate compared with other methods. For 6-9 mm polyps (193 true polyps contained in test set), when the number of false
Generalized dimensional reduction of supergravity with eight supercharges
Andrianopoli, L.; Lledo', M.A.
2005-01-01
We describe some recent investigation about the structure of generic D=4,5 theories obtained by generalized dimensional reduction of D=5,6 theories with eight supercharges. We relate the Scherk-Schwarz reduction to a special class of N=2 no-scale gauged supergravities.
Similarity reductions of the (2+1)-dimensional Burgers system
Liu, Dang-bo; Chu, Kai-qin
2001-08-01
In this paper, using the direct method of the (2+1)-dimensional multi-component Burgers system, some types of similarity reductions are obtained. The corresponding group explanations of the reductions, Virasoro integrability and soliton solutions of Burgers system are also discussed.
Generalized Dimensional Reduction of Supergravity with Eight Supercharges
Andrianopoli, L.; Ferrara, S.; Lledó, M. A.
2005-08-01
We describe some recent investigation about the structure of generic D = 4, 5 theories obtained by generalized dimensional reduction of D = 5, 6 theories with eight supercharges. We relate the Scherk-Schwarz reduction to a special class of N = 2 no-scale gauged supergravities.
Coupling running through the looking-glass of dimensional reduction
Shirkov, D. V.
2010-11-01
The dimensional reduction, in a form of transition from four to two dimensions, was used in the 90s of the past century in a context of the HE Regge scattering. Recently, it has got a new impetus in quantum gravity where it opens the way to renormalizability and finite short-distance behaviour. We consider a QFT model gφ4 with running coupling defined in both domains of different dimensionality; the bar g ( q 2) evolutions being duly correlated at the reduction scale q ˜ M. Beyond this scale, in the deep UV 2-dimensional region, the running coupling does not increase any more. Instead, it slightly decreases and tends to a finite value bar g 2(∞) < bar g 2( M 2) from above. As a result, the global evolution picture looks quite peculiar and proposes a base for the modified scenario of gauge couplings behavior with UV fixed points provided by dimensional reduction instead of leptoquarks.
Method of dimensionality reduction in contact mechanics and friction
Popov, Valentin L
2015-01-01
This book describes for the first time a simulation method for the fast calculation of contact properties and friction between rough surfaces in a complete form. In contrast to existing simulation methods, the method of dimensionality reduction (MDR) is based on the exact mapping of various types of three-dimensional contact problems onto contacts of one-dimensional foundations. Within the confines of MDR, not only are three dimensional systems reduced to one-dimensional, but also the resulting degrees of freedom are independent from another. Therefore, MDR results in an enormous reduction of the development time for the numerical implementation of contact problems as well as the direct computation time and can ultimately assume a similar role in tribology as FEM has in structure mechanics or CFD methods, in hydrodynamics. Furthermore, it substantially simplifies analytical calculation and presents a sort of “pocket book edition” of the entirety contact mechanics. Measurements of the rheology of bodies in...
High temperature 3D QCD: dimensional reduction at work
Bialas, P.; Morel, A.; Petersson, B.; Petrov, K.; Reisz, T.
2000-08-01
We investigate the three-dimensional SU(3) gauge theory at finite temperature in the framework of dimensional reduction. The large scale properties of this theory are expected to be conceptually more complicated than in four dimensions. The dimensionally reduced action is computed in closed analytical form. The resulting effective two-dimensional theory is studied numerically both in the electric and magnetic sector. We find that dimensional reduction works excellently down to temperatures of 1.5 times the deconfinement phase transition temperature and even on rather short length scales. We obtain strong evidence that for QCD 3, even at high temperature the colour averaged potential is represented by the exchange of a single state, at variance with the usual Debye screening picture involving a pair of electric gluons.
Cortical spatiotemporal dimensionality reduction for visual grouping
Cocci, Giacomo; Barbieri, Davide; Citti, Giovanna; Sarti, Alessandro
2015-01-01
The visual systems of many mammals, including humans, are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at the single cell level and geometric processing by means of cell connectivity. We present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visua...
Dimensionality reduction of SDPs through sketching
Bluhm, Andreas; Franca, Daniel Stilck
2017-01-01
We show how to sketch semidefinite programs (SDPs) using positive maps in order to reduce their dimension. More precisely, we use Johnson-Lindenstrauss transforms to produce a smaller SDP whose solution preserves feasibility or approximates the value of the original problem with high probability. These techniques allow to improve both complexity and storage space requirements. They apply to problems in which the Schatten 1-norm of the matrices specifying the SDP and of a solution to the probl...
Dimensional reduction near the deconfinement transition
International Nuclear Information System (INIS)
Kurkela, A.
2009-01-01
It is expected that incorporating the center symmetry in the conventional dimensionally reduced effective theory for high-temperature SU(N) Yang-Mills theory, EQCD, will considerably extend its applicability towards the deconfinement transition. In this talk, I will discuss the construction of such center-symmetric effective theories and present results from their lattice simulations in the case of two colors. The simulations demonstrate that unlike EQCD, the new center symmetric theory undergoes a second order confining phase transition in complete analogy with the full theory. I will also describe the perturbative and non-perturbative matching of the parameters of the effective theory, and outline ways to further improve its description of the physics near the deconfinement transition. (author)
Dimensionality reduction of quality of life indicators
Directory of Open Access Journals (Sweden)
Andrea Jindrová
2012-01-01
Full Text Available Selecting indicators for assessing the quality of life at the regional level is not unambigous. Currently, there are no precisely defined indicators that would give comprehensive information about the quality of life on a local level. In this paper we focus on the determination (selection of groups of indicators that can be interpreted, on the basis of studied literature, as factors characterizing the quality of life. Furthermore, on the application of methods to reduce the dimensionality of these indicators, from the source of the database CULS KROK, which provides statistics on the regional and districts level. To reduce the number of indicators and the subsequent creation of derived variables that capture the relationships between selected indicators multivariate statistical analysis methods, especially method of principal components and factor analysis were used. This paper also indicates the methodology grant project “Methodological Approaches to assess Subjective Aspects of the life quality in regions of the Czech Republic”.
Zenchuk, A. I.
2009-11-01
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1 + 1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with the scalar and three-dimensional reductions. Variants of the Kadomtsev-Petviashvili-type and Korteweg-de Vries-type equations are represented among them. Our algorithm is based on the combination of two Frobenius-type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.
Moon, Sangwoo; Qi, Hairong
2012-05-01
This paper presents a new hybrid dimensionality reduction method to seek projection through optimization of both structural risk (supervised criterion) and data independence (unsupervised criterion). Classification accuracy is used as a metric to evaluate the performance of the method. By minimizing the structural risk, projection originated from the decision boundaries directly improves the classification performance from a supervised perspective. From an unsupervised perspective, projection can also be obtained based on maximum independence among features (or attributes) in data to indirectly achieve better classification accuracy over more intrinsic representation of the data. Orthogonality interrelates the two sets of projections such that minimum redundancy exists between the projections, leading to more effective dimensionality reduction. Experimental results show that the proposed hybrid dimensionality reduction method that satisfies both criteria simultaneously provides higher classification performance, especially for noisy data sets, in relatively lower dimensional space than various existing methods.
Consistent dimensional reduction of five-dimensional off-shell supergravity
Abe, Hiroyuki; Sakamura, Yutaka
2006-06-01
There are some points to notice in the dimensional reduction of off-shell supergravity. We discuss a consistent way of dimensional reduction of five-dimensional off-shell supergravity compactified on S1/Z2. There are two approaches to the four-dimensional effective action, which are complementary to each other. Their essential difference is the treatment of the compensator and the radion superfields. We explain these approaches in detail and examine their consistency. Comments on related works are also provided.
MFV reductions of MSSM parameter space
Energy Technology Data Exchange (ETDEWEB)
AbdusSalam, S.S. [INFN - Sezione di Roma,P.le A. Moro 2, I-00185 Roma (Italy); The Abdus Salam ICTP,Trieste (Italy); Burgess, C.P. [Department of Physics & Astronomy, McMaster University,Hamilton ON (Canada); Perimeter Institute for Theoretical Physics,Waterloo, ON (Canada); Division PH -TH, CERN,CH-1211, Genève 23 (Switzerland); Quevedo, F. [The Abdus Salam ICTP,Trieste (Italy); DAMTP, Cambridge University,Cambridge (United Kingdom)
2015-02-11
The 100+ free parameters of the minimal supersymmetric standard model (MSSM) make it computationally difficult to compare systematically with data, motivating the study of specific parameter reductions such as the cMSSM and pMSSM. Here we instead study the reductions of parameter space implied by using minimal flavour violation (MFV) to organise the R-parity conserving MSSM, with a view towards systematically building in constraints on flavour-violating physics. Within this framework the space of parameters is reduced by expanding soft supersymmetry-breaking terms in powers of the Cabibbo angle, leading to a 24-, 30- or 42-parameter framework (which we call MSSM-24, MSSM-30, and MSSM-42 respectively), depending on the order kept in the expansion. We provide a Bayesian global fit to data of the MSSM-30 parameter set to show that this is manageable with current tools. We compare the MFV reductions to the 19-parameter pMSSM choice and show that the pMSSM is not contained as a subset. The MSSM-30 analysis favours a relatively lighter TeV-scale pseudoscalar Higgs boson and tan β∼10 with multi-TeV sparticles.
MFV reductions of MSSM parameter space
AbdusSalam, S. S.; Burgess, C. P.; Quevedo, F.
2015-02-01
The 100+ free parameters of the minimal supersymmetric standard model (MSSM) make it computationally difficult to compare systematically with data, motivating the study of specific parameter reductions such as the cMSSM and pMSSM. Here we instead study the reductions of parameter space implied by using minimal flavour violation (MFV) to organise the R-parity conserving MSSM, with a view towards systematically building in constraints on flavour-violating physics. Within this framework the space of parameters is reduced by expanding soft supersymmetry-breaking terms in powers of the Cabibbo angle, leading to a 24-, 30- or 42-parameter framework (which we call MSSM-24, MSSM-30, and MSSM-42 respectively), depending on the order kept in the expansion. We provide a Bayesian global fit to data of the MSSM-30 parameter set to show that this is manageable with current tools. We compare the MFV reductions to the 19-parameter pMSSM choice and show that the pMSSM is not contained as a subset. The MSSM-30 analysis favours a relatively lighter TeV-scale pseudoscalar Higgs boson and tan β ˜ 10 with multi-TeV sparticles.
MFV Reductions of MSSM Parameter Space
AbdusSalam, S.S.; Quevedo, F.
2015-01-01
The 100+ free parameters of the minimal supersymmetric standard model (MSSM) make it computationally difficult to compare systematically with data, motivating the study of specific parameter reductions such as the cMSSM and pMSSM. Here we instead study the reductions of parameter space implied by using minimal flavour violation (MFV) to organise the R-parity conserving MSSM, with a view towards systematically building in constraints on flavour-violating physics. Within this framework the space of parameters is reduced by expanding soft supersymmetry-breaking terms in powers of the Cabibbo angle, leading to a 24-, 30- or 42-parameter framework (which we call MSSM-24, MSSM-30, and MSSM-42 respectively), depending on the order kept in the expansion. We provide a Bayesian global fit to data of the MSSM-30 parameter set to show that this is manageable with current tools. We compare the MFV reductions to the 19-parameter pMSSM choice and show that the pMSSM is not contained as a subset. The MSSM-30 analysis favours...
Naturally light fermions from dimensional reduction
Bietenholz, W.; Gfeller, A.; Wiese, U.-J.
2004-03-01
We consider the 3-d Gross-Neveu model in the broken phase and construct a stable brane world by means of a domain wall and an anti-wall. Fermions of opposite chirality are localized on the walls and coupled through the 3-d bulk. At large wall separation β the 2-d correlation length diverges exponentially, hence a 2-d observer cannot distinguish this situation from a 2-d space-time. The 3-d 4-fermion coupling and β fix the effective 2-d coupling such that the asymptotic freedom of the 2-d model arises. This mechanism provides criticality without fine tuning.
Symmetries, dimensional reduction, and topological quantum order
Nussinov, Zohar; Ortiz, Gerardo
2009-12-01
We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. All known examples of Topological Quantum Order display Gauge-Like Symmetries. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and, most importantly, show the insufficiency of the energy spectrum, (recently defined) entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. Duality mappings illustrate that not withstanding the existence of spectral gaps, thermal fluctuations may impose restrictions on suggested topological quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer new systems with Topological Quantum Order.
A Global Geometric Framework for Nonlinear Dimensionality Reduction
Tenenbaum, Joshua B.; de Silva, Vin; Langford, John C.
2000-12-01
Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs-30,000 auditory nerve fibers or 106 optic nerve fibers-a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.
3-dimensional interactive space (3DIS)
International Nuclear Information System (INIS)
Veitch, S.; Veitch, J.; West, S.J.
1991-01-01
This paper reports on the 3DIS security system which uses standard CCTV cameras to create 3-Dimensional detection zones around valuable assets within protected areas. An intrusion into a zone changes light values and triggers an alarm that is annunciated, while images from multiple cameras are recorded. 3DIS lowers nuisance alarm rates and provides superior automated surveillance capability. Performance is improved over 2-D systems because activity around, above or below the zone does to cause an alarm. Invisible 3-D zones protect assets as small as a pin or as large as a 747 jetliner. Detection zones are created by excising subspaces from the overlapping fields of view of two or more video cameras. Hundred of zones may co-exist, operating simultaneously. Intrusion into any 3-D zone will cause a coincidental change in light values, triggering an alarm specific to that space
Electron in three-dimensional momentum space
Bacchetta, Alessandro; Mantovani, Luca; Pasquini, Barbara
2016-01-01
We study the electron as a system composed of an electron and a photon, using lowest-order perturbation theory. We derive the leading-twist transverse-momentum-dependent distribution functions for both the electron and photon in the dressed electron, thereby offering a three-dimensional description of the dressed electron in momentum space. To obtain the distribution functions, we apply both the formalism of the light-front wave function overlap representation and the diagrammatic approach. We perform the calculations both in light-cone gauge and Feynman gauge, and we present a detailed discussion of the role of the Wilson lines to obtain gauge-independent results. We provide numerical results and plots for many of the computed distributions.
Non-linear dimensionality reduction of signaling networks
Ivakhno, Sergii; Armstrong, J Douglas
2007-01-01
Background Systems wide modeling and analysis of signaling networks is essential for understanding complex cellular behaviors, such as the biphasic responses to different combinations of cytokines and growth factors. For example, tumor necrosis factor (TNF) can act as a proapoptotic or prosurvival factor depending on its concentration, the current state of signaling network and the presence of other cytokines. To understand combinatorial regulation in such systems, new computational approaches are required that can take into account non-linear interactions in signaling networks and provide tools for clustering, visualization and predictive modeling. Results Here we extended and applied an unsupervised non-linear dimensionality reduction approach, Isomap, to find clusters of similar treatment conditions in two cell signaling networks: (I) apoptosis signaling network in human epithelial cancer cells treated with different combinations of TNF, epidermal growth factor (EGF) and insulin and (II) combination of signal transduction pathways stimulated by 21 different ligands based on AfCS double ligand screen data. For the analysis of the apoptosis signaling network we used the Cytokine compendium dataset where activity and concentration of 19 intracellular signaling molecules were measured to characterise apoptotic response to TNF, EGF and insulin. By projecting the original 19-dimensional space of intracellular signals into a low-dimensional space, Isomap was able to reconstruct clusters corresponding to different cytokine treatments that were identified with graph-based clustering. In comparison, Principal Component Analysis (PCA) and Partial Least Squares – Discriminant analysis (PLS-DA) were unable to find biologically meaningful clusters. We also showed that by using Isomap components for supervised classification with k-nearest neighbor (k-NN) and quadratic discriminant analysis (QDA), apoptosis intensity can be predicted for different combinations of TNF, EGF
Non-linear dimensionality reduction of signaling networks
Directory of Open Access Journals (Sweden)
Ivakhno Sergii
2007-06-01
Full Text Available Abstract Background Systems wide modeling and analysis of signaling networks is essential for understanding complex cellular behaviors, such as the biphasic responses to different combinations of cytokines and growth factors. For example, tumor necrosis factor (TNF can act as a proapoptotic or prosurvival factor depending on its concentration, the current state of signaling network and the presence of other cytokines. To understand combinatorial regulation in such systems, new computational approaches are required that can take into account non-linear interactions in signaling networks and provide tools for clustering, visualization and predictive modeling. Results Here we extended and applied an unsupervised non-linear dimensionality reduction approach, Isomap, to find clusters of similar treatment conditions in two cell signaling networks: (I apoptosis signaling network in human epithelial cancer cells treated with different combinations of TNF, epidermal growth factor (EGF and insulin and (II combination of signal transduction pathways stimulated by 21 different ligands based on AfCS double ligand screen data. For the analysis of the apoptosis signaling network we used the Cytokine compendium dataset where activity and concentration of 19 intracellular signaling molecules were measured to characterise apoptotic response to TNF, EGF and insulin. By projecting the original 19-dimensional space of intracellular signals into a low-dimensional space, Isomap was able to reconstruct clusters corresponding to different cytokine treatments that were identified with graph-based clustering. In comparison, Principal Component Analysis (PCA and Partial Least Squares – Discriminant analysis (PLS-DA were unable to find biologically meaningful clusters. We also showed that by using Isomap components for supervised classification with k-nearest neighbor (k-NN and quadratic discriminant analysis (QDA, apoptosis intensity can be predicted for different
Non-linear dimensionality reduction of signaling networks.
Ivakhno, Sergii; Armstrong, J Douglas
2007-06-08
Systems wide modeling and analysis of signaling networks is essential for understanding complex cellular behaviors, such as the biphasic responses to different combinations of cytokines and growth factors. For example, tumor necrosis factor (TNF) can act as a proapoptotic or prosurvival factor depending on its concentration, the current state of signaling network and the presence of other cytokines. To understand combinatorial regulation in such systems, new computational approaches are required that can take into account non-linear interactions in signaling networks and provide tools for clustering, visualization and predictive modeling. Here we extended and applied an unsupervised non-linear dimensionality reduction approach, Isomap, to find clusters of similar treatment conditions in two cell signaling networks: (I) apoptosis signaling network in human epithelial cancer cells treated with different combinations of TNF, epidermal growth factor (EGF) and insulin and (II) combination of signal transduction pathways stimulated by 21 different ligands based on AfCS double ligand screen data. For the analysis of the apoptosis signaling network we used the Cytokine compendium dataset where activity and concentration of 19 intracellular signaling molecules were measured to characterise apoptotic response to TNF, EGF and insulin. By projecting the original 19-dimensional space of intracellular signals into a low-dimensional space, Isomap was able to reconstruct clusters corresponding to different cytokine treatments that were identified with graph-based clustering. In comparison, Principal Component Analysis (PCA) and Partial Least Squares - Discriminant analysis (PLS-DA) were unable to find biologically meaningful clusters. We also showed that by using Isomap components for supervised classification with k-nearest neighbor (k-NN) and quadratic discriminant analysis (QDA), apoptosis intensity can be predicted for different combinations of TNF, EGF and insulin. Prediction
Dimensionality reduction of hyperspectral images using kernel ICA
Khan, Asif; Kim, Intaek; Kong, Seong G.
2009-05-01
Computational burden due to high dimensionality of Hyperspectral images is an obstacle in efficient analysis and processing of Hyperspectral images. In this paper, we use Kernel Independent Component Analysis (KICA) for dimensionality reduction of Hyperspectraql images based on band selection. Commonly used ICA and PCA based dimensionality reduction methods do not consider non linear transformations and assumes that data has non-gaussian distribution. When the relation of source signals (pure materials) and observed Hyperspectral images is nonlinear then these methods drop a lot of information during dimensionality reduction process. Recent research shows that kernel-based methods are effective in nonlinear transformations. KICA is robust technique of blind source separation and can even work on near-gaussina data. We use Kernel Independent Component Analysis (KICA) for the selection of minimum number of bands that contain maximum information for detection in Hyperspectral images. The reduction of bands is basd on the evaluation of weight matrix generated by KICA. From the selected lower number of bands, we generate a new spectral image with reduced dimension and use it for hyperspectral image analysis. We use this technique as preprocessing step in detection and classification of poultry skin tumors. The hyperspectral iamge samples of chicken tumors used contain 65 spectral bands of fluorescence in the visible region of the spectrum. Experimental results show that KICA based band selection has high accuracy than that of fastICA based band selection for dimensionality reduction and analysis for Hyperspectral images.
Metric dimensional reduction at singularities with implications to Quantum Gravity
Stoica, Ovidiu Cristinel
2014-08-01
A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimensional reduction would still be desirable. The main contribution of this article is to show that dimensional reduction effects are due to General Relativity at singularities, and do not need to be postulated ad-hoc. Recent advances in understanding the geometry of singularities do not require modification of General Relativity, being just non-singular extensions of its mathematics to the limit cases. They turn out to work fine for some known types of cosmological singularities (black holes and FLRW Big-Bang), allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities. One consequence of this mathematical approach to the singularities in General Relativity is a special, (geo)metric type of dimensional reduction: at singularities, the metric tensor becomes degenerate in certain spacetime directions, and some properties of the fields become independent of those directions. Effectively, it is like one or more dimensions of spacetime just vanish at singularities. This suggests that it is worth exploring the possibility that the geometry of singularities leads naturally to the spontaneous dimensional reduction needed by Quantum Gravity.
Mappings with closed range and finite dimensional linear spaces
International Nuclear Information System (INIS)
Iyahen, S.O.
1984-09-01
This paper looks at two settings, each of continuous linear mappings of linear topological spaces. In one setting, the domain space is fixed while the range space varies over a class of linear topological spaces. In the second setting, the range space is fixed while the domain space similarly varies. The interest is in when the requirement that the mappings have a closed range implies that the domain or range space is finite dimensional. Positive results are obtained for metrizable spaces. (author)
Regularized Embedded Multiple Kernel Dimensionality Reduction for Mine Signal Processing
Li, Shuang; Liu, Bing; Zhang, Chen
2016-01-01
Traditional multiple kernel dimensionality reduction models are generally based on graph embedding and manifold assumption. But such assumption might be invalid for some high-dimensional or sparse data due to the curse of dimensionality, which has a negative influence on the performance of multiple kernel learning. In addition, some models might be ill-posed if the rank of matrices in their objective functions was not high enough. To address these issues, we extend the traditional graph embedding framework and propose a novel regularized embedded multiple kernel dimensionality reduction method. Different from the conventional convex relaxation technique, the proposed algorithm directly takes advantage of a binary search and an alternative optimization scheme to obtain optimal solutions efficiently. The experimental results demonstrate the effectiveness of the proposed method for supervised, unsupervised, and semisupervised scenarios. PMID:27247562
Regularized Embedded Multiple Kernel Dimensionality Reduction for Mine Signal Processing.
Li, Shuang; Liu, Bing; Zhang, Chen
2016-01-01
Traditional multiple kernel dimensionality reduction models are generally based on graph embedding and manifold assumption. But such assumption might be invalid for some high-dimensional or sparse data due to the curse of dimensionality, which has a negative influence on the performance of multiple kernel learning. In addition, some models might be ill-posed if the rank of matrices in their objective functions was not high enough. To address these issues, we extend the traditional graph embedding framework and propose a novel regularized embedded multiple kernel dimensionality reduction method. Different from the conventional convex relaxation technique, the proposed algorithm directly takes advantage of a binary search and an alternative optimization scheme to obtain optimal solutions efficiently. The experimental results demonstrate the effectiveness of the proposed method for supervised, unsupervised, and semisupervised scenarios.
Regularized Embedded Multiple Kernel Dimensionality Reduction for Mine Signal Processing
Directory of Open Access Journals (Sweden)
Shuang Li
2016-01-01
Full Text Available Traditional multiple kernel dimensionality reduction models are generally based on graph embedding and manifold assumption. But such assumption might be invalid for some high-dimensional or sparse data due to the curse of dimensionality, which has a negative influence on the performance of multiple kernel learning. In addition, some models might be ill-posed if the rank of matrices in their objective functions was not high enough. To address these issues, we extend the traditional graph embedding framework and propose a novel regularized embedded multiple kernel dimensionality reduction method. Different from the conventional convex relaxation technique, the proposed algorithm directly takes advantage of a binary search and an alternative optimization scheme to obtain optimal solutions efficiently. The experimental results demonstrate the effectiveness of the proposed method for supervised, unsupervised, and semisupervised scenarios.
Dimensionality reduction of RKHS model parameters.
Taouali, Okba; Elaissi, Ilyes; Messaoud, Hassani
2015-07-01
This paper proposes a new method to reduce the parameter number of models developed in the Reproducing Kernel Hilbert Space (RKHS). In fact, this number is equal to the number of observations used in the learning phase which is assumed to be high. The proposed method entitled Reduced Kernel Partial Least Square (RKPLS) consists on approximating the retained latent components determined using the Kernel Partial Least Square (KPLS) method by their closest observation vectors. The paper proposes the design and the comparative study of the proposed RKPLS method and the Support Vector Machines on Regression (SVR) technique. The proposed method is applied to identify a nonlinear Process Trainer PT326 which is a physical process available in our laboratory. Moreover as a thermal process with large time response may help record easily effective observations which contribute to model identification. Compared to the SVR technique, the results from the proposed RKPLS method are satisfactory. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Applications of dimensional reduction to electroweak and QCD matter
Vepsalainen, M.
2007-09-01
This paper is a slightly modified version of the introductory part of a doctoral dissertation also containing the articles hep-ph/0311268, hep-ph/0510375, hep-ph/0512177 and hep-ph/0701250. The thesis discusses effective field theory methods, in particular dimensional reduction, in the context of finite temperature field theory. We first briefly review the formalism of thermal field theory and show how dimensional reduction emerges as the high-temperature limit for static quantities. Then we apply dimensional reduction to two distinct problems, the pressure of electroweak theory and the screening masses of mesonic operators in hot QCD, and point out the similarities. We summarize the results and discuss their validity, while leaving all details to original research articles.
Exploring the CAESAR database using dimensionality reduction techniques
Mendoza-Schrock, Olga; Raymer, Michael L.
2012-06-01
The Civilian American and European Surface Anthropometry Resource (CAESAR) database containing over 40 anthropometric measurements on over 4000 humans has been extensively explored for pattern recognition and classification purposes using the raw, original data [1-4]. However, some of the anthropometric variables would be impossible to collect in an uncontrolled environment. Here, we explore the use of dimensionality reduction methods in concert with a variety of classification algorithms for gender classification using only those variables that are readily observable in an uncontrolled environment. Several dimensionality reduction techniques are employed to learn the underlining structure of the data. These techniques include linear projections such as the classical Principal Components Analysis (PCA) and non-linear (manifold learning) techniques, such as Diffusion Maps and the Isomap technique. This paper briefly describes all three techniques, and compares three different classifiers, Naïve Bayes, Adaboost, and Support Vector Machines (SVM), for gender classification in conjunction with each of these three dimensionality reduction approaches.
A Tannakian approach to dimensional reduction of principal bundles
Álvarez-Cónsul, Luis; Biswas, Indranil; García-Prada, Oscar
2017-08-01
Let P be a parabolic subgroup of a connected simply connected complex semisimple Lie group G. Given a compact Kähler manifold X, the dimensional reduction of G-equivariant holomorphic vector bundles over X × G / P was carried out in Álvarez-Cónsul and García-Prada (2003). This raises the question of dimensional reduction of holomorphic principal bundles over X × G / P. The method of Álvarez-Cónsul and García-Prada (2003) is special to vector bundles; it does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of G-equivariant principal bundles over X × G / P, and to establish a Hitchin-Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that X is a complex projective manifold.
The spatial string tension and dimensional reduction in QCD
Cheng, M.; Datta, S.; van der Heide, J.; Huebner, K.; Karsch, F.; Kaczmarek, O.; Laermann, E.; Liddle, J.; Mawhinney, R. D.; Miao, C.; Petreczky, P.; Petrov, K.; Schmidt, C.; Soeldner, W.; Umeda, T.
2008-08-01
We calculate the spatial string tension in (2+1) flavor QCD with physical strange quark mass and almost physical light quark masses using lattices with temporal extent Nτ=4, 6 and 8. We compare our results on the spatial string tension with predictions of dimensionally reduced QCD. This suggests that also in the presence of light dynamical quarks dimensional reduction works well down to temperatures 1.5Tc.
The spatial string tension and dimensional reduction in QCD
Rbc-Bielefeld Collaboration; Laermann, E.; Liddle, J.; RBC-Bielefeld Collaboration
2009-04-01
The spatial string tension is calculated in 2+1 flavor QCD with a physical strange quark mass and almost physical light quark masses on lattices with N=4,6 and 8. The results are compared with predictions of dimensionally reduced QCD. They suggest that dimensional reduction works also in the presence of light dynamical quarks down to temperatures of about 1.5T.
Local Dimensionality Reduction for Non-Parametric Regression
Hoffmann, Heiko; Schaal, Stefan; Vijayakumar, Sethu
2009-01-01
Locally-weighted regression is a computationally-efficient technique for non-linear regression. However, for high-dimensional data, this technique becomes numerically brittle and computationally too expensive if many local models need to be maintained simultaneously. Thus, local linear dimensionality reduction combined with locally-weighted regression seems to be a promising solution. In this context, we review linear dimensionalityreduction methods, compare their performance o...
Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity?
Carlip, Steven
2010-02-01
Several lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional. I will summarize the evidence for such ``spontaneous dimensional reduction,'' and suggest an additional argument coming from the strong-coupling limit of the Wheeler-DeWitt equation. If this description proves to be correct, it suggests an interesting relationship between small-scale quantum spacetime and the behavior of cosmologies near an asymptotically silent singularity. )
An R package implementation of multifactor dimensionality reduction
Winham, Stacey J; Motsinger-Reif, Alison A
2011-01-01
Abstract Background A breadth of high-dimensional data is now available with unprecedented numbers of genetic markers and data-mining approaches to variable selection are increasingly being utilized to uncover associations, including potential gene-gene and gene-environment interactions. One of the most commonly used data-mining methods for case-control data is Multifactor Dimensionality Reduction (MDR), which has displayed success in both simulations and real data applications. Additional so...
Dimensionality Reduction of Laplacian Embedding for 3D Mesh Reconstruction
Mardhiyah, I.; Madenda, S.; Salim, R. A.; Wiryana, I. M.
2016-06-01
Laplacian eigenbases are the important thing that we have to process from 3D mesh information. The information of geometric 3D mesh are include vertices locations and the connectivity of graph. Due to spectral analysis, geometric 3D mesh for large and sparse graphs with thousands of vertices is not practical to compute all the eigenvalues and eigenvector. Because of that, in this paper we discuss how to build 3D mesh reconstruction by reducing dimensionality on null eigenvalue but retain the corresponding eigenvector of Laplacian Embedding to simplify mesh processing. The result of reducing information should have to retained the connectivity of graph. The advantages of dimensionality reduction is for computational eficiency and problem simplification. Laplacian eigenbases is the point of dimensionality reduction for 3D mesh reconstruction. In this paper, we show how to reconstruct geometric 3D mesh after approximation step of 3D mesh by dimensionality reduction. Dimensionality reduction shown by Laplacian Embedding matrix. Furthermore, the effectiveness of 3D mesh reconstruction method will evaluated by geometric error, differential error, and final error. Numerical approximation error of our result are small and low complexity of computational.
Propagators and dimensional reduction of hot SU(2) gauge theory
Cucchieri, A.; Karsch, F.; Petreczky, P.
2001-08-01
We investigate the large distance behavior of the electric and magnetic propagators of hot SU(2) gauge theory in different gauges using lattice simulations of the full four-dimensional (4D) theory and the effective, dimensionally reduced, 3D theory. A comparison of the 3D and 4D propagators suggests that dimensional reduction works surprisingly well down to the temperature T=2Tc. Within statistical uncertainty the electric screening mass is found to be gauge independent. The magnetic propagator, on the other hand, exhibits a complicated gauge dependent structure at low momentum.
Dimensional reduction in 6D standing waves braneworld
Sakhelashvili, Otari
2015-11-01
We found cosmological solution of the 6D standing wave braneworld model generated by gravity coupled to a massless scalar phantom-like field. By obtaining a full exact solution of the model, we found a novel dynamical mechanism in which the anisotropic nature of the primordial metric gives rise to expansion of three spatial brane dimensions and affectively reduction of other spatial directions. This dynamical mechanism can be relevant for dimensional reduction in string and other higher-dimensional theories in the attempt of getting a 4D isotropic expanding spacetime.
Ni, Shengqiao; Lv, Jiancheng; Cheng, Zhehao; Li, Mao
2015-01-01
This paper presents improvements to the conventional Topology Representing Network to build more appropriate topology relationships. Based on this improved Topology Representing Network, we propose a novel method for online dimensionality reduction that integrates the improved Topology Representing Network and Radial Basis Function Network. This method can find meaningful low-dimensional feature structures embedded in high-dimensional original data space, process nonlinear embedded manifolds, and map the new data online. Furthermore, this method can deal with large datasets for the benefit of improved Topology Representing Network. Experiments illustrate the effectiveness of the proposed method. PMID:26161960
Directory of Open Access Journals (Sweden)
Shengqiao Ni
Full Text Available This paper presents improvements to the conventional Topology Representing Network to build more appropriate topology relationships. Based on this improved Topology Representing Network, we propose a novel method for online dimensionality reduction that integrates the improved Topology Representing Network and Radial Basis Function Network. This method can find meaningful low-dimensional feature structures embedded in high-dimensional original data space, process nonlinear embedded manifolds, and map the new data online. Furthermore, this method can deal with large datasets for the benefit of improved Topology Representing Network. Experiments illustrate the effectiveness of the proposed method.
Ni, Shengqiao; Lv, Jiancheng; Cheng, Zhehao; Li, Mao
2015-01-01
This paper presents improvements to the conventional Topology Representing Network to build more appropriate topology relationships. Based on this improved Topology Representing Network, we propose a novel method for online dimensionality reduction that integrates the improved Topology Representing Network and Radial Basis Function Network. This method can find meaningful low-dimensional feature structures embedded in high-dimensional original data space, process nonlinear embedded manifolds, and map the new data online. Furthermore, this method can deal with large datasets for the benefit of improved Topology Representing Network. Experiments illustrate the effectiveness of the proposed method.
Rigorous Model Reduction for a Damped-Forced Nonlinear Beam Model: An Infinite-Dimensional Analysis
Kogelbauer, Florian; Haller, George
2018-01-01
We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction in the governing PDE to SSMs provides an explicit low-dimensional model which captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathematically self-consistent modeling of damping in PDEs describing structural vibrations.
A finite-dimensional reduction method for slightly supercritical elliptic problems
Directory of Open Access Journals (Sweden)
Riccardo Molle
2004-01-01
Full Text Available We describe a finite-dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite-dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.
Dimensional reductions of M-theory S-branes to string theory S-branes
Roy, Shibaji
2003-12-01
We study both the direct and the double-dimensional reduction of space-like branes of M-theory and point out some peculiarities in the process unlike their time-like counterpart. In particular, we show how starting from SM2 and SM5-brane solutions we can obtain SD2 and SNS5-brane as well as SNS1 and SD4-brane solutions of string theory by direct and double-dimensional reductions, respectively. In the former case we need to use delocalized SM-brane solutions, whereas in the latter case we need to use anisotropic SM-brane solutions in the directions which are compactified.
Nussinov, Zohar; Batista, Cristian D.; Fradkin, Eduardo
2006-09-01
We discuss symmetries intermediate between global and local and formalize the notion of dimensional reduction adduced from such symmetries. We apply this generalization to several systems including liquid crystalline phases of Quantum Hall systems, transition metal orbital systems, frustrated spin systems, (p+ip) superconducting arrays, and sliding Luttinger liquids. By considering space-time reflection symmetries, we illustrate that several of these systems are dual to each other. In some systems exhibiting these symmetries, low temperature local orders emerge by an "order out of disorder" effect while in other systems, the dimensional reduction precludes standard orders yet allows for multiparticle orders (including those of a topological nature).
Anisotropic inflation in a 5D standing wave braneworld and effective dimensional reduction
Gogberashvili, Merab; Herrera-Aguilar, Alfredo; Malagón-Morejón, Dagoberto; Mora-Luna, Refugio Rigel
2013-10-01
We investigate a cosmological solution within the framework of a 5D standing wave braneworld model generated by gravity coupled to a massless scalar phantom-like field. By obtaining a full exact solution of the model we found a novel dynamical mechanism in which the anisotropic nature of the primordial metric gives rise to (i) inflation along certain spatial dimensions, and (ii) deflation and a shrinking reduction of the number of spatial dimensions along other directions. This dynamical mechanism can be relevant for dimensional reduction in string and other higher-dimensional theories in the attempt of getting a 4D isotropic expanding space-time.
Dimensionality reduction for uncertainty quantification of nuclear engineering models.
Energy Technology Data Exchange (ETDEWEB)
Roderick, O.; Wang, Z.; Anitescu, M. (Mathematics and Computer Science)
2011-01-01
The task of uncertainty quantification consists of relating the available information on uncertainties in the model setup to the resulting variation in the outputs of the model. Uncertainty quantification plays an important role in complex simulation models of nuclear engineering, where better understanding of uncertainty results in greater confidence in the model and in the improved safety and efficiency of engineering projects. In our previous work, we have shown that the effect of uncertainty can be approximated by polynomial regression with derivatives (PRD): a hybrid regression method that uses first-order derivatives of the model output as additional fitting conditions for a polynomial expansion. Numerical experiments have demonstrated the advantage of this approach over classical methods of uncertainty analysis: in precision, computational efficiency, or both. To obtain derivatives, we used automatic differentiation (AD) on the simulation code; hand-coded derivatives are acceptable for simpler models. We now present improvements on the method. We use a tuned version of the method of snapshots, a technique based on proper orthogonal decomposition (POD), to set up the reduced order representation of essential information on uncertainty in the model inputs. The automatically obtained sensitivity information is required to set up the method. Dimensionality reduction in combination with PRD allows analysis on a larger dimension of the uncertainty space (>100), at modest computational cost.
Wilson loop and dimensional reduction in noncommutative gauge theories
Lee, Sunggeun; Sin, Sang-Jin
2001-10-01
Using the anti-de Sitter (AdS) conformal field theory correspondence we study the UV behavior of Wilson loops in various noncommutative gauge theories. We get an area law in most cases and try to identify its origin. In the D3 case, we may identify the the origin as the D1 dominance over the D3: as we go to the boundary of AdS space, the effect of the flux of the D3 charge is highly suppressed, while the flux due to the D1 charge is enhanced. So near the boundary the theory is more like a theory on a D1-brane than that on a D3-brane. This phenomena is closely related to dimensional reduction due to the strong magnetic field in the charged particle in the magnetic field. The linear potential is not due to the confinement by IR effect but is the analogue of Coulomb's potential in 1+1 dimensions.
Dimensional reduction, avalanches and disorder in artificial kagome spin ice
Hugli, Remo V.; Duff, Gerard; Braun, Hans-Benjamin
2012-02-01
In collaboration with an experimental team at the Swiss Light Source we have recently demonstrated that emergent monopoles and associated Dirac strings can directly be observed in real space via x-ray circular dichroism in a kagome lattice geometry. Here we build on the fact that the experimental results are in excellent agreement with MC simulations of a lattice of point dipoles with disorder realized in the form of random switching fields. We demonstrate that within a large range of physical parameters such as interdipolar coupling and randomness, magnetization reversal proceeds via a novel 1D avalanche behaviour whose hallmark is an exponential avalanche size distribution. After presenting simple arguments for the origin of such dimensional reduction we demonstrate that such 1D avalanche behavior also occurs in a model where the dipoles are stretched into magnetic charge dumbbells which provides a more realistic model for nanolithographic islands. Finally we demonstrate how a judicious design of the island anisotropy can be used to achieve controlled switching and avalanche propagation which paves the way for spintronic applications
N=6 gauged supergravities from generalized dimensional reduction
Villadoro, Giovanni
2004-01-01
We construct new N=6 gauged supergravities in four and five dimensions using generalized dimensional reduction. Supersymmetry is spontaneously broken to N=4,2,0 with vanishing cosmological constant. We discuss the gaugings of the broken phases, the scalar geometries and the spectrum. Generalized orbifold reduction is also considered and an N=3 no-scale model is obtained with three independent mass parameters.
N=6 gauged supergravities from generalized dimensional reduction
Villadoro, Giovanni
2004-11-01
We construct new N=6 gauged supergravities in four and five dimensions using generalized dimensional reduction. Supersymmetry is spontaneously broken to N=4,2,0 with vanishing cosmological constant. We discuss the gaugings of the broken phases, the scalar geometries and the spectrum. Generalized orbifold reduction is also considered and an N=3 no-scale model is obtained with three independent mass parameters.
Pseudospectral reduction of incompressible two-dimensional turbulence
Bowman, John C.; Roberts, Malcolm
2012-05-01
Spectral reduction was originally formulated entirely in the wavenumber domain as a coarse-grained wavenumber convolution in which bins of modes interact with enhanced coupling coefficients. A Liouville theorem leads to inviscid equipartition solutions when each bin contains the same number of modes. A pseudospectral implementation of spectral reduction which enjoys the efficiency of the fast Fourier transform is described. The model compares well with full pseudospectral simulations of the two-dimensional forced-dissipative energy and enstrophy cascades.
Enhancing Dimensionality Reduction Methods for Side-Channel Attacks
Cagli, Eleonora; Dumas, Cécile; Prouff, Emmanuel
2015-01-01
International audience; Advanced Side-Channel Analyses make use of dimensionality reduction techniques to reduce both the memory and timing complexity of the attacks. The most popular methods to effectuate such a reduction are the Principal Component Analysis (PCA) and the Linear Discrim-inant Analysis (LDA). They indeed lead to remarkable efficiency gains but their use in side-channel context also raised some issues. The PCA provides a set of vectors (the principal components) onto which pro...
Two kinds of finite-dimensional integrable reduction to the Harry-Dym hierarchy
Chen, Jinbing
2016-11-01
In this paper, two kinds of finite-dimensional integrable reduction are studied for the Harry-Dym (HD) hierarchy. From the nonlinearization of Lax pair, the HD hierarchy is reduced to a class of finite-dimensional Hamiltonian systems (FDHSs) in view of a Bargmann map and a set of Neumann type systems by a Neumann map, which separate temporal and spatial variables on the symplectic space (ℝ2N,ω2) and the tangent bundle of ellipsoid (TSN-1,ω2), respectively. It turns out that involutive solutions of the resulted finite-dimensional integrable systems (FDISs) directly give rise to finite parametric solutions of HD hierarchy through the Bargmann and Neumann maps. The finite-gap potential to the high-order stationary HD equation is obtained that cuts out a finite-dimensional invariant subspace for the HD flows. Finally, some comparisons of two kinds of integrable reductions are then discussed.
Charged fluid distribution in higher dimensional spheroidal space-time
Indian Academy of Sciences (India)
A general solution of Einstein field equations corresponding to a charged fluid distribution on the background of higher dimensional spheroidal space-time is obtained. The solution generates several known solutions for superdense star having spheroidal space-time geometry.
International Nuclear Information System (INIS)
Tripathy, Rohit; Bilionis, Ilias; Gonzalez, Marcial
2016-01-01
Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the
Energy Technology Data Exchange (ETDEWEB)
Tripathy, Rohit, E-mail: rtripath@purdue.edu; Bilionis, Ilias, E-mail: ibilion@purdue.edu; Gonzalez, Marcial, E-mail: marcial-gonzalez@purdue.edu
2016-09-15
Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the
Tripathy, Rohit; Bilionis, Ilias; Gonzalez, Marcial
2016-09-01
Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the
Avalanches and Dimensional Reduction Breakdown in the Critical Behavior of Disordered Systems
Tarjus, Gilles; Baczyk, Maxime; Tissier, Matthieu
2013-03-01
We investigate the connection between a formal property of the critical behavior of several disordered systems, known as “dimensional reduction,” and the presence in these systems at zero temperature of collective events known as “avalanches.” Avalanches generically produce nonanalyticities in the functional dependence of the cumulants of the renormalized disorder. We show that this leads to a breakdown of the dimensional reduction predictions if and only if the fractal dimension characterizing the scaling properties of the avalanches is exactly equal to the difference between the dimension of space and the scaling dimension of the primary field. This is proven by combining scaling theory and the functional renormalization group. We therefore clarify the puzzle of why dimensional reduction remains valid in random field systems above a nontrivial dimension (but fails below), always applies to the statistics of branched polymer, and is always wrong in elastic models of interfaces in a random environment.
Dimensionality Reduction on Multi-Dimensional Transfer Functions for Multi-Channel Volume Data Sets
Kim, Han Suk; Schulze, Jürgen P.; Cone, Angela C.; Sosinsky, Gina E.; Martone, Maryann E.
2011-01-01
The design of transfer functions for volume rendering is a non-trivial task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel, which requires multi-dimensional transfer functions. In this paper, we propose a new method for multi-dimensional transfer function design. Our new method provides a framework to combine multiple computational approaches and pushes the boundary of gradient-based multi-dimensional transfer functions to multiple channels, while keeping the dimensionality of transfer functions at a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. Applying recently developed nonlinear dimensionality reduction algorithms reduces the high-dimensional data of the domain. In this paper, we use Isomap and Locally Linear Embedding as well as a traditional algorithm, Principle Component Analysis. Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. We demonstrate the effectiveness of our new dimensionality reduction algorithms with two volumetric confocal microscopy data sets. PMID:21841914
Dimensionality Reduction on Multi-Dimensional Transfer Functions for Multi-Channel Volume Data Sets.
Kim, Han Suk; Schulze, Jürgen P; Cone, Angela C; Sosinsky, Gina E; Martone, Maryann E
2010-09-21
The design of transfer functions for volume rendering is a non-trivial task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel, which requires multi-dimensional transfer functions. In this paper, we propose a new method for multi-dimensional transfer function design. Our new method provides a framework to combine multiple computational approaches and pushes the boundary of gradient-based multi-dimensional transfer functions to multiple channels, while keeping the dimensionality of transfer functions at a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. Applying recently developed nonlinear dimensionality reduction algorithms reduces the high-dimensional data of the domain. In this paper, we use Isomap and Locally Linear Embedding as well as a traditional algorithm, Principle Component Analysis. Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. We demonstrate the effectiveness of our new dimensionality reduction algorithms with two volumetric confocal microscopy data sets.
Multi-Channel Transfer Function with Dimensionality Reduction
Kim, Han Suk; Schulze, Jürgen P.; Cone, Angela C.; Sosinsky, Gina E.; Martone, Maryann E.
2010-01-01
The design of transfer functions for volume rendering is a difficult task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel. In this paper, we propose a new method for transfer function design. Our new method provides a framework to combine multiple approaches and pushes the boundary of gradient-based transfer functions to multiple channels, while still keeping the dimensionality of transfer functions to a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. The high-dimensional data of the domain is reduced by applying recently developed nonlinear dimensionality reduction algorithms. In this paper, we used Isomap as well as a traditional algorithm, Principle Component Analysis (PCA). Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. In this publication we report on the impact of the dimensionality reduction algorithms on transfer function design for confocal microscopy data. PMID:20582228
Multi-Channel Transfer Function with Dimensionality Reduction.
Kim, Han Suk; Schulze, Jürgen P; Cone, Angela C; Sosinsky, Gina E; Martone, Maryann E
2010-01-18
The design of transfer functions for volume rendering is a difficult task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel. In this paper, we propose a new method for transfer function design. Our new method provides a framework to combine multiple approaches and pushes the boundary of gradient-based transfer functions to multiple channels, while still keeping the dimensionality of transfer functions to a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. The high-dimensional data of the domain is reduced by applying recently developed nonlinear dimensionality reduction algorithms. In this paper, we used Isomap as well as a traditional algorithm, Principle Component Analysis (PCA). Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. In this publication we report on the impact of the dimensionality reduction algorithms on transfer function design for confocal microscopy data.
Dimensionality Reduction and Uncertainty Quantification for Inverse Problems
van Leeuwen, Tristan
2015-01-01
Many inverse problems in science and engineering involve multi-experiment data and thus require a large number of forward simulations. Dimensionality reduction techniques aim at reducing the number of forward solves by (randomly) subsampling the data. In the special case of non-linear least-squares
Pole solution in six dimensions as a dimensional reduction model
Ichinose, Shoichi
2002-01-01
A solution with the pole configuration in six dimensions is analyzed. It is a dimensional reduction model of Randall-Sundrum type. The soliton configuration is induced by the bulk Higgs mechanism. The boundary condition is systematically solved up to the 6th order. The Riemann curvature is finite everywhere.
Supervised nonlinear dimensionality reduction for visualization and classification.
Geng, Xin; Zhan, De-Chuan; Zhou, Zhi-Hua
2005-12-01
When performing visualization and classification, people often confront the problem of dimensionality reduction. Isomap is one of the most promising nonlinear dimensionality reduction techniques. However, when Isomap is applied to real-world data, it shows some limitations, such as being sensitive to noise. In this paper, an improved version of Isomap, namely S-Isomap, is proposed. S-Isomap utilizes class information to guide the procedure of nonlinear dimensionality reduction. Such a kind of procedure is called supervised nonlinear dimensionality reduction. In S-Isomap, the neighborhood graph of the input data is constructed according to a certain kind of dissimilarity between data points, which is specially designed to integrate the class information. The dissimilarity has several good properties which help to discover the true neighborhood of the data and, thus, makes S-Isomap a robust technique for both visualization and classification, especially for real-world problems. In the visualization experiments, S-Isomap is compared with Isomap, LLE, and WeightedIso. The results show that S-Isomap performs the best. In the classification experiments, S-Isomap is used as a preprocess of classification and compared with Isomap, WeightedIso, as well as some other well-established classification methods, including the K-nearest neighbor classifier, BP neural network, J4.8 decision tree, and SVM. The results reveal that S-Isomap excels compared to Isomap and WeightedIso in classification, and it is highly competitive with those well-known classification methods.
Some infinite dimensional representations of reductive groups with Frobenius maps
Xi, NanHua
2014-06-01
In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.
DRACULA: Dimensionality Reduction And Clustering for Unsupervised Learning in Astronomy
Aguena, Michel; Busti, Vinicius C.; Camacho, Hugo; Sasdelli, Michele; Ishida, Emille E. O.; Vilalta, Ricardo; Trindade, Arlindo M. M.; Gieseke, Fabien; de Souza, Rafael S.; Fantaye, Yabebal T.; Mazzali, Paolo A.
2015-12-01
DRACULA classifies objects using dimensionality reduction and clustering. The code has an easy interface and can be applied to separate several types of objects. It is based on tools developed in scikit-learn, with some usage requiring also the H2O package.
Dimensionality Reduction for Sensorimotor Learning in Mobile Robotics
Lee, Daniel D.
2002-12-01
Mobile robotic systems with a wide variety of sensors, actuators, and onboard high-speed processors are commercially and readily available. The information processing capabilities of these system presently lack the robustness and sophistication of biological systems. One challenge is that the high-dimensional input signals from the sensors need to be converted into a smaller number of perceptually relevant features. This dimensionality reduction can be performed on static signals such as a single image or on dynamic data such as a speech spectrogram. This proceedings discusses several different models for dimensionality reduction that differ only on the constraints on the variables and parameters of the models. In particular, nonnegativity constraints are shown to give rise to distributed yet sparse representations of both static and dynamic data.
Learning linear discriminant projections for dimensionality reduction of image descriptors.
Cai, Hongping; Mikolajczyk, Krystian; Matas, Jiri
2011-02-01
In this paper, we present Linear Discriminant Projections (LDP) for reducing dimensionality and improving discriminability of local image descriptors. We place LDP into the context of state-of-the-art discriminant projections and analyze its properties. LDP requires a large set of training data with point-to-point correspondence ground truth. We demonstrate that training data produced by a simulation of image transformations leads to nearly the same results as the real data with correspondence ground truth. This makes it possible to apply LDP as well as other discriminant projection approaches to the problems where the correspondence ground truth is not available, such as image categorization. We perform an extensive experimental evaluation on standard data sets in the context of image matching and categorization. We demonstrate that LDP enables significant dimensionality reduction of local descriptors and performance increases in different applications. The results improve upon the state-of-the-art recognition performance with simultaneous dimensionality reduction from 128 to 30.
Charged fluid distribution in higher dimensional spheroidal space-time
Indian Academy of Sciences (India)
Charged fluid distribution; higher dimensional space-time. PACS Nos 04.40.Dg; 04.20.-q; 04.20.Jb. 1. Introduction. Higher dimensional view of the world geometry suggests that the universe started in (4 + D)-dimensional phase with extra D dimensions either collapsing and stabi- lizing or remain at a size close to the Plank ...
Distance-preserving projection of high-dimensional data for nonlinear dimensionality reduction.
Yang, Li
2004-09-01
A distance-preserving method is presented to map high-dimensional data sequentially to low-dimensional space. It preserves exact distances of each data point to its nearest neighbor and to some other near neighbors. Intrinsic dimensionality of data is estimated by examining the preservation of interpoint distances. The method has no user-selectable parameter. It can successfully project data when the data points are spread among multiple clusters. Results of experiments show its usefulness in projecting high-dimensional data.
Two-Dilaton Theories in Two Dimensions from Dimensional Reduction
Grumiller, D.; Hofmann, D.; Kummer, W.
2001-06-01
Dimensional reduction of generalized gravity theories or string theories generically yields dilaton fields in the lower-dimensional effective theory. Thus at the level of D=4 theories and cosmology, many models contain more than just one scalar field (e.g., inflaton, Higgs, quintessence). Our present work is restricted to two-dimensional gravity theories with only two dilatons which nevertheless allow a large class of physical applications. The notions of factorizability, simplicity and conformal simplicity, Einstein form, and Jordan form are the basis of an adequate classification. We show that practically all physically motivated models belong either to the class of factorizable simple theories (e.g., dimensionally reduced gravity, bosonic string) or to factorizable conformally simple theories (e.g., spherically reduced scalar-tensor theories). For these theories a first order formulation is constructed straightforwardly. As a consequence an absolute conservation law can be established.
Two-dimensional black holes and non-commutative spaces
International Nuclear Information System (INIS)
Sadeghi, J.
2008-01-01
We study the effects of non-commutative spaces on two-dimensional black hole. The event horizon of two-dimensional black hole is obtained in non-commutative space up to second order of perturbative calculations. A lower limit for the non-commutativity parameter is also obtained. The observer in that limit in contrast to commutative case see two horizon
Borsuk-Ulam theorem in infinite-dimensional Banach spaces
International Nuclear Information System (INIS)
Gel'man, B D
2002-01-01
The well-known classical Borsuk-Ulam theorem has a broad range of applications to various problems. Its generalization to infinite-dimensional spaces runs across substantial difficulties because its statement is essentially finite-dimensional. A result established in the paper is a natural generalization of the Borsuk-Ulam theorem to infinite-dimensional Banach spaces. Applications of this theorem to various problems are discussed
Multicriteria classification method for dimensionality reduction adapted to hyperspectral images
Khoder, Mahdi; Kashana, Serge; Khoder, Jihan; Younes, Rafic
2017-04-01
Due to the incredible growth of high dimensional datasets, we address the problem of unsupervised methods sensitive to undergoing different variations, such as noise degradation, and to preserving rare information. Therefore, researchers nowadays are forced to develop techniques to meet the needed requirements. In this work, we introduce a dimensionality reduction method that focuses on the multiobjectives of multiple images taken from multiple frequency bands, which form a hyperspectral image. The multicriteria classification algorithm technique compares and classifies these images based on multiple similarity criteria, which allows the selection of particular images from the whole set of images. The selected images are the ones chosen to represent the original set of data while respecting certain quality thresholds. Knowing that the number of images in a hyperspectral image signifies its dimension, choosing a smaller number of images to represent the data leads to dimensionality reduction. Also, results of tests of the developed algorithm on multiple hyperspectral image samples are shown. A comparative study later on will show the advantages of this technique compared to other common methods used in the field of dimensionality reduction.
Dimensional reduction, Seiberg-Witten map, and supersymmetry
International Nuclear Information System (INIS)
Saka, E. Ulas; Uelker, Kayhan
2007-01-01
It is argued that dimensional reduction of the Seiberg-Witten map for a gauge field induces Seiberg-Witten maps for the other noncommutative fields of a gauge invariant theory. We demonstrate this observation by dimensionally reducing the noncommutative N=1 super Yang-Mills (SYM) theory in 6 dimensions to obtain noncommutative N=2 SYM in 4 dimensions. We explicitly derive Seiberg-Witten maps of the component fields in 6 and 4 dimensions. Moreover, we give a general method to define the deformed supersymmetry transformations that leave the actions invariant after performing Seiberg-Witten maps
Superfluid hydrodynamics of polytropic gases: dimensional reduction and sound velocity
International Nuclear Information System (INIS)
Bellomo, N; Mazzarella, G; Salasnich, L
2014-01-01
Motivated by the fact that two-component confined fermionic gases in Bardeen–Cooper–Schrieffer–Bose–Einstein condensate (BCS–BEC) crossover can be described through an hydrodynamical approach, we study these systems—both in the cigar-shaped configuration and in the disc-shaped one—by using a polytropic Lagrangian density. We start from the Popov Lagrangian density and obtain, after a dimensional reduction process, the equations that control the dynamics of such systems. By solving these equations we study the sound velocity as a function of the density by analyzing how the dimensionality affects this velocity. (paper)
Dimensional reduction, Seiberg-Witten map, and supersymmetry
Saka, E. Ulaş; Ülker, Kayhan
2007-04-01
It is argued that dimensional reduction of the Seiberg-Witten map for a gauge field induces Seiberg-Witten maps for the other noncommutative fields of a gauge invariant theory. We demonstrate this observation by dimensionally reducing the noncommutative N=1 super Yang-Mills (SYM) theory in 6 dimensions to obtain noncommutative N=2 SYM in 4 dimensions. We explicitly derive Seiberg-Witten maps of the component fields in 6 and 4 dimensions. Moreover, we give a general method to define the deformed supersymmetry transformations that leave the actions invariant after performing Seiberg-Witten maps.
Superfluid hydrodynamics of polytropic gases: dimensional reduction and sound velocity
Bellomo, N.; Mazzarella, G.; Salasnich, L.
2014-03-01
Motivated by the fact that two-component confined fermionic gases in Bardeen-Cooper-Schrieffer-Bose-Einstein condensate (BCS-BEC) crossover can be described through an hydrodynamical approach, we study these systems—both in the cigar-shaped configuration and in the disc-shaped one—by using a polytropic Lagrangian density. We start from the Popov Lagrangian density and obtain, after a dimensional reduction process, the equations that control the dynamics of such systems. By solving these equations we study the sound velocity as a function of the density by analyzing how the dimensionality affects this velocity.
Consistent S2 Pauli reduction of six-dimensional chiral gauged Einstein-Maxwell supergravity
Gibbons, G. W.; Pope, C. N.
2004-10-01
Six-dimensional N=(1,0) Einstein-Maxwell gauged supergravity is known to admit a (Minkowski) 4× S2 vacuum solution with four-dimensional N=1 chiral supersymmetry. The massless sector comprises a supergravity multiplet, an SU(2) Yang-Mills vector multiplet, and a scalar multiplet. In this paper it is shown that, remarkably, the six-dimensional theory admits a fully consistent dimensional reduction on the 2-sphere, implying that all solutions of the four-dimensional N=1 supergravity can be lifted back to solutions in six dimensions. This provides a striking realisation of the idea, first proposed by Pauli, of obtaining a theory that includes Yang-Mills fields by dimensional reduction on a coset space. We address the cosmological constant problem within this model, and argue that, contrary to recent suggestions, fine-tuning is still required. We also suggest a link between a modification of the model with 3-branes, and a five-dimensional model based on an S/Z orbifold.
Vacuum polarization in two-dimensional static spacetimes and dimensional reduction
Balbinot, Roberto; Fabbri, Alessandro; Nicolini, Piero; Sutton, Patrick J.
2002-07-01
We obtain an analytic approximation for the effective action of a quantum scalar field in a general static two-dimensional spacetime. We apply this to the dilaton gravity model resulting from the spherical reduction of a massive, non-minimally coupled scalar field in the four-dimensional Schwarzschild geometry. Careful analysis near the event horizon shows the resulting two-dimensional system to be regular in the Hartle-Hawking state for general values of the field mass, coupling, and angular momentum, while at spatial infinity it reduces to a thermal gas at the black-hole temperature.
Hayashi, Hideaki; Shibanoki, Taro; Shima, Keisuke; Kurita, Yuichi; Tsuji, Toshio
2015-12-01
This paper proposes a probabilistic neural network (NN) developed on the basis of time-series discriminant component analysis (TSDCA) that can be used to classify high-dimensional time-series patterns. TSDCA involves the compression of high-dimensional time series into a lower dimensional space using a set of orthogonal transformations and the calculation of posterior probabilities based on a continuous-density hidden Markov model with a Gaussian mixture model expressed in the reduced-dimensional space. The analysis can be incorporated into an NN, which is named a time-series discriminant component network (TSDCN), so that parameters of dimensionality reduction and classification can be obtained simultaneously as network coefficients according to a backpropagation through time-based learning algorithm with the Lagrange multiplier method. The TSDCN is considered to enable high-accuracy classification of high-dimensional time-series patterns and to reduce the computation time taken for network training. The validity of the TSDCN is demonstrated for high-dimensional artificial data and electroencephalogram signals in the experiments conducted during the study.
Optimized maximum noise fraction for dimensionality reduction of Chinese HJ-1A hyperspectral data
Gao, Lianru; Zhang, Bing; Sun, Xu; Li, Shanshan; Du, Qian; Wu, Changshan
2013-12-01
The important techniques in processing hyperspectral data acquired by interference imaging spectrometer onboard Small Satellite Constellation for Environment and Disaster mitigation (HJ-1A) are studied in this article. First, a new noise estimation method, named residual-scaled local standard deviations, is used to analyze the noise condition of HJ-1A hyperspectral images. Then, an optimized maximum noise fraction (OMNF) transform is proposed for dimensionality reduction of HJ-1A images, which adopts an accurately estimated noise covariance matrix for noise whitening. The proposed OMNF method is less sensitive to noise distribution and interference existence, thus it can more efficiently compact useful data information in a low-dimensional space. The proposed OMNF is evaluated through two applications, i.e., spectral unmixing and classification, using the HJ-1A image acquired at the Bohai Sea area in China. It demonstrates that the proposed OMNF provides better performance in comparison with other traditional dimensionality reduction methods.
Kernel Based Nonlinear Dimensionality Reduction and Classification for Genomic Microarray
Li, Xuehua; Shu, Lan
2008-01-01
Genomic microarrays are powerful research tools in bioinformatics and modern medicinal research because they enable massively-parallel assays and simultaneous monitoring of thousands of gene expression of biological samples. However, a simple microarray experiment often leads to very high-dimensional data and a huge amount of information, the vast amount of data challenges researchers into extracting the important features and reducing the high dimensionality. In this paper, a nonlinear dimensionality reduction kernel method based locally linear embedding(LLE) is proposed, and fuzzy K-nearest neighbors algorithm which denoises datasets will be introduced as a replacement to the classical LLE's KNN algorithm. In addition, kernel method based support vector machine (SVM) will be used to classify genomic microarray data sets in this paper. We demonstrate the application of the techniques to two published DNA microarray data sets. The experimental results confirm the superiority and high success rates of the presented method. PMID:27879930
Nonlinear Dimensionality Reduction via Path-Based Isometric Mapping.
Najafi, Amir; Joudaki, Amir; Fatemizadeh, Emad
2016-07-01
Nonlinear dimensionality reduction methods have demonstrated top-notch performance in many pattern recognition and image classification tasks. Despite their popularity, they suffer from highly expensive time and memory requirements, which render them inapplicable to large-scale datasets. To leverage such cases we propose a new method called "Path-Based Isomap". Similar to Isomap, we exploit geodesic paths to find the low-dimensional embedding. However, instead of preserving pairwise geodesic distances, the low-dimensional embedding is computed via a path-mapping algorithm. Due to the much fewer number of paths compared to number of data points, a significant improvement in time and memory complexity with a comparable performance is achieved. The method demonstrates state-of-the-art performance on well-known synthetic and real-world datasets, as well as in the presence of noise.
Novel signal shape descriptors through wavelet transforms and dimensionality reduction
Hughes, Nicholas P.; Tarassenko, Lionel
2003-11-01
The wavelet transform is a powerful tool for capturing the joint time-frequency characteristics of a signal. However, the resulting wavelet coefficients are typically high-dimensional, since at each time sample the wavelet transform is evaluated at a number of distinct scales. Unfortunately, modelling these coefficients can be problematic because of the large number of parameters needed to capture the dependencies between different scales. In this paper we investigate the use of algorithms from the field of dimensionality reduction to extract informative and compact descriptions of shape from wavelet coefficients. These low-dimensional shape descriptors lead to models that are governed by only a small number of parameters and can be learnt successfully from limited amounts of data. The validity of our approach is demonstrated on the task of automatically segmenting an electrocardiogram signal into its constituent waveform features.
Higher derivative corrections, dimensional reduction and Ehlers duality
Michel, Yann; Pioline, Boris
2007-09-01
Motivated by applications to black hole physics and duality, we study the effect of higher derivative corrections on the dimensional reduction of four-dimensional Einstein, Einstein-Liouville and Einstein-Maxwell gravity to one direction, as appropriate for stationary, spherically symmetric solutions. We construct a field redefinition scheme such that the one-dimensional Lagrangian is corrected only by powers of first derivatives of the fields, eliminating spurious modes and providing a suitable starting point for quantization. We show that the Ehlers symmetry, broken by the leading R2 corrections in Einstein-Liouville gravity, can be restored by including contributions of Taub-NUT instantons. Finally, we give a preliminary discussion of the duality between higher-derivative F-term corrections on the vector and hypermultiplet branches in N = 2 supergravity in four dimensions.
Kernel Based Nonlinear Dimensionality Reduction and Classification for Genomic Microarray.
Li, Xuehua; Shu, Lan
2008-07-15
Genomic microarrays are powerful research tools in bioinformatics and modern medicinal research because they enable massively-parallel assays and simultaneous monitoring of thousands of gene expression of biological samples. However, a simple microarray experiment often leads to very high-dimensional data and a huge amount of information, the vast amount of data challenges researchers into extracting the important features and reducing the high dimensionality. In this paper, a nonlinear dimensionality reduction kernel method based locally linear embedding(LLE) is proposed, and fuzzy K-nearest neighbors algorithm which denoises datasets will be introduced as a replacement to the classical LLE's KNN algorithm. In addition, kernel method based support vector machine (SVM) will be used to classify genomic microarray data sets in this paper. We demonstrate the application of the techniques to two published DNA microarray data sets. The experimental results confirm the superiority and high success rates of the presented method.
Nonlinear dimensionality reduction in molecular simulation: The diffusion map approach
Ferguson, Andrew L.; Panagiotopoulos, Athanassios Z.; Kevrekidis, Ioannis G.; Debenedetti, Pablo G.
2011-06-01
Molecular simulation is an important and ubiquitous tool in the study of microscopic phenomena in fields as diverse as materials science, protein folding and drug design. While the atomic-level resolution provides unparalleled detail, it can be non-trivial to extract the important motions underlying simulations of complex systems containing many degrees of freedom. The diffusion map is a nonlinear dimensionality reduction technique with the capacity to systematically extract the essential dynamical modes of high-dimensional simulation trajectories, furnishing a kinetically meaningful low-dimensional framework with which to develop insight and understanding of the underlying dynamics and thermodynamics. We survey the potential of this approach in the field of molecular simulation, consider its challenges, and discuss its underlying concepts and means of application. We provide examples drawn from our own work on the hydrophobic collapse mechanism of n-alkane chains, folding pathways of an antimicrobial peptide, and the dynamics of a driven interface.
Dimensionality reduction of medical image descriptors for multimodal image registration
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Degen Johanna
2015-09-01
Full Text Available Defining similarity forms a challenging and relevant research topic in multimodal image registration. The frequently used mutual information disregards contextual information, which is shared across modalities. A recent popular approach, called modality independent neigh-bourhood descriptor, is based on local self-similarities of image patches and is therefore able to capture spatial information. This image descriptor generates vectorial representations, i.e. it is multidimensional, which results in a disadvantage in terms of computation time. In this work, we present a problem-adapted solution for dimensionality reduction, by using principal component analysis and Horn’s parallel analysis. Furthermore, the influence of dimensionality reduction in global rigid image registration is investigated. It is shown that the registration results obtained from the reduced descriptor have the same high quality in comparison to those found for the original descriptor.
Enhancement of satellite precipitation estimation via unsupervised dimensionality reduction
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Mahrooghy, Majid [Mississippi State University (MSU); Younan, Nicolas H. [Mississippi State University (MSU); Anantharaj, Valentine G [ORNL; Aanstoos, James [Mississippi State University (MSU)
2012-01-01
A methodology to enhance Satellite Precipitation Estimation (SPE) using unsupervised dimensionality reduction (UDR) techniques is developed. This enhanced technique is an extension to the Precipitation Estimation from Remotely Sensed Imagery using an Artificial Neural Network (PERSIANN) and Cloud Classification System (CCS) method (PERSIANN-CCS) enriched using wavelet features combined with dimensionality reduction. Cloud-top brightness temperature measurements from Geostationary Operational Environmental Satellite (GOES-12) are used for precipitation estimation at 4 km 4 km spatial resolutions every 30 min. The study area in the continental United States covers parts of Louisiana, Arkansas, Kansas, Tennessee, Mississippi, and Alabama. Based on quantitative measures, root mean square error (RMSE) and Heidke skill score (HSS), the results show that the UDR techniques can improve the precipitation estimation accuracy. In addition, ICA is shown to have better performance than other UDR techniques; and in some cases, it achieves 10% improvement in the HSS.
Approaches to dimensionality reduction in proteomic biomarker studies.
Hilario, Melanie; Kalousis, Alexandros
2008-03-01
Mass-spectra based proteomic profiles have received widespread attention as potential tools for biomarker discovery and early disease diagnosis. A major data-analytical problem involved is the extremely high dimensionality (i.e. number of features or variables) of proteomic data, in particular when the sample size is small. This article reviews dimensionality reduction methods that have been used in proteomic biomarker studies. It then focuses on the problem of selecting the most appropriate method for a specific task or dataset, and proposes method combination as a potential alternative to single-method selection. Finally, it points out the potential of novel dimension reduction techniques, in particular those that incorporate domain knowledge through the use of informative priors or causal inference.
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Xiomara Patricia BLANCO VALENCIA
2017-03-01
Full Text Available This work outlines a unified formulation to represent spectral approaches for both dimensionality reduction and clustering. Proposed formulation starts with a generic latent variable model in terms of the projected input data matrix.Particularly, such a projection maps data onto a unknown high-dimensional space. Regarding this model, a generalized optimization problem is stated using quadratic formulations and a least-squares support vector machine.The solution of the optimization is addressed through a primal-dual scheme.Once latent variables and parameters are determined, the resultant model outputs a versatile projected matrix able to represent data in a low-dimensional space, as well as to provide information about clusters. Particularly, proposedformulation yields solutions for kernel spectral clustering and weighted-kernel principal component analysis.
Qi, Zhao-Hui; Wei, Ruo-Yan
2011-03-07
Graphical techniques have become powerful tools for the visualization and analysis of complicated biological systems. However, we cannot give such a graphical representation in a 2D/3D space when the dimensions of the represented data are more than three dimensions. The proposed method, a combination dimensionality reduction approach (CDR), consists of two parts: (i) principal component analysis (PCA) with a newly defined parameter ρ and (ii) locally linear embedding (LLE) with a proposed graphical selection for its optional parameter k. The CDR approach with ρ and k not only avoids loss of principal information, but also sufficiently well preserves the global high-dimensional structures in low-dimensional space such as 2D or 3D. The applications of the CDR on characteristic analysis at different codon positions in genome show that the method is a useful tool by which biologists could find useful biological knowledge. Copyright © 2010 Elsevier Ltd. All rights reserved.
Dimensionality Reduction Methods: Comparative Analysis of methods PCA, PPCA and KPCA
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Jorge Arroyo-Hernández
2016-01-01
Full Text Available The dimensionality reduction methods are algorithms mapping the set of data in subspaces derived from the original space, of fewer dimensions, that allow a description of the data at a lower cost. Due to their importance, they are widely used in processes associated with learning machine. This article presents a comparative analysis of PCA, PPCA and KPCA dimensionality reduction methods. A reconstruction experiment of worm-shape data was performed through structures of landmarks located in the body contour, with methods having different number of main components. The results showed that all methods can be seen as alternative processes. Nevertheless, thanks to the potential for analysis in the features space and the method for calculation of its preimage presented, KPCA offers a better method for recognition process and pattern extraction
Vacuum polarization in the Schwarzschild spacetime and dimensional reduction
Balbinot, R.; Fabbri, A.; Nicolini, P.; Frolov, V.; Sutton, P.; Zelnikov, A.
2001-04-01
A massless scalar field minimally coupled to gravity and propagating in Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviors and analytical approximations are given for in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down.
Grouping and dimensionality reduction by locally linear embedding
Polito, Marzia; Perona, Pietro
2002-01-01
Locally Linear Embedding (LLE) is an elegant nonlinear dimensionality-reduction technique recently introduced by Roweis and Saul 2]. It fails when the data is divided into separate groups. We study a variant of LLE that can simultaneously group the data and calculate local embedding of each group. An estimate for the upper bound on the intrinsic dimension of the data set is obtained automatically.
A duality view of spectral methods for dimensionality reduction
Xiao, Lin; Sun, Jun; Boyd, Stephen
2006-01-01
We present a unified duality view of several recently emerged spectral methods for nonlinear dimensionality reduction, including Isomap, locally linear embedding, Laplacian eigenmaps, and maximum variance unfolding. We discuss the duality theory for the maximum variance unfolding problem, and show that other methods are directly related to either its primal formulation or its dual formulation, or can be interpreted from the optimality conditions. This duality framework reveals close connectio...
Dimensional reduction applied to QCD at three loops
Harlander, Robert V.; Kant, Phillip; Mihaila, Luminita; Steinhauser, Matthias
2006-09-01
Dimensional Reduction is applied to QCD in order to compute various renormalization constants in the DR-bar scheme at higher orders in perturbation theory. In particular, the β function and the anomalous dimension of the quark masses are derived to three-loop order. Special emphasis is put on the proper treatment of the so-called ɛ-scalars and the additional couplings which have to be considered.
RNA folding inside a virus capsid and dimensional reduction.
Ghafouri, Rouzbeh; Bruinsma, Robijn; Rudnick, Joseph
2006-03-01
As RNA folds on itself , in certain conditions, it takes the form of a branched polymer. So the problem of RNA folding in a virus capsid is essentially the problem of a branched polymer in a confined environment. In this paper we attack the problem using the technique of dimensional reduction which relates a branched polymer with self interation in D dimension to a hardcore classical gas in (D-2) dimension. We look for phase transitions and intersting physical quantities such as pressure.
Dimensionality Reduction of very large document collections by Semantic Mapping
Corrêa, Renato Fernandes; Ludermir, Teresa Bernarda
2007-01-01
This paper describes improving in Semantic Mapping, a feature extraction method useful to dimensionality reduction of vectors representing documents of large text collections. This method may be viewed as a specialization of the Random Mapping, method proposed in WEBSOM project. Semantic Mapping, Random Mapping and Principal Component Analysis (PCA) are applied to categorization of document collections using Self-Organizing Maps (SOM). Semantic Mapping generated document representation as goo...
Dimensionality Reduction and Channel Selection of Motor Imagery Electroencephalographic Data
Naeem, Muhammad; Brunner, Clemens; Pfurtscheller, Gert
2009-01-01
The performance of spatial filters based on independent components analysis (ICA) was evaluated by employing principal component analysis (PCA) preprocessing for dimensional reduction. The PCA preprocessing was not found to be a suitable method that could retain motor imagery information in a smaller set of components. In contrast, 6 ICA components selected on the basis of visual inspection performed comparably (61.9%) to the full range of 22 components (63.9%). An automated selection of ICA ...
DMLLE: a large-scale dimensionality reduction method for detection of polyps in CT colonography
Wang, Shijun; Yao, Jianhua; Summers, Ronald M.
2008-03-01
Computer-aided diagnosis systems have been shown to be feasible for polyp detection on computed tomography (CT) scans. After 3-D image segmentation and feature extraction, the dataset of colonic polyp candidates has large-scale and high dimension characteristics. In this paper, we propose a large-scale dimensionality reduction method based on Diffusion Map and Locally Linear Embedding for detection of polyps in CT colonography. By selecting partial data as landmarks, we first map the landmarks into a low dimensional embedding space using Diffusion Map. Then by using Locally Linear Embedding algorithm, non-landmark samples are embedded into the same low dimensional space according to their nearest landmark samples. The local geometry of samples is preserved in both the original space and the embedding space. We applied the proposed method called DMLLE to a colonic polyp dataset which contains 58336 candidates (including 85 6-9mm true polyps) with 155 features. Visual inspection shows that true polyps with similar shapes are mapped to close vicinity in the low dimensional space. FROC analysis shows that SVM with DMLLE achieves higher sensitivity with lower false positives per patient than that of SVM using all features. At the false positives of 8 per patient, SVM with DMLLE improves the average sensitivity from 64% to 75% for polyps whose sizes are in the range from 6 mm to 9 mm (p < 0.05). This higher sensitivity is comparable to unaided readings by trained radiologists.
Dimensionality reduction for large-scale neural recordings.
Cunningham, John P; Yu, Byron M
2014-11-01
Most sensory, cognitive and motor functions depend on the interactions of many neurons. In recent years, there has been rapid development and increasing use of technologies for recording from large numbers of neurons, either sequentially or simultaneously. A key question is what scientific insight can be gained by studying a population of recorded neurons beyond studying each neuron individually. Here, we examine three important motivations for population studies: single-trial hypotheses requiring statistical power, hypotheses of population response structure and exploratory analyses of large data sets. Many recent studies have adopted dimensionality reduction to analyze these populations and to find features that are not apparent at the level of individual neurons. We describe the dimensionality reduction methods commonly applied to population activity and offer practical advice about selecting methods and interpreting their outputs. This review is intended for experimental and computational researchers who seek to understand the role dimensionality reduction has had and can have in systems neuroscience, and who seek to apply these methods to their own data.
Dimensionality reduction by supervised neighbor embedding using laplacian search.
Zheng, Jianwei; Zhang, Hangke; Cattani, Carlo; Wang, Wanliang
2014-01-01
Dimensionality reduction is an important issue for numerous applications including biomedical images analysis and living system analysis. Neighbor embedding, those representing the global and local structure as well as dealing with multiple manifolds, such as the elastic embedding techniques, can go beyond traditional dimensionality reduction methods and find better optima. Nevertheless, existing neighbor embedding algorithms can not be directly applied in classification as suffering from several problems: (1) high computational complexity, (2) nonparametric mappings, and (3) lack of class labels information. We propose a supervised neighbor embedding called discriminative elastic embedding (DEE) which integrates linear projection matrix and class labels into the final objective function. In addition, we present the Laplacian search direction for fast convergence. DEE is evaluated in three aspects: embedding visualization, training efficiency, and classification performance. Experimental results on several benchmark databases present that the proposed DEE exhibits a supervised dimensionality reduction approach which not only has strong pattern revealing capability, but also brings computational advantages over standard gradient based methods.
Dimensionality Reduction by Supervised Neighbor Embedding Using Laplacian Search
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Jianwei Zheng
2014-01-01
Full Text Available Dimensionality reduction is an important issue for numerous applications including biomedical images analysis and living system analysis. Neighbor embedding, those representing the global and local structure as well as dealing with multiple manifolds, such as the elastic embedding techniques, can go beyond traditional dimensionality reduction methods and find better optima. Nevertheless, existing neighbor embedding algorithms can not be directly applied in classification as suffering from several problems: (1 high computational complexity, (2 nonparametric mappings, and (3 lack of class labels information. We propose a supervised neighbor embedding called discriminative elastic embedding (DEE which integrates linear projection matrix and class labels into the final objective function. In addition, we present the Laplacian search direction for fast convergence. DEE is evaluated in three aspects: embedding visualization, training efficiency, and classification performance. Experimental results on several benchmark databases present that the proposed DEE exhibits a supervised dimensionality reduction approach which not only has strong pattern revealing capability, but also brings computational advantages over standard gradient based methods.
Graph embedding and extensions: a general framework for dimensionality reduction.
Yan, Shuicheng; Xu, Dong; Zhang, Benyu; Zhang, Hong-Jiang; Yang, Qiang; Lin, Stephen
2007-01-01
Over the past few decades, a large family of algorithms - supervised or unsupervised; stemming from statistics or geometry theory - has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called Marginal Fisher Analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional Linear Discriminant Analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions.
Semisupervised dimensionality reduction and classification through virtual label regression.
Nie, Feiping; Xu, Dong; Li, Xuelong; Xiang, Shiming
2011-06-01
Semisupervised dimensionality reduction has been attracting much attention as it not only utilizes both labeled and unlabeled data simultaneously, but also works well in the situation of out-of-sample. This paper proposes an effective approach of semisupervised dimensionality reduction through label propagation and label regression. Different from previous efforts, the new approach propagates the label information from labeled to unlabeled data with a well-designed mechanism of random walks, in which outliers are effectively detected and the obtained virtual labels of unlabeled data can be well encoded in a weighted regression model. These virtual labels are thereafter regressed with a linear model to calculate the projection matrix for dimensionality reduction. By this means, when the manifold or the clustering assumption of data is satisfied, the labels of labeled data can be correctly propagated to the unlabeled data; and thus, the proposed approach utilizes the labeled and the unlabeled data more effectively than previous work. Experimental results are carried out upon several databases, and the advantage of the new approach is well demonstrated.
Second order dimensionality reduction using minimum and maximum mutual information models.
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Jeffrey D Fitzgerald
2011-10-01
Full Text Available Conventional methods used to characterize multidimensional neural feature selectivity, such as spike-triggered covariance (STC or maximally informative dimensions (MID, are limited to Gaussian stimuli or are only able to identify a small number of features due to the curse of dimensionality. To overcome these issues, we propose two new dimensionality reduction methods that use minimum and maximum information models. These methods are information theoretic extensions of STC that can be used with non-Gaussian stimulus distributions to find relevant linear subspaces of arbitrary dimensionality. We compare these new methods to the conventional methods in two ways: with biologically-inspired simulated neurons responding to natural images and with recordings from macaque retinal and thalamic cells responding to naturalistic time-varying stimuli. With non-Gaussian stimuli, the minimum and maximum information methods significantly outperform STC in all cases, whereas MID performs best in the regime of low dimensional feature spaces.
Second order dimensionality reduction using minimum and maximum mutual information models.
Fitzgerald, Jeffrey D; Rowekamp, Ryan J; Sincich, Lawrence C; Sharpee, Tatyana O
2011-10-01
Conventional methods used to characterize multidimensional neural feature selectivity, such as spike-triggered covariance (STC) or maximally informative dimensions (MID), are limited to Gaussian stimuli or are only able to identify a small number of features due to the curse of dimensionality. To overcome these issues, we propose two new dimensionality reduction methods that use minimum and maximum information models. These methods are information theoretic extensions of STC that can be used with non-Gaussian stimulus distributions to find relevant linear subspaces of arbitrary dimensionality. We compare these new methods to the conventional methods in two ways: with biologically-inspired simulated neurons responding to natural images and with recordings from macaque retinal and thalamic cells responding to naturalistic time-varying stimuli. With non-Gaussian stimuli, the minimum and maximum information methods significantly outperform STC in all cases, whereas MID performs best in the regime of low dimensional feature spaces.
International Nuclear Information System (INIS)
Fiziev, P P; Shirkov, D V
2012-01-01
The paper presents a generalization and further development of our recent publications, where solutions of the Klein–Fock–Gordon equation defined on a few particular D = (2 + 1)-dimensional static spacetime manifolds were considered. The latter involve toy models of two-dimensional spaces with axial symmetry, including dimensional reduction to the one-dimensional space as a singular limiting case. Here, the non-static models of space geometry with axial symmetry are under consideration. To make these models closer to physical reality, we define a set of ‘admissible’ shape functions ρ(t, z) as the (2 + 1)-dimensional Einstein equation solutions in the vacuum spacetime, in the presence of the Λ-term and for the spacetime filled with the standard ‘dust’. It is curious that in the last case the Einstein equations reduce to the well-known Monge–Ampère equation, thus enabling one to obtain the general solution of the Cauchy problem, as well as a set of other specific solutions involving one arbitrary function. A few explicit solutions of the Klein–Fock–Gordon equation in this set are given. An interesting qualitative feature of these solutions relates to the dimensional reduction points, their classification and time behavior. In particular, these new entities could provide us with novel insight into the nature of P- and T-violations and of the Big Bang. A short comparison with other attempts to utilize the dimensional reduction of the spacetime is given. (paper)
Fiziev, P. P.; Shirkov, D. V.
2012-02-01
The paper presents a generalization and further development of our recent publications, where solutions of the Klein-Fock-Gordon equation defined on a few particular D = (2 + 1)-dimensional static spacetime manifolds were considered. The latter involve toy models of two-dimensional spaces with axial symmetry, including dimensional reduction to the one-dimensional space as a singular limiting case. Here, the non-static models of space geometry with axial symmetry are under consideration. To make these models closer to physical reality, we define a set of ‘admissible’ shape functions ρ(t, z) as the (2 + 1)-dimensional Einstein equation solutions in the vacuum spacetime, in the presence of the Λ-term and for the spacetime filled with the standard ‘dust’. It is curious that in the last case the Einstein equations reduce to the well-known Monge-Ampère equation, thus enabling one to obtain the general solution of the Cauchy problem, as well as a set of other specific solutions involving one arbitrary function. A few explicit solutions of the Klein-Fock-Gordon equation in this set are given. An interesting qualitative feature of these solutions relates to the dimensional reduction points, their classification and time behavior. In particular, these new entities could provide us with novel insight into the nature of P- and T-violations and of the Big Bang. A short comparison with other attempts to utilize the dimensional reduction of the spacetime is given.
Holography for Einstein-Maxwell-dilaton theories from generalized dimensional reduction
Goutéraux, Blaise; Smolic, Jelena; Smolic, Milena; Skenderis, Kostas; Taylor, Marika
2012-01-01
We show that a class of Einstein-Maxwell-Dilaton (EMD) theories are re- lated to higher dimensional AdS-Maxwell gravity via a dimensional reduction over com- pact Einstein spaces combined with continuation in the dimension of the compact space to non-integral values (`generalized dimensional reduction'). This relates (fairly complicated) black hole solutions of EMD theories to simple black hole/brane solutions of AdS-Maxwell gravity and explains their properties. The generalized dimensional reduction is used to infer the holographic dictionary and the hydrodynamic behavior for this class of theories from those of AdS. As a specific example, we analyze the case of a black brane carrying a wave whose universal sector is described by gravity coupled to a Maxwell field and two neutral scalars. At thermal equilibrium and finite chemical potential the two operators dual to the bulk scalar fields acquire expectation values characterizing the breaking of con- formal and generalized conformal invariance. We compute holographically the first order transport coefficients (conductivity, shear and bulk viscosity) for this system.
Nonlinear dimensionality reduction of electroencephalogram (EEG) for Brain Computer interfaces.
Teli, Mohammad Nayeem; Anderson, Charles
2009-01-01
Patterns in electroencephalogram (EEG) signals are analyzed for a Brain Computer Interface (BCI). An important aspect of this analysis is the work on transformations of high dimensional EEG data to low dimensional spaces in which we can classify the data according to mental tasks being performed. In this research we investigate how a Neural Network (NN) in an auto-encoder with bottleneck configuration can find such a transformation. We implemented two approximate second-order methods to optimize the weights of these networks, because the more common first-order methods are very slow to converge for networks like these with more than three layers of computational units. The resulting non-linear projections of time embedded EEG signals show interesting separations that are related to tasks. The bottleneck networks do indeed discover nonlinear transformations to low-dimensional spaces that capture much of the information present in EEG signals. However, the resulting low-dimensional representations do not improve classification rates beyond what is possible using Quadratic Discriminant Analysis (QDA) on the original time-lagged EEG.
Pesenson, Meyer; Pesenson, I. Z.; McCollum, B.
2009-05-01
The complexity of multitemporal/multispectral astronomical data sets together with the approaching petascale of such datasets and large astronomical surveys require automated or semi-automated methods for knowledge discovery. Traditional statistical methods of analysis may break down not only because of the amount of data, but mostly because of the increase of the dimensionality of data. Image fusion (combining information from multiple sensors in order to create a composite enhanced image) and dimension reduction (finding lower-dimensional representation of high-dimensional data) are effective approaches to "the curse of dimensionality,” thus facilitating automated feature selection, classification and data segmentation. Dimension reduction methods greatly increase computational efficiency of machine learning algorithms, improve statistical inference and together with image fusion enable effective scientific visualization (as opposed to mere illustrative visualization). The main approach of this work utilizes recent advances in multidimensional image processing, as well as representation of essential structure of a data set in terms of its fundamental eigenfunctions, which are used as an orthonormal basis for the data visualization and analysis. We consider multidimensional data sets and images as manifolds or combinatorial graphs and construct variational splines that minimize certain Sobolev norms. These splines allow us to reconstruct the eigenfunctions of the combinatorial Laplace operator by using only a small portion of the graph. We use the first two or three eigenfunctions for embedding large data sets into two- or three-dimensional Euclidean space. Such reduced data sets allow efficient data organization, retrieval, analysis and visualization. We demonstrate applications of the algorithms to test cases from the Spitzer Space Telescope. This work was carried out with funding from the National Geospatial-Intelligence Agency University Research Initiative
Das, Payel; Moll, Mark; Stamati, Hernán; Kavraki, Lydia E.; Clementi, Cecilia
2006-01-01
The definition of reaction coordinates for the characterization of a protein-folding reaction has long been a controversial issue, even for the “simple” case in which one single free-energy barrier separates the folded and unfolded ensemble. We propose a general approach to this problem to obtain a few collective coordinates by using nonlinear dimensionality reduction. We validate the usefulness of this method by characterizing the folding landscape associated with a coarse-grained protein model of src homology 3 as sampled by molecular dynamics simulations. The folding free-energy landscape projected on the few relevant coordinates emerging from the dimensionality reduction can correctly identify the transition-state ensemble of the reaction. The first embedding dimension efficiently captures the evolution of the folding process along the main folding route. These results clearly show that the proposed method can efficiently find a low-dimensional representation of a complex process such as protein folding. PMID:16785435
Lyapunov Schmidt reduction algorithm for three-dimensional discrete vortices
Lukas, Mike; Pelinovsky, Dmitry; Kevrekidis, P. G.
2008-03-01
We address the persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger equation and develop a symbolic package based on Wolfram’s MATHEMATICA for computations of the Lyapunov-Schmidt reduction method. The Lyapunov-Schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations. The method was developed earlier in the context of the two-dimensional lattice and applied to the onsite and offsite configurations (called the vortex cross and the vortex cell) by using semianalytical computations [D.E. Pelinovsky, P.G. Kevrekidis, D. Frantzeskakis, Physica D 212 (2005) 20-53; P.G. Kevrekidis, D.E. Pelinovsky, Proc. R. Soc. A 462 (2006) 2671-2694]. The present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites, performs a required number of Lyapunov-Schmidt reductions and outputs the predictions on whether the configuration persists, for finite coupling, in the three-dimensional lattice and whether it is stable or unstable. It also provides approximations for the eigenvalues of the linearized stability problem. We report a number of applications of the algorithm to important multisite three-dimensional configurations, such as the simple cube, the double cross and the diamond. For each configuration, we identify exactly one solution, which is stable for small coupling between lattice nodes.
Krivov, Sergei V.
2011-07-01
Dimensionality reduction is ubiquitous in the analysis of complex dynamics. The conventional dimensionality reduction techniques, however, focus on reproducing the underlying configuration space, rather than the dynamics itself. The constructed low-dimensional space does not provide a complete and accurate description of the dynamics. Here I describe how to perform dimensionality reduction while preserving the essential properties of the dynamics. The approach is illustrated by analyzing the chess game—the archetype of complex dynamics. A variable that provides complete and accurate description of chess dynamics is constructed. The winning probability is predicted by describing the game as a random walk on the free-energy landscape associated with the variable. The approach suggests a possible way of obtaining a simple yet accurate description of many important complex phenomena. The analysis of the chess game shows that the approach can quantitatively describe the dynamics of processes where human decision-making plays a central role, e.g., financial and social dynamics.
Dimensional Analysis with space discrimination applied to Fickian difussion phenomena
International Nuclear Information System (INIS)
Diaz Sanchidrian, C.; Castans, M.
1989-01-01
Dimensional Analysis with space discrimination is applied to Fickian difussion phenomena in order to transform its partial differen-tial equations into ordinary ones, and also to obtain in a dimensionl-ess fom the Ficks second law. (Author)
A trace ratio maximization approach to multiple kernel-based dimensionality reduction.
Jiang, Wenhao; Chung, Fu-lai
2014-01-01
Most dimensionality reduction techniques are based on one metric or one kernel, hence it is necessary to select an appropriate kernel for kernel-based dimensionality reduction. Multiple kernel learning for dimensionality reduction (MKL-DR) has been recently proposed to learn a kernel from a set of base kernels which are seen as different descriptions of data. As MKL-DR does not involve regularization, it might be ill-posed under some conditions and consequently its applications are hindered. This paper proposes a multiple kernel learning framework for dimensionality reduction based on regularized trace ratio, termed as MKL-TR. Our method aims at learning a transformation into a space of lower dimension and a corresponding kernel from the given base kernels among which some may not be suitable for the given data. The solutions for the proposed framework can be found based on trace ratio maximization. The experimental results demonstrate its effectiveness in benchmark datasets, which include text, image and sound datasets, for supervised, unsupervised as well as semi-supervised settings. Copyright © 2013 Elsevier Ltd. All rights reserved.
Execution spaces for simple higher dimensional automata
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Raussen, Martin
2012-01-01
Higher dimensional automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek (Theor Comput Sci 368(1–2): 168–194, 2006). For a topologist, they are attractive since they can be modeled as cubical complexes—with an inbuilt restriction for directions of allowa......Higher dimensional automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek (Theor Comput Sci 368(1–2): 168–194, 2006). For a topologist, they are attractive since they can be modeled as cubical complexes—with an inbuilt restriction for directions...
An R package implementation of multifactor dimensionality reduction
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Winham Stacey J
2011-08-01
Full Text Available Abstract Background A breadth of high-dimensional data is now available with unprecedented numbers of genetic markers and data-mining approaches to variable selection are increasingly being utilized to uncover associations, including potential gene-gene and gene-environment interactions. One of the most commonly used data-mining methods for case-control data is Multifactor Dimensionality Reduction (MDR, which has displayed success in both simulations and real data applications. Additional software applications in alternative programming languages can improve the availability and usefulness of the method for a broader range of users. Results We introduce a package for the R statistical language to implement the Multifactor Dimensionality Reduction (MDR method for nonparametric variable selection of interactions. This package is designed to provide an alternative implementation for R users, with great flexibility and utility for both data analysis and research. The 'MDR' package is freely available online at http://www.r-project.org/. We also provide data examples to illustrate the use and functionality of the package. Conclusions MDR is a frequently-used data-mining method to identify potential gene-gene interactions, and alternative implementations will further increase this usage. We introduce a flexible software package for R users.
Dimensional reduction, hard thermal loops, and the renormalization group
Stephens, C. R.; Weber, Axel; Hess, Peter O.; Astorga, Francisco
2004-08-01
We study the realization of dimensional reduction and the validity of the hard thermal loop expansion for λφ4 theory at finite temperature, using an environmentally friendly finite-temperature renormalization group with a fiducial temperature as flow parameter. The one-loop renormalization group allows for a consistent description of the system at low and high temperatures, and, in particular, of the phase transition. The main results are that dimensional reduction applies, apart from a range of temperatures around the phase transition, at high temperatures (compared to the zero temperature mass) only for sufficiently small coupling constants, while the hard thermal loop expansion is valid below (and rather far from) the phase transition, and, again, at high temperatures only in the case of sufficiently small coupling constants. We emphasize that close to the critical temperature, physics is completely dominated by thermal fluctuations that are not resummed in the hard thermal loop approach and where universal quantities are independent of the parameters of the fundamental four-dimensional theory.
Nonlinear dimensionality reduction of data lying on the multicluster manifold.
Meng, Deyu; Leung, Yee; Fung, Tung; Xu, Zongben
2008-08-01
A new method, which is called decomposition-composition (D-C) method, is proposed for the nonlinear dimensionality reduction (NLDR) of data lying on the multicluster manifold. The main idea is first to decompose a given data set into clusters and independently calculate the low-dimensional embeddings of each cluster by the decomposition procedure. Based on the intercluster connections, the embeddings of all clusters are then composed into their proper positions and orientations by the composition procedure. Different from other NLDR methods for multicluster data, which consider associatively the intracluster and intercluster information, the D-C method capitalizes on the separate employment of the intracluster neighborhood structures and the intercluster topologies for effective dimensionality reduction. This, on one hand, isometrically preserves the rigid-body shapes of the clusters in the embedding process and, on the other hand, guarantees the proper locations and orientations of all clusters. The theoretical arguments are supported by a series of experiments performed on the synthetic and real-life data sets. In addition, the computational complexity of the proposed method is analyzed, and its efficiency is theoretically analyzed and experimentally demonstrated. Related strategies for automatic parameter selection are also examined.
An R package implementation of multifactor dimensionality reduction.
Winham, Stacey J; Motsinger-Reif, Alison A
2011-08-16
A breadth of high-dimensional data is now available with unprecedented numbers of genetic markers and data-mining approaches to variable selection are increasingly being utilized to uncover associations, including potential gene-gene and gene-environment interactions. One of the most commonly used data-mining methods for case-control data is Multifactor Dimensionality Reduction (MDR), which has displayed success in both simulations and real data applications. Additional software applications in alternative programming languages can improve the availability and usefulness of the method for a broader range of users. We introduce a package for the R statistical language to implement the Multifactor Dimensionality Reduction (MDR) method for nonparametric variable selection of interactions. This package is designed to provide an alternative implementation for R users, with great flexibility and utility for both data analysis and research. The 'MDR' package is freely available online at http://www.r-project.org/. We also provide data examples to illustrate the use and functionality of the package. MDR is a frequently-used data-mining method to identify potential gene-gene interactions, and alternative implementations will further increase this usage. We introduce a flexible software package for R users.
Execution spaces for simple higher dimensional automata
DEFF Research Database (Denmark)
Raussen, Martin
Higher Dimensional Automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek [26]. For a topologist, they are attractive since they can be modeled as cubical complexes - with an inbuilt restriction for directions´of allowable (d-)paths. In Raussen [25], we...
Ji, Shuiwang
2013-07-11
The structured organization of cells in the brain plays a key role in its functional efficiency. This delicate organization is the consequence of unique molecular identity of each cell gradually established by precise spatiotemporal gene expression control during development. Currently, studies on the molecular-structural association are beginning to reveal how the spatiotemporal gene expression patterns are related to cellular differentiation and structural development. In this article, we aim at a global, data-driven study of the relationship between gene expressions and neuroanatomy in the developing mouse brain. To enable visual explorations of the high-dimensional data, we map the in situ hybridization gene expression data to a two-dimensional space by preserving both the global and the local structures. Our results show that the developing brain anatomy is largely preserved in the reduced gene expression space. To provide a quantitative analysis, we cluster the reduced data into groups and measure the consistency with neuroanatomy at multiple levels. Our results show that the clusters in the low-dimensional space are more consistent with neuroanatomy than those in the original space. Gene expression patterns and developing brain anatomy are closely related. Dimensionality reduction and visual exploration facilitate the study of this relationship.
Interpreting the dimensions of neural feature representations revealed by dimensionality reduction.
Goddard, Erin; Klein, Colin; Solomon, Samuel G; Hogendoorn, Hinze; Carlson, Thomas A
2017-06-27
Recent progress in understanding the structure of neural representations in the cerebral cortex has centred around the application of multivariate classification analyses to measurements of brain activity. These analyses have proved a sensitive test of whether given brain regions provide information about specific perceptual or cognitive processes. An exciting extension of this approach is to infer the structure of this information, thereby drawing conclusions about the underlying neural representational space. These approaches rely on exploratory data-driven dimensionality reduction to extract the natural dimensions of neural spaces, including natural visual object and scene representations, semantic and conceptual knowledge, and working memory. However, the efficacy of these exploratory methods is unknown, because they have only been applied to representations in brain areas for which we have little or no secondary knowledge. One of the best-understood areas of the cerebral cortex is area MT of primate visual cortex, which is known to be important in motion analysis. To assess the effectiveness of dimensionality reduction for recovering neural representational space we applied several dimensionality reduction methods to multielectrode measurements of spiking activity obtained from area MT of marmoset monkeys, made while systematically varying the motion direction and speed of moving stimuli. Despite robust tuning at individual electrodes, and high classifier performance, dimensionality reduction rarely revealed dimensions for direction and speed. We use this example to illustrate important limitations of these analyses, and suggest a framework for how to best apply such methods to data where the structure of the neural representation is unknown. Copyright © 2017 Elsevier Inc. All rights reserved.
Distribution-based dimensionality reduction applied to articulated motion recognition.
Nayak, Sunita; Sarkar, Sudeep; Loeding, Barbara
2009-05-01
Some articulated motion representations rely on frame-wise abstractions of the statistical distribution of low-level features such as orientation, color, or relational distributions. As configuration among parts changes with articulated motion, the distribution changes, tracing a trajectory in the latent space of distributions, which we call the configuration space. These trajectories can then be used for recognition using standard techniques such as dynamic time warping. The core theory in this paper concerns embedding the frame-wise distributions, which can be looked upon as probability functions, into a low-dimensional space so that we can estimate various meaningful probabilistic distances such as the Chernoff, Bhattacharya, Matusita, Kullback-Leibler (KL) or symmetric-KL distances based on dot products between points in this space. Apart from computational advantages, this representation also affords speed-normalized matching of motion signatures. Speed normalized representations can be formed by interpolating the configuration trajectories along their arc lengths, without using any knowledge of the temporal scale variations between the sequences. We experiment with five different probabilistic distance measures and show the usefulness of the representation in three different contexts-sign recognition (with large number of possible classes), gesture recognition (with person variations), and classification of human-human interaction sequences (with segmentation problems). We find the importance of using the right distance measure for each situation. The low-dimensional embedding makes matching two to three times faster, while achieving recognition accuracies that are close to those obtained without using a low-dimensional embedding. We also empirically establish the robustness of the representation with respect to low-level parameters, embedding parameters, and temporal-scale parameters.
Dimensionality reduction oriented toward the feature visualization for ischemia detection.
Delgado-Trejos, Edilson; Perera-Lluna, Alexandre; Vallverdú-Ferrer, Montserrat; Caminal-Magrans, Peré; Castellanos-Domínguez, Germán
2009-07-01
An effective data representation methodology on high-dimension feature spaces is presented, which allows a better interpretation of subjacent physiological phenomena (namely, cardiac behavior related to cardiovascular diseases), and is based on search criteria over a feature set resulting in an increase in the detection capability of ischemic pathologies, but also connecting these features with the physiologic representation of the ECG. The proposed dimension reduction scheme consists of three levels: projection, interpretation, and visualization. First, a hybrid algorithm is described that projects the multidimensional data to a lower dimension space, gathering the features that contribute similarly in the meaning of the covariance reconstruction in order to find information of clinical relevance over the initial training space. Next, an algorithm of variable selection is provided that further reduces the dimension, taking into account only the variables that offer greater class separability, and finally, the selected feature set is projected to a 2-D space in order to verify the performance of the suggested dimension reduction algorithm in terms of the discrimination capability for ischemia detection. The ECG recordings used in this study are from the European ST-T database and from the Universidad Nacional de Colombia database. In both cases, over 99% feature reduction was obtained, and classification precision was over 99% using a five-nearest-neighbor classifier (5-NN).
Chen, Xi; Diez, Matteo; Kandasamy, Manivannan; Zhang, Zhiguo; Campana, Emilio F.; Stern, Frederick
2015-04-01
Advances in high-fidelity shape optimization for industrial problems are presented, based on geometric variability assessment and design-space dimensionality reduction by Karhunen-Loève expansion, metamodels and deterministic particle swarm optimization (PSO). Hull-form optimization is performed for resistance reduction of the high-speed Delft catamaran, advancing in calm water at a given speed, and free to sink and trim. Two feasible sets (A and B) are assessed, using different geometric constraints. Dimensionality reduction for 95% confidence is applied to high-dimensional free-form deformation. Metamodels are trained by design of experiments with URANS; multiple deterministic PSOs achieve a resistance reduction of 9.63% for A and 6.89% for B. Deterministic PSO is found to be effective and efficient, as shown by comparison with stochastic PSO. The optimum for A has the best overall performance over a wide range of speed. Compared with earlier optimization, the present studies provide an additional resistance reduction of 6.6% at 1/10 of the computational cost.
Monsoon convection dynamics and nonlinear dimensionality reduction vis Isomap
Hannachi, A.; Turner, A.
2012-04-01
The Asian summer monsoon is a high dimensional and highly nonlinear phenomenon involving considerable moisture transport into land from ocean, and is critical for the whole region. We have used the European Reanalysis ERA-40 sea-level pressure (SLP) anomalies, with respect to the seasonal cycle, over the region (50E-145E, 20S-35N) to study the nonlinearity of the Asian monsoon using Isomap. We have focussed on the two-dimensional embedding of the SLP anomalies. Unlike the unimodality obtained from the empirical orthogonal function space, the probability density function computed within the two-dimensional Isomap space is shown to be bimodal. A clustering procedure is applied and reveals that the data support three clusters, which are identified using a three-component bivariate Gaussian mixture model. Two modes are associated with an active phase over India/Bay of Bengal and East China sea respectively, whereas the third mode is associated witha break over East/South China sea. Using the low-level wind field anomalies the (first mode) active phase is found to be characterised by a strengthening and an eastward extension of the Somali jet whereas during the (second mode) break phase the Somali jet is weakened and reversed by an easterly flow emanating from the West Pacific. The effect of large scale seasonal mean monsoon and lower boundary forcing is also investigated and discussed.
Dynamics and predictability of Asian Monsoon and nonlinear dimensionality reduction
Hannachi, Abdel; Turner, Andy
2013-04-01
The Asian summer monsoon is a high dimensional and highly nonlinear phenomenon involving considerable moisture transport into land from ocean, and is critical for the whole region. We have used the European Reanalysis ERA-40 sea-level pressure (SLP) anomalies, with respect to the seasonal cycle, over the region (50E-145E, 20S-35N) to study the nonlinearity of the Asian monsoon using Isomap. We have focussed on the two-dimensional embedding of the SLP anomalies. Unlike the unimodality obtained from the empirical orthogonal function space, the probability density function, within the two-dimensional Isomap space, turns out to be bimodal. A clustering procedure is applied and reveals that the data support three clusters, which are identified using a three-component bivariate Gaussian mixture model. The modes are found to be associated respectively with the break and the active phases of the monsoon in addition to a third phase: the China sea active phase. Using the low-level wind field anomalies the active phase is found to be characterised by a strengthening and an eastward extension of the Somali jet whereas during the break phase the Somali jet is weakened and reversed by an easterly flow emanating from the West Pacific. The effect of large scale seasonal mean monsoon and lower boundary forcing is also investigated and discussed.
Dimensionality reduction using Principal Component Analysis for network intrusion detection
Directory of Open Access Journals (Sweden)
K. Keerthi Vasan
2016-09-01
Full Text Available Intrusion detection is the identification of malicious activities in a given network by analyzing its traffic. Data mining techniques used for this analysis study the traffic traces and identify hostile flows in the traffic. Dimensionality reduction in data mining focuses on representing data with minimum number of dimensions such that its properties are not lost and hence reducing the underlying complexity in processing the data. Principal Component Analysis (PCA is one of the prominent dimensionality reduction techniques widely used in network traffic analysis. In this paper, we focus on the efficiency of PCA for intrusion detection and determine its Reduction Ratio (RR, ideal number of Principal Components needed for intrusion detection and the impact of noisy data on PCA. We carried out experiments with PCA using various classifier algorithms on two benchmark datasets namely, KDD CUP and UNB ISCX. Experiments show that the first 10 Principal Components are effective for classification. The classification accuracy for 10 Principal Components is about 99.7% and 98.8%, nearly same as the accuracy obtained using original 41 features for KDD and 28 features for ISCX, respectively.
Dimensionality reduction based on fuzzy rough sets oriented to ischemia detection.
Orrego, Diana A; Becerra, Miguel A; Delgado-Trejos, Edilson
2012-01-01
This paper presents a dimensionality reduction study based on fuzzy rough sets with the aim of increasing the discriminant capability of the representation of normal ECG beats and those that contain ischemic events. A novel procedure is proposed to obtain the fuzzy equivalence classes based on entropy and neighborhood techniques and a modification of the Quick Reduct Algorithm is used to select the relevant features from a large feature space by a dependency function. The tests were carried out on a feature space made up by 840 wavelet features extracted from 900 ECG normal beats and 900 ECG beats with evidence of ischemia. Results of around 99% classification accuracy are obtained. This methodology provides a reduced feature space with low complexity and high representation capability. Additionally, the discriminant strength of entropy in terms of representing ischemic disorders from time-frequency information in ECG signals is highlighted.
An online incremental orthogonal component analysis method for dimensionality reduction.
Zhu, Tao; Xu, Ye; Shen, Furao; Zhao, Jinxi
2017-01-01
In this paper, we introduce a fast linear dimensionality reduction method named incremental orthogonal component analysis (IOCA). IOCA is designed to automatically extract desired orthogonal components (OCs) in an online environment. The OCs and the low-dimensional representations of original data are obtained with only one pass through the entire dataset. Without solving matrix eigenproblem or matrix inversion problem, IOCA learns incrementally from continuous data stream with low computational cost. By proposing an adaptive threshold policy, IOCA is able to automatically determine the dimension of feature subspace. Meanwhile, the quality of the learned OCs is guaranteed. The analysis and experiments demonstrate that IOCA is simple, but efficient and effective. Copyright © 2016 Elsevier Ltd. All rights reserved.
Condensation of F-Actin by Dimensional Reduction
Bruinsma, Robijn; Christian, Cyron; Mueller, Kei; Bausch, Andreas; Wall, Wolfgang
2012-02-01
We present a Brownian Dynamics simulation of the equilibrium condensation of F-actin in the presence of linker molecules. The filaments are modeled as worm-like chains, using finite element analysis. At low linker concentrations, the systems forms a gel whose physical properties do not depend on the linker molecules. If the linker concentration is increased then for isotropic linkers only a single mode of condensation is encountered: bundle formation. If the linker molecules impose a preferential angle between F-actin filaments, then condensation takes place either into a either a hexatic or squaratic two-dimensional liquid crystal phase or into a heterogeneous cluster. Condensation is driven by competition between linker and filament entropy, which imposes dimensional reduction on the F-actin aggregate.
Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection
Hannachi, A.; Turner, A. G.
2013-04-01
It is known that the empirical orthogonal function method is unable to detect possible nonlinear structure in climate data. Here, isometric feature mapping (Isomap), as a tool for nonlinear dimensionality reduction, is applied to 1958-2001 ERA-40 sea-level pressure anomalies to study nonlinearity of the Asian summer monsoon intraseasonal variability. Using the leading two Isomap time series, the probability density function is shown to be bimodal. A two-dimensional bivariate Gaussian mixture model is then applied to identify the monsoon phases, the obtained regimes representing enhanced and suppressed phases, respectively. The relationship with the large-scale seasonal mean monsoon indicates that the frequency of monsoon regime occurrence is significantly perturbed in agreement with conceptual ideas, with preference for enhanced convection on intraseasonal time scales during large-scale strong monsoons. Trend analysis suggests a shift in concentration of monsoon convection, with less emphasis on South Asia and more on the East China Sea.
Multi-dimensional reduction using self-organizing map
Kim, Kho Pui; Yusof, Fadhilah; Daud, Zalina binti Mohd
2014-07-01
Self-Organising Map (SOM) is found to be a useful tool for climatological synoptic, analysis in extreme and rainfall pattern, cloud classification and climate change analysis. In data preprocessing for use in statistical downscaling, Principal Component Analysis (PCA) or empirical orthogonal function (EOF) analysis is used to select the mode criterion for the predictor and predictand fields for building a model. However, EOF contributes less total variance for most cases of which 70% to 90% of total population variance is accounted in the analysis. Therefore, SOM is proposed to obtain a nonlinear mapping for the preprocessing process. This study examines the dimension reduction of NCEP variable using SOM during the periods of November-December-January-February (NDJF). The NCEP data used is the 20 grids point atmospheric data for variable Sea Level Pressure (SLP). The result showed that SOM had extracted the high dimensional data onto a low dimensional representation.
Differentiation Theory over Infinite-Dimensional Banach Spaces
Directory of Open Access Journals (Sweden)
Claudio Asci
2016-01-01
Full Text Available We study, for any positive integer k and for any subset I of N⁎, the Banach space EI of the bounded real sequences xnn∈I and a measure over RI,B(I that generalizes the k-dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables’ formula for the integration of the measurable real functions on RI,B(I. This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.
Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods.
Harandi, Mehrtash; Salzmann, Mathieu; Hartley, Richard
2018-01-01
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.
Dimensionally Stable Structural Space Cable, Phase II
National Aeronautics and Space Administration — Jet Propulsion Laboratory (JPL) is involved in an ongoing effort to design and demonstrate a full-scale (30-32m diameter) Starshade engineering demonstrator that...
Dimensionally Stable Structural Space Cable, Phase I
National Aeronautics and Space Administration — In response to the need for an affordable exoplanet-analysis science mission, NASA has recently embarked on the ROSES Technology Development for Exoplanet Missions...
Mihaila, L.
2009-10-01
The two-loop relations between the running gluino-quark-squark coupling, the gluino and the quark mass defined in dimensional regularization (DREG) and dimensional reduction (DRED) in the framework of SUSY-QCD are presented. Furthermore, we verify with the help of these relations that the three-loop β-functions derived in the minimal subtraction scheme combined with DREG or DRED transform into each other. This result confirms the equivalence of the two schemes at the three-loop order, if applied to SUSY-QCD.
Embedding of attitude determination in n-dimensional spaces
Bar-Itzhack, Itzhack Y.; Markley, F. Landis
1988-01-01
The problem of attitude determination in n-dimensional spaces is addressed. The proper parameters are found, and it is shown that not all three-dimensional methods have useful extensions to higher dimensions. It is demonstrated that Rodriguez parameters are conveniently extendable to other dimensions. An algorithm for using these parameters in the general n-dimensional case is developed and tested with a four-dimensional example. The correct mathematical description of angular velocities is addressed, showing that angular velocity in n dimensions cannot be represented by a vector but rather by a tensor of the second rank. Only in three dimensions can the angular velocity be described by a vector.
Hyperspectral dimensionality reduction for biophysical variable statistical retrieval
Rivera-Caicedo, Juan Pablo; Verrelst, Jochem; Muñoz-Marí, Jordi; Camps-Valls, Gustau; Moreno, José
2017-10-01
Current and upcoming airborne and spaceborne imaging spectrometers lead to vast hyperspectral data streams. This scenario calls for automated and optimized spectral dimensionality reduction techniques to enable fast and efficient hyperspectral data processing, such as inferring vegetation properties. In preparation of next generation biophysical variable retrieval methods applicable to hyperspectral data, we present the evaluation of 11 dimensionality reduction (DR) methods in combination with advanced machine learning regression algorithms (MLRAs) for statistical variable retrieval. Two unique hyperspectral datasets were analyzed on the predictive power of DR + MLRA methods to retrieve leaf area index (LAI): (1) a simulated PROSAIL reflectance data (2101 bands), and (2) a field dataset from airborne HyMap data (125 bands). For the majority of MLRAs, applying first a DR method leads to superior retrieval accuracies and substantial gains in processing speed as opposed to using all bands into the regression algorithm. This was especially noticeable for the PROSAIL dataset: in the most extreme case, using the classical linear regression (LR), validation results RCV2 (RMSECV) improved from 0.06 (12.23) without a DR method to 0.93 (0.53) when combining it with a best performing DR method (i.e., CCA or OPLS). However, these DR methods no longer excelled when applied to noisy or real sensor data such as HyMap. Then the combination of kernel CCA (KCCA) with LR, or a classical PCA and PLS with a MLRA showed more robust performances (RCV2 of 0.93). Gaussian processes regression (GPR) uncertainty estimates revealed that LAI maps as trained in combination with a DR method can lead to lower uncertainties, as opposed to using all HyMap bands. The obtained results demonstrated that, in general, biophysical variable retrieval from hyperspectral data can largely benefit from dimensionality reduction in both accuracy and computational efficiency.
Dimensional reduction of Markov state models from renormalization group theory
Orioli, S.; Faccioli, P.
2016-09-01
Renormalization Group (RG) theory provides the theoretical framework to define rigorous effective theories, i.e., systematic low-resolution approximations of arbitrary microscopic models. Markov state models are shown to be rigorous effective theories for Molecular Dynamics (MD). Based on this fact, we use real space RG to vary the resolution of the stochastic model and define an algorithm for clustering microstates into macrostates. The result is a lower dimensional stochastic model which, by construction, provides the optimal coarse-grained Markovian representation of the system's relaxation kinetics. To illustrate and validate our theory, we analyze a number of test systems of increasing complexity, ranging from synthetic toy models to two realistic applications, built form all-atom MD simulations. The computational cost of computing the low-dimensional model remains affordable on a desktop computer even for thousands of microstates.
UV dimensional reduction to two from group valued momenta
Arzano, Michele; Nettel, Francisco
2017-04-01
We describe a new model of deformed relativistic kinematics based on the group manifold U (1) × SU (2) as a four-momentum space. We discuss the action of the Lorentz group on such space and illustrate the deformed composition law for the group-valued momenta. Due to the geometric structure of the group, the deformed kinematics is governed by two energy scales λ and κ. A relevant feature of the model is that it exhibits a running spectral dimension ds with the characteristic short distance reduction to ds = 2 found in most quantum gravity scenarios.
A Dimensionality Reduction Technique for Efficient Time Series Similarity Analysis
Wang, Qiang; Megalooikonomou, Vasileios
2008-01-01
We propose a dimensionality reduction technique for time series analysis that significantly improves the efficiency and accuracy of similarity searches. In contrast to piecewise constant approximation (PCA) techniques that approximate each time series with constant value segments, the proposed method--Piecewise Vector Quantized Approximation--uses the closest (based on a distance measure) codeword from a codebook of key-sequences to represent each segment. The new representation is symbolic and it allows for the application of text-based retrieval techniques into time series similarity analysis. Experiments on real and simulated datasets show that the proposed technique generally outperforms PCA techniques in clustering and similarity searches. PMID:18496587
A Dimensionality Reduction Technique for Efficient Time Series Similarity Analysis.
Wang, Qiang; Megalooikonomou, Vasileios
2008-03-01
We propose a dimensionality reduction technique for time series analysis that significantly improves the efficiency and accuracy of similarity searches. In contrast to piecewise constant approximation (PCA) techniques that approximate each time series with constant value segments, the proposed method--Piecewise Vector Quantized Approximation--uses the closest (based on a distance measure) codeword from a codebook of key-sequences to represent each segment. The new representation is symbolic and it allows for the application of text-based retrieval techniques into time series similarity analysis. Experiments on real and simulated datasets show that the proposed technique generally outperforms PCA techniques in clustering and similarity searches.
[Detecting interaction for quantitative trait by generalized multifactor dimensionality reduction].
Chen, Qing; Tang, Xun; Hu, Yong-Hua
2010-08-01
To introduce the application of generalized multifactor dimensionality reduction (GMDR) method for detecting interactions, especially gene-gene interactions for quantitative traits. Principles, basic steps as well as features of GMDR were discussed, illustrated with a practical research case. As an interaction analysis method, GMDR was model-free, available for studies on different outcome variables including continuous ones, and permitted adjustment for covariates to improve prediction accuracy. Evidences of its capacity had been supposed by research on different diseases, e.g. nicotine dependence. GMDR method was applicable to different types of samples and outcome variables, which was superior to other statistical approaches for continuous variables in some aspects.
Charged fluid distribution in higher dimensional spheroidal space-time
Indian Academy of Sciences (India)
analogue of Tolman's solutions. Recently, Ponce de Leon and Cruz [11] have con- sidered higher dimensional Schwarzschild space-time and studied the influence of the extra dimensions on the equilibrium configuration of stars. Vaidya and Tikekar. [12] have discussed spheroidal space-time and obtained an exact model ...
Simplicial models for trace spaces II: General higher dimensional automata
DEFF Research Database (Denmark)
Raussen, Martin
Higher Dimensional Automata (HDA) are topological models for the study of concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space...
Dimensionality reduction oriented toward the feature visualization for ischemia detection
Delgado-Tejos, Edilson; Perera Lluna, Alexandre; Vallverdú Ferrer, Montserrat; Caminal Magrans, Pere; Castellanos Dominguez, German
2009-01-01
An effective data representation methodology on high-dimension feature spaces is presented, which allows a better interpretation of subjacent physiological phenomena (namely, cardiac behavior related to cardiovascular diseases), and is based on search criteria over a feature set resulting in an increase in the detection capability of ischemic pathologies, but also connecting these features with the physiologic representation of the ECG. The proposed dimension reduction scheme consists o...
Dimensionality Reduction Applied to Spam Filtering using Bayesian Classifiers
Directory of Open Access Journals (Sweden)
Tiago A. Almeida
2011-04-01
Full Text Available In recent years, e-mail spam has become an increasingly important problem with a big economic impact in society. Fortunately, there are different approaches able to automatically detect and remove most of these messages, and the best-known ones are based on Bayesian decision theory. However, the most of these probabilistic approaches have the same difficulty: the high dimensionality of the feature space. Many term selection methods have been proposed in the literature. In this paper, we revise the most popular methods used as term selection techniques with seven different versions of Naive Bayes spam filters.
Investigation on methods for dimensionality reduction on hyperspectral image data
Robin T. Clarke; Vitor Haertel; Maciel Zortea
2005-01-01
In the present study, we propose a new simple approach to reduce the dimensionality in hyperspectral image data. The basic assumption consists in assuming that a pixel’s curve of spectral response, as defined in the spectral space by the recorded digital numbers (DNs) at the available spectral bands, can be segmented and each segment can be replaced by a smaller number of statistics, e.g., the mean and the variance, describing the main characteristics of a pixel’s spectral response. Results s...
Localization of a mobile laser scanner via dimensional reduction
Lehtola, Ville V.; Virtanen, Juho-Pekka; Vaaja, Matti T.; Hyyppä, Hannu; Nüchter, Andreas
2016-11-01
We extend the concept of intrinsic localization from a theoretical one-dimensional (1D) solution onto a 2D manifold that is embedded in a 3D space, and then recover the full six degrees of freedom for a mobile laser scanner with a simultaneous localization and mapping algorithm (SLAM). By intrinsic localization, we mean that no reference coordinate system, such as global navigation satellite system (GNSS), nor inertial measurement unit (IMU) are used. Experiments are conducted with a 2D laser scanner mounted on a rolling prototype platform, VILMA. The concept offers potential in being extendable to other wheeled platforms.
Rebhan, A.; van Nieuwenhuizen, P.; Wimmer, R.
2003-01-01
We show that the anomalous contribution to the central charge of the (1+1)-dimensional N=1 supersymmetric kink that is required for BPS saturation at the quantum level can be linked to an analogous term in the extra momentum operator of a (2+1)-dimensional kink domain wall with spontaneous parity violation and chiral domain wall fermions. In the quantization of the domain wall, BPS saturation is preserved by nonvanishing quantum corrections to the momentum density in the extra space dimension. Dimensional reduction from 2+1 to 1+1 dimensions preserves the unbroken N=1/2 supersymmetry and turns these parity-violating contributions into the anomaly of the central charge of the supersymmetric kink. On the other hand, standard dimensional regularization by dimensional reduction from 1 to (1- ɛ) spatial dimensions, which also preserves supersymmetry, obtains the anomaly from an evanescent counterterm. We identify the anomaly in the ordinary central charge as an anomalous contribution to the divergence of the conformal central-charge current.
Directory of Open Access Journals (Sweden)
Fubiao Feng
2017-03-01
Full Text Available Recently, graph embedding has drawn great attention for dimensionality reduction in hyperspectral imagery. For example, locality preserving projection (LPP utilizes typical Euclidean distance in a heat kernel to create an affinity matrix and projects the high-dimensional data into a lower-dimensional space. However, the Euclidean distance is not sufficiently correlated with intrinsic spectral variation of a material, which may result in inappropriate graph representation. In this work, a graph-based discriminant analysis with spectral similarity (denoted as GDA-SS measurement is proposed, which fully considers curves changing description among spectral bands. Experimental results based on real hyperspectral images demonstrate that the proposed method is superior to traditional methods, such as supervised LPP, and the state-of-the-art sparse graph-based discriminant analysis (SGDA.
Sparse kernel entropy component analysis for dimensionality reduction of neuroimaging data.
Jiang, Qikun; Shi, Jun
2014-01-01
The neuroimaging data typically has extremely high dimensions. Therefore, dimensionality reduction is commonly used to extract discriminative features. Kernel entropy component analysis (KECA) is a newly developed data transformation method, where the key idea is to preserve the most estimated Renyi entropy of the input space data set via a kernel-based estimator. Despite its good performance, KECA still suffers from the problem of low computational efficiency for large-scale data. In this paper, we proposed a sparse KECA (SKECA) algorithm with the recursive divide-and-conquer based solution, and then applied it to perform dimensionality reduction of neuroimaging data for classification of the Alzheimer's disease (AD). We compared the SKECA with KECA, principal component analysis (PCA), kernel PCA (KPCA) and sparse KPCA. The experimental results indicate that the proposed SKECA has most superior performance to all other algorithms when extracting discriminative features from neuroimaging data for AD classification.
Electromagnetic Fields and Waves in Fractional Dimensional Space
Zubair, Muhammad; Naqvi, Qaisar Abbas
2012-01-01
This book presents the concept of fractional dimensional space applied to the use of electromagnetic fields and waves. It provides demonstrates the advantages in studying the behavior of electromagnetic fields and waves in fractal media. The book presents novel fractional space generalization of the differential electromagnetic equations is provided as well as a new form of vector differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell's electromagnetic equations are worked out. The Laplace's, Poisson's and Helmholtz's
Das, Payel; Moll, Mark; Stamati, Hernán; Kavraki, Lydia E.; Clementi, Cecilia
2006-01-01
The definition of reaction coordinates for the characterization of a protein-folding reaction has long been a controversial issue, even for the “simple” case in which one single free-energy barrier separates the folded and unfolded ensemble. We propose a general approach to this problem to obtain a few collective coordinates by using nonlinear dimensionality reduction. We validate the usefulness of this method by characterizing the folding landscape associated with a coarse-grained protein mo...
Kernel Based Nonlinear Dimensionality Reduction and Classification for Genomic Microarray
Directory of Open Access Journals (Sweden)
Lan Shu
2008-07-01
Full Text Available Genomic microarrays are powerful research tools in bioinformatics and modern medicinal research because they enable massively-parallel assays and simultaneous monitoring of thousands of gene expression of biological samples. However, a simple microarray experiment often leads to very high-dimensional data and a huge amount of information, the vast amount of data challenges researchers into extracting the important features and reducing the high dimensionality. In this paper, a nonlinear dimensionality reduction kernel method based locally linear embedding(LLE is proposed, and fuzzy K-nearest neighbors algorithm which denoises datasets will be introduced as a replacement to the classical LLEÃ¢Â€Â™s KNN algorithm. In addition, kernel method based support vector machine (SVM will be used to classify genomic microarray data sets in this paper. We demonstrate the application of the techniques to two published DNA microarray data sets. The experimental results confirm the superiority and high success rates of the presented method.
Comparison of dimensionality reduction methods for wood surface inspection
Niskanen, Matti; Silven, Olli
2003-04-01
Dimensionality reduction methods for visualization map the original high-dimensional data typically into two dimensions. Mapping preserves the important information of the data, and in order to be useful, fulfils the needs of a human observer. We have proposed a self-organizing map (SOM)- based approach for visual surface inspection. The method provides the advantages of unsupervised learning and an intuitive user interface that allows one to very easily set and tune the class boundaries based on observations made on visualization, for example, to adapt to changing conditions or material. There are, however, some problems with a SOM. It does not address the true distances between data, and it has a tendency to ignore rare samples in the training set at the expense of more accurate representation of common samples. In this paper, some alternative methods for a SOM are evaluated. These methods, PCA, MDS, LLE, ISOMAP, and GTM, are used to reduce dimensionality in order to visualize the data. Their principal differences are discussed and performances quantitatively evaluated in a few special classification cases, such as in wood inspection using centile features. For the test material experimented with, SOM and GTM outperform the others when classification performance is considered. For data mining kinds of applications, ISOMAP and LLE appear to be more promising methods.
Dimensionality Reduction for Damage Detection in Engineering Structures
Prada, Miguel A.; Domínguez, Manuel; Barrientos, Pablo; García, Sergio
2012-10-01
The detection of damages in engineering structures by means of the changes in their vibration response is called structural health monitoring (SHM). It is a promising field but presents fundamental challenges. Accurate theoretical models of the structure are generally unfeasible, so data-based approaches are required. Indeed, only data from the undamaged condition are usually available, so the approach needs to be framed as novelty detection. Data are acquired from a network of sensors to measure local changes in the operating condition of the structures. In order to distinguish changes produced by damages from those caused by the environmental conditions, several physically meaningful features have been proposed, most of them in the frequency domain. Nevertheless, multiple measurement locations and the absence of a principled criterion to select among the potentially damage-sensitive features contribute to increase data dimensionality. Since high dimensionality affects the effectiveness of damage detection, we evaluate the effect of a dimensionality reduction approach in the diagnostic accuracy of damage detection.
Multispectral face recognition using non linear dimensionality reduction
Akhloufi, Moulay A.; Bendada, Abdelhakim; Batsale, Jean-Christophe
2009-05-01
Face recognition in the infrared spectrum has attracted a lot of interest in recent years. Many of the techniques used in infrared are based on their visible counterpart, especially linear techniques like PCA (Principal Component Analysis) and LDA (Linear Discriminant Analysis). In this work, we introduce non linear dimensionality reduction approaches for multispectral face recognition. For this purpose, the following techniques were developed: global non linear techniques (Kernel-PCA, Kernel-LDA) and local non linear techniques (Local Linear Embedding, Locality Preserving Projection). The performances of these techniques were compared to classical linear techniques for face recognition like PCA and LDA. Two multispectral face recognition databases were used in our experiments: Equinox Face Recognition Database and Laval University Database. Equinox database contains images in the Visible, Short, Mid and Long waves infrared spectrums. Laval database contains images in the Visible, Near, Mid and Long waves infrared spectrums with variations in time and metabolic activity of the subjects. The obtained results are interesting and show the increase in recognition performance using local non linear dimensionality reduction techniques for infrared face recognition, particularly in near and short wave infrared spectrums.
DISEASE CLASSIFICATION AND PREDICTION VIA SEMI-SUPERVISED DIMENSIONALITY REDUCTION.
Batmanghelich, Kayhan N; Ye, Dong H; Pohl, Kilian M; Taskar, Ben; Davatzikos, Christos
2011-01-01
We present a new semi-supervised algorithm for dimensionality reduction which exploits information of unlabeled data in order to improve the accuracy of image-based disease classification based on medical images. We perform dimensionality reduction by adopting the formalism of constrained matrix decomposition of [1] to semi-supervised learning. In addition, we add a new regularization term to the objective function to better captur the affinity between labeled and unlabeled data. We apply our method to a data set consisting of medical scans of subjects classified as Normal Control (CN) and Alzheimer (AD). The unlabeled data are scans of subjects diagnosed with Mild Cognitive Impairment (MCI), which are at high risk to develop AD in the future. We measure the accuracy of our algorithm in classifying scans as AD and NC. In addition, we use the classifier to predict which subjects with MCI will converge to AD and compare those results to the diagnosis given at later follow ups. The experiments highlight that unlabeled data greatly improves the accuracy of our classifier.
Dimensionality reduction for the analysis of brain oscillations.
Haufe, Stefan; Dähne, Sven; Nikulin, Vadim V
2014-11-01
Neuronal oscillations have been shown to be associated with perceptual, motor and cognitive brain operations. While complex spatio-temporal dynamics are a hallmark of neuronal oscillations, they also represent a formidable challenge for the proper extraction and quantification of oscillatory activity with non-invasive recording techniques such as EEG and MEG. In order to facilitate the study of neuronal oscillations we present a general-purpose pre-processing approach, which can be applied for a wide range of analyses including but not restricted to inverse modeling and multivariate single-trial classification. The idea is to use dimensionality reduction with spatio-spectral decomposition (SSD) instead of the commonly and almost exclusively used principal component analysis (PCA). The key advantage of SSD lies in selecting components explaining oscillations-related variance instead of just any variance as in the case of PCA. For the validation of SSD pre-processing we performed extensive simulations with different inverse modeling algorithms and signal-to-noise ratios. In all these simulations SSD invariably outperformed PCA often by a large margin. Moreover, using a database of multichannel EEG recordings from 80 subjects we show that pre-processing with SSD significantly increases the performance of single-trial classification of imagined movements, compared to the classification with PCA pre-processing or without any dimensionality reduction. Our simulations and analysis of real EEG experiments show that, while not being supervised, the SSD algorithm is capable of extracting components primarily relating to the signal of interest often using as little as 20% of the data variance, instead of > 90% variance as in case of PCA. Given its ease of use, absence of supervision, and capability to efficiently reduce the dimensionality of multivariate EEG/MEG data, we advocate the application of SSD pre-processing for the analysis of spontaneous and induced neuronal
Object-based Dimensionality Reduction in Land Surface Phenology Classification
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Brian E. Bunker
2016-11-01
Full Text Available Unsupervised classification or clustering of multi-decadal land surface phenology provides a spatio-temporal synopsis of natural and agricultural vegetation response to environmental variability and anthropogenic activities. Notwithstanding the detailed temporal information available in calibrated bi-monthly normalized difference vegetation index (NDVI and comparable time series, typical pre-classification workflows average a pixel’s bi-monthly index within the larger multi-decadal time series. While this process is one practical way to reduce the dimensionality of time series with many hundreds of image epochs, it effectively dampens temporal variation from both intra and inter-annual observations related to land surface phenology. Through a novel application of object-based segmentation aimed at spatial (not temporal dimensionality reduction, all 294 image epochs from a Moderate Resolution Imaging Spectroradiometer (MODIS bi-monthly NDVI time series covering the northern Fertile Crescent were retained (in homogenous landscape units as unsupervised classification inputs. Given the inherent challenges of in situ or manual image interpretation of land surface phenology classes, a cluster validation approach based on transformed divergence enabled comparison between traditional and novel techniques. Improved intra-annual contrast was clearly manifest in rain-fed agriculture and inter-annual trajectories showed increased cluster cohesion, reducing the overall number of classes identified in the Fertile Crescent study area from 24 to 10. Given careful segmentation parameters, this spatial dimensionality reduction technique augments the value of unsupervised learning to generate homogeneous land surface phenology units. By combining recent scalable computational approaches to image segmentation, future work can pursue new global land surface phenology products based on the high temporal resolution signatures of vegetation index time series.