Eigenvalues properties of terms correspondences matrix
Bondarchuk, Dmitry; Timofeeva, Galina
2016-12-01
Vector model representations of text documents are widely used in the intelligent search. In this approach a collection of documents is represented in the form of the term-document matrix, reflecting the frequency of terms. In the latent semantic analysis the dimension of the vector space is reduced by the singular value decomposition of the term-document matrix. Authors use a matrix of terms correspondences, reflecting the relationship between the terms, to allocate a semantic core and to obtain more simple presentation of the documents. With this approach, reducing the number of terms is based on the orthogonal decomposition of the matrix of terms correspondences. Properties of singular values of the term-document matrix and eigenvalues of the matrix of terms correspondences are studied in the case when documents differ substantially in length.
On the eigenvalue and eigenvector derivatives of a general matrix
Juang, Jer-Nan; Lim, Kyong B.
1987-01-01
The existence of differentiable eigenvalues and eigenvectors for a general matrix is addressed. The eigenspace which contains differentiable eigenvectors is determined and computed by using the concept of subspace intersection in conjunction with the singular value decomposition algorithm. The differentiable eigenvectors associated with repeated eigenvalues should be simultaneously the eigenvectors of the general matrix and its corresponding sensitivity matrix. Furthermore, the derivatives for differentiable eigenvectors associated with repeated eigenvalues can be computed using higher order derivatives of the matrix, whereas the corresponding eigenvalue derivatives are the eigenvalues of the sensitivity matrix.
Parallel Sparse Matrix - Vector Product
DEFF Research Database (Denmark)
Alexandersen, Joe; Lazarov, Boyan Stefanov; Dammann, Bernd
This technical report contains a case study of a sparse matrix-vector product routine, implemented for parallel execution on a compute cluster with both pure MPI and hybrid MPI-OpenMP solutions. C++ classes for sparse data types were developed and the report shows how these class can be used...
Sparse matrix test collections
Energy Technology Data Exchange (ETDEWEB)
Duff, I.
1996-12-31
This workshop will discuss plans for coordinating and developing sets of test matrices for the comparison and testing of sparse linear algebra software. We will talk of plans for the next release (Release 2) of the Harwell-Boeing Collection and recent work on improving the accessibility of this Collection and others through the World Wide Web. There will only be three talks of about 15 to 20 minutes followed by a discussion from the floor.
Improving the accuracy of computed matrix eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J J
1980-08-01
A computational method is described for improving the accuracy of a given eigenvalue and its associated eigenvector, arrived at through a computation in a lower precision. The method to be described will increase the accuracy of the pair and do so at a relatively low cost. The technique used is similar to iterative refinement for the solution of a linear system; that is, through the factorization from the low-precision computation, an iterative algorithm is applied to increase the accuracy of the eigenpair. Extended precision arithmetic is used at critical points in the algorithm. The iterative algorithm requires O(n/sup 2/) operations for each iteration.
Energy Technology Data Exchange (ETDEWEB)
Gene Golub; Kwok Ko
2009-03-30
The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.
Derivatives of eigenvalues and eigenvectors of a general complex matrix
Murthy, Durbha V.; Haftka, Raphael T.
1988-01-01
A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.
Matrix perturbations: bounding and computing eigenvalues
Reis da Silva, R.J.
2011-01-01
Despite the somewhat negative connotation of the word, not every perturbation is a bad perturbation. In fact, while disturbing the matrix entries, many perturbations still preserve useful properties such as the orthonormality of the basis of eigenvectors or the Hermicity of the original matrix. In
Ecosystem stability and the distribution of community matrix eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Harte, J.
1977-07-01
A definition or measure of ecological stability is introduced which is advantageous in that it may be empirically accessible and in that it relates to practical concerns in situations of environmental stress. It is shown how this measure can be analyzed, for an arbitrary system described by a community matrix, by model-independent mathematical methods. The relationship between the stability of any such system and both the distribution of community matrix eigenvalues and the pattern of pathway linkages is discussed.
Recursive Approach in Sparse Matrix LU Factorization
Directory of Open Access Journals (Sweden)
Jack Dongarra
2001-01-01
Full Text Available This paper describes a recursive method for the LU factorization of sparse matrices. The recursive formulation of common linear algebra codes has been proven very successful in dense matrix computations. An extension of the recursive technique for sparse matrices is presented. Performance results given here show that the recursive approach may perform comparable to leading software packages for sparse matrix factorization in terms of execution time, memory usage, and error estimates of the solution.
An Implementation and Evaluation of the AMLS Method for SparseEigenvalue Problems
Energy Technology Data Exchange (ETDEWEB)
Gao, Weiguo; Li, Xiaoye S.; Yang, Chao; Bai, Zhaojun
2006-02-14
We describe an efficient implementation and present aperformance study of an algebraic multilevel sub-structuring (AMLS)method for sparse eigenvalue problems. We assess the time and memoryrequirements associated with the key steps of the algorithm, and compareitwith the shift-and-invert Lanczos algorithm in computational cost. Oureigenvalue problems come from two very different application areas: theaccelerator cavity design and the normal mode vibrational analysis of thepolyethylene particles. We show that the AMLS method, when implementedcarefully, is very competitive with the traditional method in broadapplication areas, especially when large numbers of eigenvalues aresought.
Finding Nonoverlapping Substructures of a Sparse Matrix
Energy Technology Data Exchange (ETDEWEB)
Pinar, Ali; Vassilevska, Virginia
2005-08-11
Many applications of scientific computing rely on computations on sparse matrices. The design of efficient implementations of sparse matrix kernels is crucial for the overall efficiency of these applications. Due to the high compute-to-memory ratio and irregular memory access patterns, the performance of sparse matrix kernels is often far away from the peak performance on a modern processor. Alternative data structures have been proposed, which split the original matrix A into A{sub d} and A{sub s}, so that A{sub d} contains all dense blocks of a specified size in the matrix, and A{sub s} contains the remaining entries. This enables the use of dense matrix kernels on the entries of A{sub d} producing better memory performance. In this work, we study the problem of finding a maximum number of nonoverlapping dense blocks in a sparse matrix, which is previously not studied in the sparse matrix community. We show that the maximum nonoverlapping dense blocks problem is NP-complete by using a reduction from the maximum independent set problem on cubic planar graphs. We also propose a 2/3-approximation algorithm that runs in linear time in the number of nonzeros in the matrix. This extended abstract focuses on our results for 2x2 dense blocks. However we show that our results can be generalized to arbitrary sized dense blocks, and many other oriented substructures, which can be exploited to improve the memory performance of sparse matrix operations.
From gap probabilities in random matrix theory to eigenvalue expansions
Bothner, Thomas
2016-02-01
We present a method to derive asymptotics of eigenvalues for trace-class integral operators K :{L}2(J;{{d}}λ )\\circlearrowleft , acting on a single interval J\\subset {{R}}, which belongs to the ring of integrable operators (Its et al 1990 Int. J. Mod. Phys. B 4 1003-37 ). Our emphasis lies on the behavior of the spectrum \\{{λ }i(J)\\}{}i=0∞ of K as | J| \\to ∞ and i is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant {det}(I-γ K){| }{L2(J)} as | J| \\to ∞ and γ \\uparrow 1 in a Stokes type scaling regime. Concrete asymptotic formulæ are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory. Dedicated to Percy Deift and Craig Tracy on the occasion of their 70th birthdays.
Huang, Tsung-Ming; Lin, Wen-Wei; Tian, Heng; Chen, Guan-Hua
2018-03-01
Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.
Better Size Estimation for Sparse Matrix Products
DEFF Research Database (Denmark)
Amossen, Rasmus Resen; Campagna, Andrea; Pagh, Rasmus
2010-01-01
We consider the problem of doing fast and reliable estimation of the number of non-zero entries in a sparse Boolean matrix product. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1 ± ε approximation (with small probability of error) in expected...
Multi-threaded Sparse Matrix Sparse Matrix Multiplication for Many-Core and GPU Architectures.
Energy Technology Data Exchange (ETDEWEB)
Deveci, Mehmet [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Trott, Christian Robert [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Rajamanickam, Sivasankaran [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2018-01-01
Sparse Matrix-Matrix multiplication is a key kernel that has applications in several domains such as scientific computing and graph analysis. Several algorithms have been studied in the past for this foundational kernel. In this paper, we develop parallel algorithms for sparse matrix- matrix multiplication with a focus on performance portability across different high performance computing architectures. The performance of these algorithms depend on the data structures used in them. We compare different types of accumulators in these algorithms and demonstrate the performance difference between these data structures. Furthermore, we develop a meta-algorithm, kkSpGEMM, to choose the right algorithm and data structure based on the characteristics of the problem. We show performance comparisons on three architectures and demonstrate the need for the community to develop two phase sparse matrix-matrix multiplication implementations for efficient reuse of the data structures involved.
On the eigenvalue and eigenvector derivatives of a non-defective matrix
Juang, Jer-Nan; Ghaemmaghami, Peiman; Lim, Kyong Been
1988-01-01
A novel approach is introduced to address the problem of existence of differentiable eigenvectors for a nondefective matrix which may have repeated eigenvalues. The existence of eigenvector derivatives for a unique set of continuous eigenvectors corresponding to a repeated eigenvalue is rigorously established for nondefective and analytic matrices. A numerically implementable method is then developed to compute the differentiable eigenvectors associated with repeated eigenvalues. The solutions of eigenvalue and eigenvector derivatives for repeated eigenvalues are then derived. An example is given to illustrate the validity of formulations developed in this paper.
SparseM: A Sparse Matrix Package for R *
Directory of Open Access Journals (Sweden)
Roger Koenker
2003-02-01
Full Text Available SparseM provides some basic R functionality for linear algebra with sparse matrices. Use of the package is illustrated by a family of linear model fitting functions that implement least squares methods for problems with sparse design matrices. Significant performance improvements in memory utilization and computational speed are possible for applications involving large sparse matrices.
Derivatives of eigenvalues and eigenvectors for a general matrix
Rudisill, C. S.
1974-01-01
Expressions are obtained for the derivatives of the eigenvalues and eigenvectors which are expressions of only one left-hand and one right-hand eigenvector. The approach described makes use of a Choleski decomposition or some other decomposition method. The method may be extended to find any order of derivative of the eigenvalue and eigenvector. The expressions obtained for finding the derivatives of eigenvalues and eigenvectors for nonself-adjoint systems may be applied to self-adjoint systems.
Accounting for Sampling Error in Genetic Eigenvalues Using Random Matrix Theory.
Sztepanacz, Jacqueline L; Blows, Mark W
2017-07-01
The distribution of genetic variance in multivariate phenotypes is characterized by the empirical spectral distribution of the eigenvalues of the genetic covariance matrix. Empirical estimates of genetic eigenvalues from random effects linear models are known to be overdispersed by sampling error, where large eigenvalues are biased upward, and small eigenvalues are biased downward. The overdispersion of the leading eigenvalues of sample covariance matrices have been demonstrated to conform to the Tracy-Widom (TW) distribution. Here we show that genetic eigenvalues estimated using restricted maximum likelihood (REML) in a multivariate random effects model with an unconstrained genetic covariance structure will also conform to the TW distribution after empirical scaling and centering. However, where estimation procedures using either REML or MCMC impose boundary constraints, the resulting genetic eigenvalues tend not be TW distributed. We show how using confidence intervals from sampling distributions of genetic eigenvalues without reference to the TW distribution is insufficient protection against mistaking sampling error as genetic variance, particularly when eigenvalues are small. By scaling such sampling distributions to the appropriate TW distribution, the critical value of the TW statistic can be used to determine if the magnitude of a genetic eigenvalue exceeds the sampling error for each eigenvalue in the spectral distribution of a given genetic covariance matrix. Copyright © 2017 by the Genetics Society of America.
Bounds for departure from normality and the Frobenius norm of matrix eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Lee, S.L.
1994-12-01
New lower and upper bounds for the departure from normality and the Frobenius norm of the eigenvalues of a matrix axe given. The significant properties of these bounds axe also described. For example, the upper bound for matrix eigenvalues improves upon the one derived by Kress, de Vries and Wegmann in [Lin. Alg. Appl., 8 (1974), pp. 109-120]. The upper bound for departure from normality is sharp for any matrix whose eigenvalues are collinear in the complex plane. Moreover, the latter formula is a practical estimate that costs (at most) 2m multiplications, where m is the number of nonzeros in the matrix. In terms of applications, the results can be used to bound from above the sensitivity of eigenvalues to matrix perturbations or bound from below the condition number of the eigenbasis of a matrix.
Comparative Analysis of Sparse Matrix Algorithms For Information Retrieval
Directory of Open Access Journals (Sweden)
Nazli Goharian
2003-02-01
Full Text Available We evaluate and compare the storage efficiency of different sparse matrix storage formats as index structure for text collection and their corresponding sparse matrixvector multiplication algorithm to perform query processing in information retrieval (IR application. We show the results of our implementations for several sparse matrix algorithms such as Coordinate Storage (COO, Compressed Sparse Column (CSC, Compressed Sparse Row (CSR, and Block Sparse Row (BSR sparse matrix algorithms, using a standard text collection. Evaluation is based on the storage space requirement for each indexing structure and the efficiency of the query-processing algorithm. Our results demonstrate that CSR is more efficient in terms of storage space requirement and query processing timing over the other sparse matrix algorithms for Information Retrieval application. Furthermore, we experimentally evaluate the mapping of various existing index compression techniques used to compress index in information retrieval systems (IR on Compressed Sparse Row Information Retrieval (CSR IR.
Analytic approximation to the largest eigenvalue distribution of a white Wishart matrix
CSIR Research Space (South Africa)
Vlok, JD
2012-08-14
Full Text Available Eigenvalue distributions of Wishart matrices are given in the literature as functions or distributions defined in terms of matrix arguments requiring numerical evaluation. As a result the relationship between parameter values and statistics...
On the eigenvalue-eigenvector method for solution of the stationary discrete matrix Riccati equation
DEFF Research Database (Denmark)
Michelsen, Michael Locht
1979-01-01
The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation.......The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation....
Predicting structure in nonsymmetric sparse matrix factorizations
Energy Technology Data Exchange (ETDEWEB)
Gilbert, J.R. (Xerox Palo Alto Research Center, CA (United States)); Ng, E.G. (Oak Ridge National Lab., TN (United States))
1992-10-01
Many computations on sparse matrices have a phase that predicts the nonzero structure of the output, followed by a phase that actually performs the numerical computation. We study structure prediction for computations that involve nonsymmetric row and column permutations and nonsymmetric or non-square matrices. Our tools are bipartite graphs, matchings, and alternating paths. Our main new result concerns LU factorization with partial pivoting. We show that if a square matrix A has the strong Hall property (i.e., is fully indecomposable) then an upper bound due to George and Ng on the nonzero structure of L + U is as tight as possible. To show this, we prove a crucial result about alternating paths in strong Hall graphs. The alternating-paths theorem seems to be of independent interest: it can also be used to prove related results about structure prediction for QR factorization that are due to Coleman, Edenbrandt, Gilbert, Hare, Johnson, Olesky, Pothen, and van den Driessche.
JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices
Bollhöfer, Matthias; Notay, Yvan
2007-12-01
A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi-Davidson method with efficient multilevel incomplete LU (ILU) preconditioning. Key features are modest memory requirements and robust convergence to accurate solutions. Parameters needed for incomplete LU preconditioning are automatically computed and may be updated at run time depending on the convergence pattern. The software is easy to use by non-experts and its top level routines are written in FORTRAN 77. Its potentialities are demonstrated on a few applications taken from computational physics. Program summaryProgram title: JADAMILU Catalogue identifier: ADZT_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZT_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 101 359 No. of bytes in distributed program, including test data, etc.: 7 493 144 Distribution format: tar.gz Programming language: Fortran 77 Computer: Intel or AMD with g77 and pgf; Intel EM64T or Itanium with ifort; AMD Opteron with g77, pgf and ifort; Power (IBM) with xlf90. Operating system: Linux, AIX RAM: problem dependent Word size: real:8; integer: 4 or 8, according to user's choice Classification: 4.8 Nature of problem: Any physical problem requiring the computation of a few eigenvalues of a symmetric matrix. Solution method: Jacobi-Davidson combined with multilevel ILU preconditioning. Additional comments: We supply binaries rather than source code because JADAMILU uses the following external packages: MC64. This software is copyrighted software and not freely available. COPYRIGHT (c) 1999
Second SIAM conference on sparse matrices: Abstracts. Final technical report
Energy Technology Data Exchange (ETDEWEB)
NONE
1996-12-31
This report contains abstracts on the following topics: invited and long presentations (IP1 & LP1); sparse matrix reordering & graph theory I; sparse matrix tools & environments I; eigenvalue computations I; iterative methods & acceleration techniques I; applications I; parallel algorithms I; sparse matrix reordering & graphy theory II; sparse matrix tool & environments II; least squares & optimization I; iterative methods & acceleration techniques II; applications II; eigenvalue computations II; least squares & optimization II; parallel algorithms II; sparse direct methods; iterative methods & acceleration techniques III; eigenvalue computations III; and sparse matrix reordering & graph theory III.
Overlap Dirac operator, eigenvalues and random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Edwards, Robert G.; Heller, Urs M.; Kiskis, Joe; Narayanan, Rajamani
2000-03-01
The properties of the spectrum of the overlap Dirac operator and their relation to random matrix theory are studied. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested.
Energy Technology Data Exchange (ETDEWEB)
Theiler, James P [Los Alamos National Laboratory; Cao, Guangzhi [PURDUE UNIV; Bouman, Charles A [PURDUE UNIV
2009-01-01
Many detection algorithms in hyperspectral image analysis, from well-characterized gaseous and solid targets to deliberately uncharacterized anomalies and anomlous changes, depend on accurately estimating the covariance matrix of the background. In practice, the background covariance is estimated from samples in the image, and imprecision in this estimate can lead to a loss of detection power. In this paper, we describe the sparse matrix transform (SMT) and investigate its utility for estimating the covariance matrix from a limited number of samples. The SMT is formed by a product of pairwise coordinate (Givens) rotations, which can be efficiently estimated using greedy optimization. Experiments on hyperspectral data show that the estimate accurately reproduces even small eigenvalues and eigenvectors. In particular, we find that using the SMT to estimate the covariance matrix used in the adaptive matched filter leads to consistently higher signal-to-noise ratios.
A framework for general sparse matrix-matrix multiplication on GPUs and heterogeneous processors
DEFF Research Database (Denmark)
Liu, Weifeng; Vinter, Brian
2015-01-01
General sparse matrix-matrix multiplication (SpGEMM) is a fundamental building block for numerous applications such as algebraic multigrid method (AMG), breadth first search and shortest path problem. Compared to other sparse BLAS routines, an efficient parallel SpGEMM implementation has to handle...
An Efficient GPU General Sparse Matrix-Matrix Multiplication for Irregular Data
DEFF Research Database (Denmark)
Liu, Weifeng; Vinter, Brian
2014-01-01
General sparse matrix-matrix multiplication (SpGEMM) is a fundamental building block for numerous applications such as algebraic multigrid method, breadth first search and shortest path problem. Compared to other sparse BLAS routines, an efficient parallel SpGEMM algorithm has to handle extra...
Use of Wishart Prior and Simple Extensions for Sparse Precision Matrix Estimation.
Directory of Open Access Journals (Sweden)
Markku Kuismin
Full Text Available A conjugate Wishart prior is used to present a simple and rapid procedure for computing the analytic posterior (mode and uncertainty of the precision matrix elements of a Gaussian distribution. An interpretation of covariance estimates in terms of eigenvalues is presented, along with a simple decision-rule step to improve the performance of the estimation of sparse precision matrices and associated graphs. In this, elements of the estimated precision matrix that are zero or near zero can be detected and shrunk to zero. Simulated data sets are used to compare posterior estimation with decision-rule with two other Wishart-based approaches and with graphical lasso. Furthermore, an empirical Bayes procedure is used to select prior hyperparameters in high dimensional cases with extension to sparsity.
Vector sparse representation of color image using quaternion matrix analysis.
Xu, Yi; Yu, Licheng; Xu, Hongteng; Zhang, Hao; Nguyen, Truong
2015-04-01
Traditional sparse image models treat color image pixel as a scalar, which represents color channels separately or concatenate color channels as a monochrome image. In this paper, we propose a vector sparse representation model for color images using quaternion matrix analysis. As a new tool for color image representation, its potential applications in several image-processing tasks are presented, including color image reconstruction, denoising, inpainting, and super-resolution. The proposed model represents the color image as a quaternion matrix, where a quaternion-based dictionary learning algorithm is presented using the K-quaternion singular value decomposition (QSVD) (generalized K-means clustering for QSVD) method. It conducts the sparse basis selection in quaternion space, which uniformly transforms the channel images to an orthogonal color space. In this new color space, it is significant that the inherent color structures can be completely preserved during vector reconstruction. Moreover, the proposed sparse model is more efficient comparing with the current sparse models for image restoration tasks due to lower redundancy between the atoms of different color channels. The experimental results demonstrate that the proposed sparse image model avoids the hue bias issue successfully and shows its potential as a general and powerful tool in color image analysis and processing domain.
Enforced Sparse Non-Negative Matrix Factorization
2016-01-23
mixture of topics constitutes a document. Other common methods for topic modeling include the following: latent semantic analysis (LSA) [1...probabilistic latent semantic analysis (PLSA) [2], and term frequency- inverse document frequency (TF-IDF) [3] analysis. More recently, non-negative matrix...3, pp. 993–1022, 2003. [2] T. Hofmann, “Probabilistic latent semantic indexing,” in Proceedings of the 22nd Annual International ACM SIGIR
Sparse Non-negative Matrix Factor 2-D Deconvolution
DEFF Research Database (Denmark)
Mørup, Morten; Schmidt, Mikkel N.
2006-01-01
We introduce the non-negative matrix factor 2-D deconvolution (NMF2D) model, which decomposes a matrix into a 2-dimensional convolution of two factor matrices. This model is an extension of the non-negative matrix factor deconvolution (NMFD) recently introduced by Smaragdis (2004). We derive...... and prove the convergence of two algorithms for NMF2D based on minimizing the squared error and the Kullback-Leibler divergence respectively. Next, we introduce a sparse non-negative matrix factor 2-D deconvolution model that gives easy interpretable decompositions and devise two algorithms for computing...
Distribution of Schmidt-like eigenvalues for Gaussian ensembles of the random matrix theory
Pato, Mauricio P.; Oshanin, Gleb
2013-03-01
We study the probability distribution function P(β)n(w) of the Schmidt-like random variable w = x21/(∑j = 1nx2j/n), where xj, (j = 1, 2, …, n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P(β)n(w) converges to the Marčenko-Pastur form, i.e. is defined as P_{n}^{( \\beta )}(w) \\sim \\sqrt{(4 - w)/w} for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P(β = 2)n(w) which are valid for arbitrary n and analyse their behaviour.
Performance Aspects of Sparse Matrix-Vector Multiplication
Directory of Open Access Journals (Sweden)
I. Šimeček
2006-01-01
Full Text Available Sparse matrix-vector multiplication (shortly SpM×V is an important building block in algorithms solving sparse systems of linear equations, e.g., FEM. Due to matrix sparsity, the memory access patterns are irregular and utilization of the cache can suffer from low spatial or temporal locality. Approaches to improve the performance of SpM×V are based on matrix reordering and register blocking [1, 2], sometimes combined with software-pipelining [3]. Due to its overhead, register blocking achieves good speedups only for a large number of executions of SpM×V with the same matrix A.We have investigated the impact of two simple SW transformation techniques (software-pipelining and loop unrolling on the performance of SpM×V, and have compared it with several implementation modifications aimed at reducing computational and memory complexity and improving the spatial locality. We investigate performance gains of these modifications on four CPU platforms.
Lower bounds for sparse matrix vector multiplication on hypercubic networks
Directory of Open Access Journals (Sweden)
Giovanni Manzini
1998-12-01
Full Text Available In this paper we consider the problem of computing on a local memory machine the product y = Ax,where A is a random n×n sparse matrix with Θ(n nonzero elements. To study the average case communication cost of this problem, we introduce four different probability measures on the set of sparse matrices. We prove that on most local memory machines with p processors, this computation requires Ω((n/p log p time on the average. We prove that the same lower bound also holds, in the worst case, for matrices with only 2n or 3n nonzero elements.
Finite sparse matrix techniques. [Solution of linear systems Ax = b
Energy Technology Data Exchange (ETDEWEB)
Bareiss, E.H.; de Peniza, C.M.
1979-11-01
A unified theory of finite sparse matrix techniques based on a literature search and new results is presented. It is intended to aid in computational work and symbolic manipulation of large sparse systems of linear equations. The theory relies on the bijection property of bipartite graph and rectangular Boolean matrix representation. The concept of perfect elimination matrices is extended from the classification under similarity transformations to that under equivalence transformations with permutation matrices. The reducibility problem is treated with a new and simpler proof than found in the literature. A number of useful algorithms are described. The minimum deficiency algorithms are extended to the new classification, where the latter required a different technique of proof. 13 figures, 7 tables.
Moments of the transmission eigenvalues, proper delay times and random matrix theory II
Mezzadri, F.; Simm, N. J.
2012-05-01
We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-Büttiker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of random matrix theory. The starting points are the finite-n formulae that we recently discovered [F. Mezzadri and N. J. Simm, "Moments of the transmission eigenvalues, proper delay times and random matrix theory," J. Math. Phys. 52, 103511 (2011)], 10.1063/1.3644378. Our analysis includes all the symmetry classes β ∈ {1, 2, 4}; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer ["Riemannian symmetric superspaces and their origin in random-matrix theory," J. Math. Phys. 37(10), 4986 (1996)], 10.1063/1.531675 and Altland and Zirnbauer ["Random matrix theory of a chaotic Andreev quantum dot," Phys. Rev. Lett. 76(18), 3420 (1996), 10.1103/PhysRevLett.76.3420; Altland and Zirnbauer "Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures," Phys. Rev. B 55(2), 1142 (1997)], 10.1103/PhysRevB.55.1142. Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. ["Full counting statistics of chaotic cavities from classical action correlations," J. Phys. A: Math. Theor. 41(36), 365102 (2008)], 10.1088/1751-8113/41/36/365102 and Berkolaiko and Kuipers ["Moments of the Wigner delay times," J. Phys. A: Math. Theor. 43(3), 035101 (2010), 10.1088/1751-8113/43/3/035101; Berkolaiko and Kuipers "Transport moments beyond the leading order," New J. Phys. 13(6), 063020 (2011)], 10.1088/1367-2630/13/6/063020. Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion
Acceleration of Sparse Matrix-Vector Multiplication by Region Traversal
Directory of Open Access Journals (Sweden)
I. Šimeček
2008-01-01
Full Text Available Sparse matrix-vector multiplication (shortly SpM×V is one of most common subroutines in numerical linear algebra. The problem is that the memory access patterns during SpM×V are irregular, and utilization of the cache can suffer from low spatial or temporal locality. Approaches to improve the performance of SpM×V are based on matrix reordering and register blocking. These matrix transformations are designed to handle randomly occurring dense blocks in a sparse matrix. The efficiency of these transformations depends strongly on the presence of suitable blocks. The overhead of reorganization of a matrix from one format to another is often of the order of tens of executions ofSpM×V. For this reason, such a reorganization pays off only if the same matrix A is multiplied by multiple different vectors, e.g., in iterative linear solvers.This paper introduces an unusual approach to accelerate SpM×V. This approach can be combined with other acceleration approaches andconsists of three steps:1 dividing matrix A into non-empty regions,2 choosing an efficient way to traverse these regions (in other words, choosing an efficient ordering of partial multiplications,3 choosing the optimal type of storage for each region.All these three steps are tightly coupled. The first step divides the whole matrix into smaller parts (regions that can fit in the cache. The second step improves the locality during multiplication due to better utilization of distant references. The last step maximizes the machine computation performance of the partial multiplication for each region.In this paper, we describe aspects of these 3 steps in more detail (including fast and time-inexpensive algorithms for all steps. Ourmeasurements prove that our approach gives a significant speedup for almost all matrices arising from various technical areas.
Graph Regularized Nonnegative Matrix Factorization with Sparse Coding
Directory of Open Access Journals (Sweden)
Chuang Lin
2015-01-01
Full Text Available In this paper, we propose a sparseness constraint NMF method, named graph regularized matrix factorization with sparse coding (GRNMF_SC. By combining manifold learning and sparse coding techniques together, GRNMF_SC can efficiently extract the basic vectors from the data space, which preserves the intrinsic manifold structure and also the local features of original data. The target function of our method is easy to propose, while the solving procedures are really nontrivial; in the paper we gave the detailed derivation of solving the target function and also a strict proof of its convergence, which is a key contribution of the paper. Compared with sparseness constrained NMF and GNMF algorithms, GRNMF_SC can learn much sparser representation of the data and can also preserve the geometrical structure of the data, which endow it with powerful discriminating ability. Furthermore, the GRNMF_SC is generalized as supervised and unsupervised models to meet different demands. Experimental results demonstrate encouraging results of GRNMF_SC on image recognition and clustering when comparing with the other state-of-the-art NMF methods.
Capacity of entanglement and the distribution of density matrix eigenvalues in gapless systems
Nakagawa, Yuya O.; Furukawa, Shunsuke
2017-11-01
We propose that the properties of the capacity of entanglement (COE) in gapless systems can efficiently be investigated through the use of the distribution of eigenvalues of the reduced density matrix (RDM). The COE is defined as the fictitious heat capacity calculated from the entanglement spectrum. Its dependence on the fictitious temperature can reflect the low-temperature behavior of the physical heat capacity and thus provide a useful probe of gapless bulk or edge excitations of the system. Assuming a power-law scaling of the COE with an exponent α at low fictitious temperatures, we derive an analytical formula for the distribution function of the RDM eigenvalues. We numerically test the effectiveness of the formula in a relativistic free scalar boson in two spatial dimensions and find that the distribution function can detect the expected α =3 scaling of the COE much more efficiently than the raw data of the COE. We also calculate the distribution function in the ground state of the half-filled Landau level with short-range interactions and find better agreement with the α =2 /3 formula than with the α =1 one, which indicates a non-Fermi-liquid nature of the system.
Joint Learning of Multiple Sparse Matrix Gaussian Graphical Models.
Huang, Feihu; Chen, Songcan
2015-11-01
We consider joint learning of multiple sparse matrix Gaussian graphical models and propose the joint matrix graphical Lasso to discover the conditional independence structures among rows (columns) in the matrix variable under distinct conditions. The proposed approach borrows strength across the different graphical models and is based on the maximum likelihood with penalized row and column precision matrices, respectively. In particular, our model is more parsimonious and flexible than the joint vector graphical models. Furthermore, we establish the asymptotic properties of our model on consistency and sparsistency. And the asymptotic analysis shows that our model enjoys a better convergence rate than the joint vector graphical models. Extensive simulation experiments demonstrate that our methods outperform state-of-the-art methods in identifying graphical structures and estimating precision matrices. Moreover, the effectiveness of our methods is also illustrated via a real data set analysis. Sparsistency is shorthand for consistency of the sparsity pattern of the parameters.
Transmission eigenvalue densities and moments in chaotic cavities from random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Vivo, Pierpaolo [School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH (United Kingdom); Vivo, Edoardo [Universita degli Studi di Parma, Dipartimento di Fisica Teorica, Viale GP Usberti n.7/A (Parco Area delle Scienze), Parma (Italy)
2008-03-28
We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads (N{sub 1}, N{sub 2}) are analytically known from the Jacobi ensemble of random matrix theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form ({lambda}{sup m}{sub 1}) for m > -|N{sub 1} - N{sub 2}| - 1 and {beta} = 2, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation function for {beta} = 2, 4 which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants. (fast track communication)
Maximizing sparse matrix vector product performance in MIMD computers
Energy Technology Data Exchange (ETDEWEB)
McLay, R.T.; Kohli, H.S.; Swift, S.L.; Carey, G.F.
1994-12-31
A considerable component of the computational effort involved in conjugate gradient solution of structured sparse matrix systems is expended during the Matrix-Vector Product (MVP), and hence it is the focus of most efforts at improving performance. Such efforts are hindered on MIMD machines due to constraints on memory, cache and speed of memory-cpu data transfer. This paper describes a strategy for maximizing the performance of the local computations associated with the MVP. The method focuses on single stride memory access, and the efficient use of cache by pre-loading it with data that is re-used while bypassing it for other data. The algorithm is designed to behave optimally for varying grid sizes and number of unknowns per gridpoint. Results from an assembly language implementation of the strategy on the iPSC/860 show a significant improvement over the performance using FORTRAN.
Optimizing Tpetra%3CU%2B2019%3Es sparse matrix-matrix multiplication routine.
Energy Technology Data Exchange (ETDEWEB)
Nusbaum, Kurtis Lee
2011-08-01
Over the course of the last year, a sparse matrix-matrix multiplication routine has been developed for the Tpetra package. This routine is based on the same algorithm that is used in EpetraExt with heavy modifications. Since it achieved a working state, several major optimizations have been made in an effort to speed up the routine. This report will discuss the optimizations made to the routine, its current state, and where future work needs to be done.
Sparse Matrix-Vector Multiplication on Multicore and Accelerators
Energy Technology Data Exchange (ETDEWEB)
Williams, Samuel W. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Bell, Nathan [NVIDIA Research, Santa Clara, CA (United States); Choi, Jee Whan [Georgia Inst. of Technology, Atlanta, GA (United States); Garland, Michael [NVIDIA Research, Santa Clara, CA (United States); Oliker, Leonid [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Vuduc, Richard [Georgia Inst. of Technology, Atlanta, GA (United States)
2010-12-07
This chapter consolidates recent work on the development of high performance multicore and accelerator-based implementations of sparse matrix-vector multiplication (SpMV). As an object of study, SpMV is an interesting computation for two key reasons. First, it appears widely in applications in scientific and engineering computing, financial and economic modeling, and information retrieval, among others, and is therefore of great practical interest. Secondly, it is both simple to describe but challenging to implement well, since its performance is limited by a variety of factors, including low computational intensity, potentially highly irregular memory access behavior, and a strong input dependence that be known only at run time. Thus, we believe SpMV is both practically important and provides important insights for understanding the algorithmic and implementation principles necessary to making effective use of state-of-the-art systems.
Dense and Sparse Matrix Operations on the Cell Processor
Energy Technology Data Exchange (ETDEWEB)
Williams, Samuel W.; Shalf, John; Oliker, Leonid; Husbands,Parry; Yelick, Katherine
2005-05-01
The slowing pace of commodity microprocessor performance improvements combined with ever-increasing chip power demands has become of utmost concern to computational scientists. Therefore, the high performance computing community is examining alternative architectures that address the limitations of modern superscalar designs. In this work, we examine STI's forthcoming Cell processor: a novel, low-power architecture that combines a PowerPC core with eight independent SIMD processing units coupled with a software-controlled memory to offer high FLOP/s/Watt. Since neither Cell hardware nor cycle-accurate simulators are currently publicly available, we develop an analytic framework to predict Cell performance on dense and sparse matrix operations, using a variety of algorithmic approaches. Results demonstrate Cell's potential to deliver more than an order of magnitude better GFLOP/s per watt performance, when compared with the Intel Itanium2 and Cray X1 processors.
Sparse Nonnegative Matrix Factorization Strategy for Cochlear Implants
Directory of Open Access Journals (Sweden)
Hongmei Hu
2015-12-01
Full Text Available Current cochlear implant (CI strategies carry speech information via the waveform envelope in frequency subbands. CIs require efficient speech processing to maximize information transfer to the brain, especially in background noise, where the speech envelope is not robust to noise interference. In such conditions, the envelope, after decomposition into frequency bands, may be enhanced by sparse transformations, such as nonnegative matrix factorization (NMF. Here, a novel CI processing algorithm is described, which works by applying NMF to the envelope matrix (envelopogram of 22 frequency channels in order to improve performance in noisy environments. It is evaluated for speech in eight-talker babble noise. The critical sparsity constraint parameter was first tuned using objective measures and then evaluated with subjective speech perception experiments for both normal hearing and CI subjects. Results from vocoder simulations with 10 normal hearing subjects showed that the algorithm significantly enhances speech intelligibility with the selected sparsity constraints. Results from eight CI subjects showed no significant overall improvement compared with the standard advanced combination encoder algorithm, but a trend toward improvement of word identification of about 10 percentage points at +15 dB signal-to-noise ratio (SNR was observed in the eight CI subjects. Additionally, a considerable reduction of the spread of speech perception performance from 40% to 93% for advanced combination encoder to 80% to 100% for the suggested NMF coding strategy was observed.
A Bootstrap Approach to Eigenvalue Correction
Hendrikse, A.J.; Spreeuwers, Lieuwe Jan; Veldhuis, Raymond N.J.
2009-01-01
Eigenvalue analysis is an important aspect in many data modeling methods. Unfortunately, the eigenvalues of the sample covariance matrix (sample eigenvalues) are biased estimates of the eigenvalues of the covariance matrix of the data generating process (population eigenvalues). We present a new
Pulse-Width-Modulation of Neutral-Point-Clamped Sparse Matrix Converter
DEFF Research Database (Denmark)
Loh, P.C.; Blaabjerg, Frede; Gao, F.
2007-01-01
input current and output voltage can be achieved with minimized rectification switching loss, rendering the sparse matrix converter as a competitive choice for interfacing the utility grid to (e.g.) defense facilities that require a different frequency supply. As an improvement, sparse matrix converter......Sparse matrix converter is an alternative "allsemiconductor" energy processor proposed recently for converting an ac source with fixed magnitude and frequency to a variable voltage and frequency supply that can meet the requirements of a particular industry application. In principle, sparse matrix...... converter is constructed by connecting a front-end bi-directional rectification stage to a conventional two-level inversion stage with no bulky capacitive or inductive energy storage connected to their intermediate dc-link. Through proper modulation and compensation of the resulting converter, sinusoidal...
A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations
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H. Yoshimura
2013-06-01
Full Text Available In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.
Optimal Sparse Matrix Dense Vector Multiplication in the I/O-Model
DEFF Research Database (Denmark)
Bender, Michael A.; Brodal, Gerth Stølting; Fagerberg, Rolf
2010-01-01
We study the problem of sparse-matrix dense-vector multiplication (SpMV) in external memory. The task of SpMV is to compute y:=Ax, where A is a sparse Nx N matrix and x is a vector. We express sparsity by a parameter k, and for each choice of k consider the class of matrices where the number...
Directory of Open Access Journals (Sweden)
G. Akinci
2014-12-01
Full Text Available Sparse matrices are occasionally encountered during solution of various problems by means of numerical methods, particularly the finite element method. ELLPACK sparse matrix storage scheme, one of the most widely used methods due to its implementation ease, is investigated in this study. The scheme uses excessive memory due to its definition. For the conventional finite element method, where the node elements are used, the excessive memory caused by redundant entries in the ELLPACK sparse matrix storage scheme becomes negligible for large scale problems. On the other hand, our analyses show that the redundancy is still considerable for the occasions where facet or edge elements have to be used.
Design Patterns for Sparse-Matrix Computations on Hybrid CPU/GPU Platforms
Directory of Open Access Journals (Sweden)
Valeria Cardellini
2014-01-01
Full Text Available We apply object-oriented software design patterns to develop code for scientific software involving sparse matrices. Design patterns arise when multiple independent developments produce similar designs which converge onto a generic solution. We demonstrate how to use design patterns to implement an interface for sparse matrix computations on NVIDIA GPUs starting from PSBLAS, an existing sparse matrix library, and from existing sets of GPU kernels for sparse matrices. We also compare the throughput of the PSBLAS sparse matrix–vector multiplication on two platforms exploiting the GPU with that obtained by a CPU-only PSBLAS implementation. Our experiments exhibit encouraging results regarding the comparison between CPU and GPU executions in double precision, obtaining a speedup of up to 35.35 on NVIDIA GTX 285 with respect to AMD Athlon 7750, and up to 10.15 on NVIDIA Tesla C2050 with respect to Intel Xeon X5650.
Directory of Open Access Journals (Sweden)
Vibha Tiwari
2015-12-01
Full Text Available Compressive sensing theory enables faithful reconstruction of signals, sparse in domain $ \\Psi $, at sampling rate lesser than Nyquist criterion, while using sampling or sensing matrix $ \\Phi $ which satisfies restricted isometric property. The role played by sensing matrix $ \\Phi $ and sparsity matrix $ \\Psi $ is vital in faithful reconstruction. If the sensing matrix is dense then it takes large storage space and leads to high computational cost. In this paper, effort is made to design sparse sensing matrix with least incurred computational cost while maintaining quality of reconstructed image. The design approach followed is based on sparse block circulant matrix (SBCM with few modifications. The other used sparse sensing matrix consists of 15 ones in each column. The medical images used are acquired from US, MRI and CT modalities. The image quality measurement parameters are used to compare the performance of reconstructed medical images using various sensing matrices. It is observed that, since Gram matrix of dictionary matrix ($ \\Phi \\Psi \\mathrm{} $ is closed to identity matrix in case of proposed modified SBCM, therefore, it helps to reconstruct the medical images of very good quality.
A Combinatorial Approach to Some Sparse Matrix Problems.
1983-06-01
were introduced, particularly after the advent of linear programming in 1947 ( sec Dantzig (1963)), it has been recognized that real-world problems are...COMBINATORIAL APPROACH TO SOME SPARSE Technical Report MATRX POBLES 6 PERFORMING ORG. REPORT NUMBR 7. AUTHOR(q) 6. CONTRACT ORt GRANT NUMBER(d) S
Weber, Valéry; Laino, Teodoro; Pozdneev, Alexander; Fedulova, Irina; Curioni, Alessandro
2015-07-14
In this paper, we present a novel, highly efficient, and massively parallel implementation of the sparse matrix-matrix multiplication algorithm inspired by the midpoint method that is suitable for matrices with decay. Compared with the state of the art in sparse matrix-matrix multiplications, the new algorithm heavily exploits data locality, yielding better performance and scalability, approaching a perfect linear scaling up to a process box size equal to a characteristic length that is intrinsic to the matrices. Moreover, the method is able to scale linearly with system size reaching constant time with proportional resources, also regarding memory consumption. We demonstrate how the proposed method can be effectively used for the construction of the density matrix in electronic structure theory, such as Hartree-Fock, density functional theory, and semiempirical Hamiltonians. We present the details of the implementation together with a performance analysis up to 185,193 processes, employing a Hamiltonian matrix generated from a semiempirical NDDO scheme.
Data-Parallel Language for Correct and Efficient Sparse Matrix Codes
Arnold, Gilad
2011-01-01
Sparse matrix formats encode very large numerical matrices with relatively few nonzeros. They are typically implemented using imperative languages, with emphasis on low-level optimization. Such implementations are far removed from the conceptual organization of the sparse format, which obscures the representation invariants. They are further specialized by hand for particular applications, machines and workloads. Consequently, they are difficult to write, maintain, and verify.In this disserta...
Sleijpen, G.L.G.; Vorst, H.A. van der
1995-01-01
We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix.The matrix may be complex and non-normal.The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors,
Subspace Methods for Eigenvalue Problems
Hochstenbach, Michiel Erik
2003-01-01
This thesis treats a number of aspects of subspace methods for various eigenvalue problems. Vibrations and their corresponding eigenvalues (or frequencies) arise in science, engineering, and daily life. Matrix eigenvalue problems come from a large number of areas, such as chemistry,
Speculative segmented sum for sparse matrix-vector multiplication on heterogeneous processors
DEFF Research Database (Denmark)
Liu, Weifeng; Vinter, Brian
2015-01-01
Sparse matrix-vector multiplication (SpMV) is a central building block for scientific software and graph applications. Recently, heterogeneous processors composed of different types of cores attracted much attention because of their flexible core configuration and high energy efficiency. In this ......Sparse matrix-vector multiplication (SpMV) is a central building block for scientific software and graph applications. Recently, heterogeneous processors composed of different types of cores attracted much attention because of their flexible core configuration and high energy efficiency...
Solution of the two-dimensional Navier-Stokes equations using sparse matrix solvers
Bender, Erich E.; Khosla, Prem K.
1987-01-01
The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equations are discretized and linearized and the resulting system of equations are solved using this package. Several viscous flow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.
Energy Technology Data Exchange (ETDEWEB)
Aktulga, Hasan Metin [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Buluc, Aydin [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Williams, Samuel [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Yang, Chao [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
2014-08-14
Obtaining highly accurate predictions on the properties of light atomic nuclei using the configuration interaction (CI) approach requires computing a few extremal Eigen pairs of the many-body nuclear Hamiltonian matrix. In the Many-body Fermion Dynamics for nuclei (MFDn) code, a block Eigen solver is used for this purpose. Due to the large size of the sparse matrices involved, a significant fraction of the time spent on the Eigen value computations is associated with the multiplication of a sparse matrix (and the transpose of that matrix) with multiple vectors (SpMM and SpMM-T). Existing implementations of SpMM and SpMM-T significantly underperform expectations. Thus, in this paper, we present and analyze optimized implementations of SpMM and SpMM-T. We base our implementation on the compressed sparse blocks (CSB) matrix format and target systems with multi-core architectures. We develop a performance model that allows us to understand and estimate the performance characteristics of our SpMM kernel implementations, and demonstrate the efficiency of our implementation on a series of real-world matrices extracted from MFDn. In particular, we obtain 3-4 speedup on the requisite operations over good implementations based on the commonly used compressed sparse row (CSR) matrix format. The improvements in the SpMM kernel suggest we may attain roughly a 40% speed up in the overall execution time of the block Eigen solver used in MFDn.
Fast sparse matrix-vector multiplication by exploiting variable block structure
Energy Technology Data Exchange (ETDEWEB)
Vuduc, R W; Moon, H
2005-07-07
We improve the performance of sparse matrix-vector multiply (SpMV) on modern cache-based superscalar machines when the matrix structure consists of multiple, irregularly aligned rectangular blocks. Matrices from finite element modeling applications often have this kind of structure. Our technique splits the matrix, A, into a sum, A{sub 1} + A{sub 2} + ... + A{sub s}, where each term is stored in a new data structure, unaligned block compressed sparse row (UBCSR) format . The classical alternative approach of storing A in a block compressed sparse row (BCSR) format yields limited performance gains because it imposes a particular alignment of the matrix non-zero structure, leading to extra work from explicitly padded zeros. Combining splitting and UBCSR reduces this extra work while retaining the generally lower memory bandwidth requirements and register-level tiling opportunities of BCSR. Using application test matrices, we show empirically that speedups can be as high as 2.1x over not blocking at all, and as high as 1.8x over the standard BCSR implementation used in prior work. When performance does not improve, split UBCSR can still significantly reduce matrix storage. Through extensive experiments, we further show that the empirically optimal number of splittings s and the block size for each matrix term A{sub i} will in practice depend on the matrix and hardware platform. Our data lay a foundation for future development of fully automated methods for tuning these parameters.
DEFF Research Database (Denmark)
Janssen, Anja; Mikosch, Thomas Valentin; Rezapour, Mohsen
2017-01-01
of the sample covariance matrix. While we show that in the case of heavy-tailed innovations the limiting behavior resembles that of completely independent observations, we also derive that in the case of a heavy-tailed volatility sequence the possible limiting behavior is more diverse, i.e. allowing...
Energy Technology Data Exchange (ETDEWEB)
1978-01-01
The program and abstracts of the SIAM 1978 fall meeting in Knoxville, Tennessee, are given, along with those of the associated symposium on sparse matrix computations. The papers dealt with both pure mathematics and mathematics applied to many different subject areas. (RWR)
Performance modeling and optimization of sparse matrix-vector multiplication on NVIDIA CUDA platform
Xu, S.; Xue, W.; Lin, H.X.
2011-01-01
In this article, we discuss the performance modeling and optimization of Sparse Matrix-Vector Multiplication (SpMV) on NVIDIA GPUs using CUDA. SpMV has a very low computation-data ratio and its performance is mainly bound by the memory bandwidth. We propose optimization of SpMV based on ELLPACK from
Sparse subspace clustering for data with missing entries and high-rank matrix completion.
Fan, Jicong; Chow, Tommy W S
2017-09-01
Many methods have recently been proposed for subspace clustering, but they are often unable to handle incomplete data because of missing entries. Using matrix completion methods to recover missing entries is a common way to solve the problem. Conventional matrix completion methods require that the matrix should be of low-rank intrinsically, but most matrices are of high-rank or even full-rank in practice, especially when the number of subspaces is large. In this paper, a new method called Sparse Representation with Missing Entries and Matrix Completion is proposed to solve the problems of incomplete-data subspace clustering and high-rank matrix completion. The proposed algorithm alternately computes the matrix of sparse representation coefficients and recovers the missing entries of a data matrix. The proposed algorithm recovers missing entries through minimizing the representation coefficients, representation errors, and matrix rank. Thorough experimental study and comparative analysis based on synthetic data and natural images were conducted. The presented results demonstrate that the proposed algorithm is more effective in subspace clustering and matrix completion compared with other existing methods. Copyright © 2017 Elsevier Ltd. All rights reserved.
Energy Technology Data Exchange (ETDEWEB)
Edwards, R.G.; Heller, U.M. [SCRI, Florida State University, Tallahassee, Florida 32306-4130 (United States); Narayanan, R. [Department of Physics, Building 510A, Brookhaven National Laboratory, P. O. Box 5000, Upton, New York 11973 (United States)
1999-10-01
The low-lying spectrum of the Dirac operator is predicted to be universal, within three classes, depending on symmetry properties specified according to random matrix theory. The three universal classes are the orthogonal, unitary and symplectic ensembles. Lattice gauge theory with staggered fermions has verified two of the cases so far, unitary and symplectic, with staggered fermions in the fundamental representation of SU(3) and SU(2). We verify the missing case here, namely orthogonal, with staggered fermions in the adjoint representation of SU(N{sub c}), N{sub c}=2,3. {copyright} {ital 1999} {ital The American Physical Society}
DEFF Research Database (Denmark)
Ding, Tao; Li, Cheng; Yang, Yongheng
2017-01-01
It was usually considered in power systems that power flow equations had multiple solutions and all the eigenvalues of Jacobian ma-trix at the high-voltage operable solution should have negative real parts. Accordingly, type-1 low-voltage power flow solutions are defined in the case...... solution may be positive and also the type-1 low-voltage solutions could have more than one positive real-part eigen-values, being a major challenge. Therefore, in this paper, the recognition of the type-1 low-voltage power flow solutions is re-examined with the presence of negative reactance. Selected......-voltage operable solution as well as the number of positive real-part eigenvalues at the type-1 low-voltage power flow solutions....
Algorithms and Application of Sparse Matrix Assembly and Equation Solvers for Aeroacoustics
Watson, W. R.; Nguyen, D. T.; Reddy, C. J.; Vatsa, V. N.; Tang, W. H.
2001-01-01
An algorithm for symmetric sparse equation solutions on an unstructured grid is described. Efficient, sequential sparse algorithms for degree-of-freedom reordering, supernodes, symbolic/numerical factorization, and forward backward solution phases are reviewed. Three sparse algorithms for the generation and assembly of symmetric systems of matrix equations are presented. The accuracy and numerical performance of the sequential version of the sparse algorithms are evaluated over the frequency range of interest in a three-dimensional aeroacoustics application. Results show that the solver solutions are accurate using a discretization of 12 points per wavelength. Results also show that the first assembly algorithm is impractical for high-frequency noise calculations. The second and third assembly algorithms have nearly equal performance at low values of source frequencies, but at higher values of source frequencies the third algorithm saves CPU time and RAM. The CPU time and the RAM required by the second and third assembly algorithms are two orders of magnitude smaller than that required by the sparse equation solver. A sequential version of these sparse algorithms can, therefore, be conveniently incorporated into a substructuring for domain decomposition formulation to achieve parallel computation, where different substructures are handles by different parallel processors.
Cakoni, Fioralba; Haddar, Houssem
2013-10-01
In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission
Directory of Open Access Journals (Sweden)
Anil S Thakur
2007-10-01
Full Text Available Crystallization is a major bottleneck in the process of macromolecular structure determination by X-ray crystallography. Successful crystallization requires the formation of nuclei and their subsequent growth to crystals of suitable size. Crystal growth generally occurs spontaneously in a supersaturated solution as a result of homogenous nucleation. However, in a typical sparse matrix screening experiment, precipitant and protein concentration are not sampled extensively, and supersaturation conditions suitable for nucleation are often missed.We tested the effect of nine potential heterogenous nucleating agents on crystallization of ten test proteins in a sparse matrix screen. Several nucleating agents induced crystal formation under conditions where no crystallization occurred in the absence of the nucleating agent. Four nucleating agents: dried seaweed; horse hair; cellulose and hydroxyapatite, had a considerable overall positive effect on crystallization success. This effect was further enhanced when these nucleating agents were used in combination with each other.Our results suggest that the addition of heterogeneous nucleating agents increases the chances of crystal formation when using sparse matrix screens.
Sparse and smooth canonical correlation analysis through rank-1 matrix approximation
Aïssa-El-Bey, Abdeldjalil; Seghouane, Abd-Krim
2017-12-01
Canonical correlation analysis (CCA) is a well-known technique used to characterize the relationship between two sets of multidimensional variables by finding linear combinations of variables with maximal correlation. Sparse CCA and smooth or regularized CCA are two widely used variants of CCA because of the improved interpretability of the former and the better performance of the later. So far, the cross-matrix product of the two sets of multidimensional variables has been widely used for the derivation of these variants. In this paper, two new algorithms for sparse CCA and smooth CCA are proposed. These algorithms differ from the existing ones in their derivation which is based on penalized rank-1 matrix approximation and the orthogonal projectors onto the space spanned by the two sets of multidimensional variables instead of the simple cross-matrix product. The performance and effectiveness of the proposed algorithms are tested on simulated experiments. On these results, it can be observed that they outperform the state of the art sparse CCA algorithms.
Energy Technology Data Exchange (ETDEWEB)
Geist, G.A. [Oak Ridge National Lab., TN (United States). Computer Science and Mathematics Div.; Howell, G.W. [Florida Inst. of Tech., Melbourne, FL (United States). Dept. of Applied Mathematics; Watkins, D.S. [Washington State Univ., Pullman, WA (United States). Dept. of Pure and Applied Mathematics
1997-11-01
The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.
Library designs for generic C++ sparse matrix computations of iterative methods
Energy Technology Data Exchange (ETDEWEB)
Pozo, R.
1996-12-31
A new library design is presented for generic sparse matrix C++ objects for use in iterative algorithms and preconditioners. This design extends previous work on C++ numerical libraries by providing a framework in which efficient algorithms can be written *independent* of the matrix layout or format. That is, rather than supporting different codes for each (element type) / (matrix format) combination, only one version of the algorithm need be maintained. This not only reduces the effort for library developers, but also simplifies the calling interface seen by library users. Furthermore, the underlying matrix library can be naturally extended to support user-defined objects, such as hierarchical block-structured matrices, or application-specific preconditioners. Utilizing optimized kernels whenever possible, the resulting performance of such framework can be shown to be competitive with optimized Fortran programs.
Xuan, Junyu; Lu, Jie; Zhang, Guangquan; Xu, Richard Yi Da; Luo, Xiangfeng
2017-04-12
Sparse nonnegative matrix factorization (SNMF) aims to factorize a data matrix into two optimized nonnegative sparse factor matrices, which could benefit many tasks, such as document-word co-clustering. However, the traditional SNMF typically assumes the number of latent factors (i.e., dimensionality of the factor matrices) to be fixed. This assumption makes it inflexible in practice. In this paper, we propose a doubly sparse nonparametric NMF framework to mitigate this issue by using dependent Indian buffet processes (dIBP). We apply a correlation function for the generation of two stick weights associated with each column pair of factor matrices while still maintaining their respective marginal distribution specified by IBP. As a consequence, the generation of two factor matrices will be columnwise correlated. Under this framework, two classes of correlation function are proposed: 1) using bivariate Beta distribution and 2) using Copula function. Compared with the single IBP-based NMF, this paper jointly makes two factor matrices nonparametric and sparse, which could be applied to broader scenarios, such as co-clustering. This paper is seen to be much more flexible than Gaussian process-based and hierarchial Beta process-based dIBPs in terms of allowing the two corresponding binary matrix columns to have greater variations in their nonzero entries. Our experiments on synthetic data show the merits of this paper compared with the state-of-the-art models in respect of factorization efficiency, sparsity, and flexibility. Experiments on real-world data sets demonstrate the efficiency of this paper in document-word co-clustering tasks.
Directory of Open Access Journals (Sweden)
Wang Wen-qin
2015-02-01
Full Text Available The waveforms used in Multiple-Input Multiple-Output (MIMO Synthetic Aperture Radar (SAR should have a large time-bandwidth product and good ambiguity function performance. A scheme to design multiple orthogonal MIMO SAR Orthogonal Frequency Division Multiplexing (OFDM chirp waveforms by combinational sparse matrix and correlation optimization is proposed. First, the problem of MIMO SAR waveform design amounts to the associated design of hopping frequency and amplitudes. Then a iterative exhaustive search algorithm is adopted to optimally design the code matrix with the constraints minimizing the block correlation coefficient of sparse matrix and the sum of cross-correlation peaks. And the amplitudes matrix are adaptively designed by minimizing the cross-correlation peaks with the genetic algorithm. Additionally, the impacts of waveform number, hopping frequency interval and selectable frequency index are also analyzed. The simulation results verify the proposed scheme can design multiple orthogonal large time-bandwidth product OFDM chirp waveforms with low cross-correlation peak and sidelobes and it improves ambiguity performance.
Tarai, Madhumita; Kumar, Keshav; Divya, O.; Bairi, Partha; Mishra, Kishor Kumar; Mishra, Ashok Kumar
2017-09-01
The present work compares the dissimilarity and covariance based unsupervised chemometric classification approaches by taking the total synchronous fluorescence spectroscopy data sets acquired for the cumin and non-cumin based herbal preparations. The conventional decomposition method involves eigenvalue-eigenvector analysis of the covariance of the data set and finds the factors that can explain the overall major sources of variation present in the data set. The conventional approach does this irrespective of the fact that the samples belong to intrinsically different groups and hence leads to poor class separation. The present work shows that classification of such samples can be optimized by performing the eigenvalue-eigenvector decomposition on the pair-wise dissimilarity matrix.
Single-Channel Speech Separation using Sparse Non-Negative Matrix Factorization
DEFF Research Database (Denmark)
Schmidt, Mikkel N.; Olsson, Rasmus Kongsgaard
2007-01-01
We apply machine learning techniques to the problem of separating multiple speech sources from a single microphone recording. The method of choice is a sparse non-negative matrix factorization algorithm, which in an unsupervised manner can learn sparse representations of the data. This is applied...... to the learning of personalized dictionaries from a speech corpus, which in turn are used to separate the audio stream into its components. We show that computational savings can be achieved by segmenting the training data on a phoneme level. To split the data, a conventional speech recognizer is used....... The performance of the unsupervised and supervised adaptation schemes result in significant improvements in terms of the target-to-masker ratio....
SelInv - An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
Energy Technology Data Exchange (ETDEWEB)
Lin, Lin; Yang, Chao; Meza, Juan C.; Lu, Jianfeng; Ying, Lexing; E, Weinan
2009-10-16
We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDL^T, where L is lower triangular and D is diagonal. Our implementation, which is called SelInv, is built on top of an efficient supernodal left-looking LDL^T factorization of A. We discuss how computational efficiency can be gained by making use of a relative index array to handle indirect addressing. We report the performance of SelInv on a collection of sparse matrices of various sizes and nonzero structures. We also demonstrate how SelInv can be used in electronic structure calculations.
Directory of Open Access Journals (Sweden)
Tingna Shi
2018-02-01
Full Text Available In a matrix converter, the frequencies of output voltage harmonics are related to the frequencies of input, output, and carrier signals, which are independent of each other. This nature may cause an inaccurate harmonic spectrum when using conventional analytical methods, such as fast Fourier transform (FFT and the double Fourier analysis. Based on triple Fourier series, this paper proposes a method to pinpoint harmonic components of output voltages of an ultra sparse matrix converter (USMC under space vector pulse width modulation (SVPWM strategy. Amplitudes and frequencies of harmonic components are determined precisely for the first time, and the distribution pattern of harmonics can be observed directly from the analytical results. The conclusions drawn in this paper may contribute to the analysis of harmonic characteristics and serve as a reference for the harmonic suppression of USMC. Besides, the proposed method is also applicable to other types of matrix converters.
Nonlocal low-rank and sparse matrix decomposition for spectral CT reconstruction
Niu, Shanzhou; Yu, Gaohang; Ma, Jianhua; Wang, Jing
2018-02-01
Spectral computed tomography (CT) has been a promising technique in research and clinics because of its ability to produce improved energy resolution images with narrow energy bins. However, the narrow energy bin image is often affected by serious quantum noise because of the limited number of photons used in the corresponding energy bin. To address this problem, we present an iterative reconstruction method for spectral CT using nonlocal low-rank and sparse matrix decomposition (NLSMD), which exploits the self-similarity of patches that are collected in multi-energy images. Specifically, each set of patches can be decomposed into a low-rank component and a sparse component, and the low-rank component represents the stationary background over different energy bins, while the sparse component represents the rest of the different spectral features in individual energy bins. Subsequently, an effective alternating optimization algorithm was developed to minimize the associated objective function. To validate and evaluate the NLSMD method, qualitative and quantitative studies were conducted by using simulated and real spectral CT data. Experimental results show that the NLSMD method improves spectral CT images in terms of noise reduction, artifact suppression and resolution preservation.
A Novel CSR-Based Sparse Matrix-Vector Multiplication on GPUs
Directory of Open Access Journals (Sweden)
Guixia He
2016-01-01
Full Text Available Sparse matrix-vector multiplication (SpMV is an important operation in scientific computations. Compressed sparse row (CSR is the most frequently used format to store sparse matrices. However, CSR-based SpMVs on graphic processing units (GPUs, for example, CSR-scalar and CSR-vector, usually have poor performance due to irregular memory access patterns. This motivates us to propose a perfect CSR-based SpMV on the GPU that is called PCSR. PCSR involves two kernels and accesses CSR arrays in a fully coalesced manner by introducing a middle array, which greatly alleviates the deficiencies of CSR-scalar (rare coalescing and CSR-vector (partial coalescing. Test results on a single C2050 GPU show that PCSR fully outperforms CSR-scalar, CSR-vector, and CSRMV and HYBMV in the vendor-tuned CUSPARSE library and is comparable with a most recently proposed CSR-based algorithm, CSR-Adaptive. Furthermore, we extend PCSR on a single GPU to multiple GPUs. Experimental results on four C2050 GPUs show that no matter whether the communication between GPUs is considered or not PCSR on multiple GPUs achieves good performance and has high parallel efficiency.
Turbo-SMT: Parallel Coupled Sparse Matrix-Tensor Factorizations and Applications.
Papalexakis, Evangelos E; Faloutsos, Christos; Mitchell, Tom M; Talukdar, Partha Pratim; Sidiropoulos, Nicholas D; Murphy, Brian
2016-08-01
How can we correlate the neural activity in the human brain as it responds to typed words, with properties of these terms (like 'edible', 'fits in hand')? In short, we want to find latent variables, that jointly explain both the brain activity, as well as the behavioral responses. This is one of many settings of the Coupled Matrix-Tensor Factorization (CMTF) problem. Can we enhance any CMTF solver, so that it can operate on potentially very large datasets that may not fit in main memory? We introduce Turbo-SMT, a meta-method capable of doing exactly that: it boosts the performance of any CMTF algorithm, produces sparse and interpretable solutions, and parallelizes any CMTF algorithm, producing sparse and interpretable solutions (up to 65 fold ). Additionally, we improve upon ALS, the work-horse algorithm for CMTF, with respect to efficiency and robustness to missing values. We apply Turbo-SMT to BrainQ, a dataset consisting of a (nouns, brain voxels, human subjects) tensor and a (nouns, properties) matrix, with coupling along the nouns dimension. Turbo-SMT is able to find meaningful latent variables, as well as to predict brain activity with competitive accuracy. Finally, we demonstrate the generality of Turbo-SMT, by applying it on a Facebook dataset (users, 'friends', wall-postings); there, Turbo-SMT spots spammer-like anomalies.
Graph Transformation and Designing Parallel Sparse Matrix Algorithms beyond Data Dependence Analysis
Directory of Open Access Journals (Sweden)
H.X. Lin
2004-01-01
Full Text Available Algorithms are often parallelized based on data dependence analysis manually or by means of parallel compilers. Some vector/matrix computations such as the matrix-vector products with simple data dependence structures (data parallelism can be easily parallelized. For problems with more complicated data dependence structures, parallelization is less straightforward. The data dependence graph is a powerful means for designing and analyzing parallel algorithms. However, for sparse matrix computations, parallelization based on solely exploiting the existing parallelism in an algorithm does not always give satisfactory results. For example, the conventional Gaussian elimination algorithm for the solution of a tri-diagonal system is inherently sequential, so algorithms specially for parallel computation has to be designed. After briefly reviewing different parallelization approaches, a powerful graph formalism for designing parallel algorithms is introduced. This formalism will be discussed using a tri-diagonal system as an example. Its application to general matrix computations is also discussed. Its power in designing parallel algorithms beyond the ability of data dependence analysis is shown by means of a new algorithm called ACER (Alternating Cyclic Elimination and Reduction algorithm.
Sparse modeling of EELS and EDX spectral imaging data by nonnegative matrix factorization
Energy Technology Data Exchange (ETDEWEB)
Shiga, Motoki, E-mail: shiga_m@gifu-u.ac.jp [Department of Electrical, Electronic and Computer Engineering, Gifu University, 1-1, Yanagido, Gifu 501-1193 (Japan); Tatsumi, Kazuyoshi; Muto, Shunsuke [Advanced Measurement Technology Center, Institute of Materials and Systems for Sustainability, Nagoya University, Chikusa-ku, Nagoya 464-8603 (Japan); Tsuda, Koji [Graduate School of Frontier Sciences, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8561 (Japan); Center for Materials Research by Information Integration, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047 (Japan); Biotechnology Research Institute for Drug Discovery, National Institute of Advanced Industrial Science and Technology, 2-4-7 Aomi Koto-ku, Tokyo 135-0064 (Japan); Yamamoto, Yuta [High-Voltage Electron Microscope Laboratory, Institute of Materials and Systems for Sustainability, Nagoya University, Chikusa-ku, Nagoya 464-8603 (Japan); Mori, Toshiyuki [Environment and Energy Materials Division, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044 (Japan); Tanji, Takayoshi [Division of Materials Research, Institute of Materials and Systems for Sustainability, Nagoya University, Chikusa-ku, Nagoya 464-8603 (Japan)
2016-11-15
Advances in scanning transmission electron microscopy (STEM) techniques have enabled us to automatically obtain electron energy-loss (EELS)/energy-dispersive X-ray (EDX) spectral datasets from a specified region of interest (ROI) at an arbitrary step width, called spectral imaging (SI). Instead of manually identifying the potential constituent chemical components from the ROI and determining the chemical state of each spectral component from the SI data stored in a huge three-dimensional matrix, it is more effective and efficient to use a statistical approach for the automatic resolution and extraction of the underlying chemical components. Among many different statistical approaches, we adopt a non-negative matrix factorization (NMF) technique, mainly because of the natural assumption of non-negative values in the spectra and cardinalities of chemical components, which are always positive in actual data. This paper proposes a new NMF model with two penalty terms: (i) an automatic relevance determination (ARD) prior, which optimizes the number of components, and (ii) a soft orthogonal constraint, which clearly resolves each spectrum component. For the factorization, we further propose a fast optimization algorithm based on hierarchical alternating least-squares. Numerical experiments using both phantom and real STEM-EDX/EELS SI datasets demonstrate that the ARD prior successfully identifies the correct number of physically meaningful components. The soft orthogonal constraint is also shown to be effective, particularly for STEM-EELS SI data, where neither the spatial nor spectral entries in the matrices are sparse. - Highlights: • Automatic resolution of chemical components from spectral imaging is considered. • We propose a new non-negative matrix factorization with two new penalties. • The first penalty is sparseness to choose the number of components from data. • Experimental results with real data demonstrate effectiveness of our method.
Robust and sparse correlation matrix estimation for the analysis of high-dimensional genomics data.
Serra, Angela; Coretto, Pietro; Fratello, Michele; Tagliaferri, Roberto
2017-10-12
Microarray technology can be used to study the expression of thousands of genes across a number of different experimental conditions, usually hundreds. The underlying principle is that genes sharing similar expression patterns, across different samples, can be part of the same co-expression system, or they may share the same biological functions. Groups of genes are usually identified based on cluster analysis. Clustering methods rely on the similarity matrix between genes. A common choice to measure similarity is to compute the sample correlation matrix. Dimensionality reduction is another popular data analysis task which is also based on covariance/correlation matrix estimates. Unfortunately, covariance/correlation matrix estimation suffers from the intrinsic noise present in high-dimensional data. Sources of noise are: sampling variations, presents of outlying sample units, and the fact that in most cases the number of units is much larger than the number of genes. In this paper we propose a robust correlation matrix estimator that is regularized based on adaptive thresholding. The resulting method jointly tames the effects of the high-dimensionality, and data contamination. Computations are easy to implement and do not require hand tunings. Both simulated and real data are analysed. A Monte Carlo experiment shows that the proposed method is capable of remarkable performances. Our correlation metric is more robust to outliers compared with the existing alternatives in two gene expression data sets. It is also shown how the regularization allows to automatically detect and filter spurious correlations. The same regularization is also extended to other less robust correlation measures. Finally, we apply the ARACNE algorithm on the SyNTreN gene expression data. Sensitivity and specificity of the reconstructed network is compared with the gold standard. We show that ARACNE performs better when it takes the proposed correlation matrix estimator as input. The R
spam: A Sparse Matrix R Package with Emphasis on MCMC Methods for Gaussian Markov Random Fields
Directory of Open Access Journals (Sweden)
Reinhard Furrer
2010-10-01
Full Text Available spam is an R package for sparse matrix algebra with emphasis on a Cholesky factorization of sparse positive definite matrices. The implemantation of spam is based on the competing philosophical maxims to be competitively fast compared to existing tools and to be easy to use, modify and extend. The first is addressed by using fast Fortran routines and the second by assuring S3 and S4 compatibility. One of the features of spam is to exploit the algorithmic steps of the Cholesky factorization and hence to perform only a fraction of the workload when factorizing matrices with the same sparsity structure. Simulations show that exploiting this break-down of the factorization results in a speed-up of about a factor 5 and memory savings of about a factor 10 for large matrices and slightly smaller factors for huge matrices. The article is motivated with Markov chain Monte Carlo methods for Gaussian Markov random fields, but many other statistical applications are mentioned that profit from an efficient Cholesky factorization as well.
Woo, Jonghye; Xing, Fangxu; Lee, Junghoon; Stone, Maureen; Prince, Jerry L
2014-01-01
Tongue motion during speech and swallowing involves synergies of locally deforming regions, or functional units. Motion clustering during tongue motion can be used to reveal the tongue's intrinsic functional organization. A novel matrix factorization and clustering method for tissues tracked using tagged magnetic resonance imaging (tMRI) is presented. Functional units are estimated using a graph-regularized sparse non-negative matrix factorization framework, learning latent building blocks and the corresponding weighting map from motion features derived from tissue displacements. Spectral clustering using the weighting map is then performed to determine the coherent regions--i.e., functional units--efined by the tongue motion. Two-dimensional image data is used to ver-fy that the proposed algorithm clusters the different types of images ac-urately. Three-dimensional tMRI data from five subjects carrying out simple non-speech/speech tasks are analyzed to show how the proposed approach defines a subject/task-specific functional parcellation of the tongue in localized regions.
Parallelism in matrix computations
Gallopoulos, Efstratios; Sameh, Ahmed H
2016-01-01
This book is primarily intended as a research monograph that could also be used in graduate courses for the design of parallel algorithms in matrix computations. It assumes general but not extensive knowledge of numerical linear algebra, parallel architectures, and parallel programming paradigms. The book consists of four parts: (I) Basics; (II) Dense and Special Matrix Computations; (III) Sparse Matrix Computations; and (IV) Matrix functions and characteristics. Part I deals with parallel programming paradigms and fundamental kernels, including reordering schemes for sparse matrices. Part II is devoted to dense matrix computations such as parallel algorithms for solving linear systems, linear least squares, the symmetric algebraic eigenvalue problem, and the singular-value decomposition. It also deals with the development of parallel algorithms for special linear systems such as banded ,Vandermonde ,Toeplitz ,and block Toeplitz systems. Part III addresses sparse matrix computations: (a) the development of pa...
Abdelfattah, Ahmad
2016-05-23
Simulations of many multi-component PDE-based applications, such as petroleum reservoirs or reacting flows, are dominated by the solution, on each time step and within each Newton step, of large sparse linear systems. The standard solver is a preconditioned Krylov method. Along with application of the preconditioner, memory-bound Sparse Matrix-Vector Multiplication (SpMV) is the most time-consuming operation in such solvers. Multi-species models produce Jacobians with a dense block structure, where the block size can be as large as a few dozen. Failing to exploit this dense block structure vastly underutilizes hardware capable of delivering high performance on dense BLAS operations. This paper presents a GPU-accelerated SpMV kernel for block-sparse matrices. Dense matrix-vector multiplications within the sparse-block structure leverage optimization techniques from the KBLAS library, a high performance library for dense BLAS kernels. The design ideas of KBLAS can be applied to block-sparse matrices. Furthermore, a technique is proposed to balance the workload among thread blocks when there are large variations in the lengths of nonzero rows. Multi-GPU performance is highlighted. The proposed SpMV kernel outperforms existing state-of-the-art implementations using matrices with real structures from different applications. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.
Spectral redemption in clustering sparse networks.
Krzakala, Florent; Moore, Cristopher; Mossel, Elchanan; Neeman, Joe; Sly, Allan; Zdeborová, Lenka; Zhang, Pan
2013-12-24
Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here, we present a class of spectral algorithms based on a nonbacktracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all of the way down to the theoretical limit. We also show the spectrum of the nonbacktracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.
Optimization of Sparse Matrix-Vector Multiplication on Emerging Multicore Platforms
Energy Technology Data Exchange (ETDEWEB)
Williams, Samuel; Oliker, Leonid; Vuduc, Richard; Shalf, John; Yelick, Katherine; Demmel, James
2008-10-16
We are witnessing a dramatic change in computer architecture due to the multicore paradigm shift, as every electronic device from cell phones to supercomputers confronts parallelism of unprecedented scale. To fully unleash the potential of these systems, the HPC community must develop multicore specific-optimization methodologies for important scientific computations. In this work, we examine sparse matrix-vector multiply (SpMV) - one of the most heavily used kernels in scientific computing - across a broad spectrum of multicore designs. Our experimental platform includes the homogeneous AMD quad-core, AMD dual-core, and Intel quad-core designs, the heterogeneous STI Cell, as well as one of the first scientific studies of the highly multithreaded Sun Victoria Falls (a Niagara2 SMP). We present several optimization strategies especially effective for the multicore environment, and demonstrate significant performance improvements compared to existing state-of-the-art serial and parallel SpMV implementations. Additionally, we present key insights into the architectural trade-offs of leading multicore design strategies, in the context of demanding memory-bound numerical algorithms.
Optimization of sparse matrix-vector multiplication on emerging multicore platforms
Energy Technology Data Exchange (ETDEWEB)
Williams, Samuel [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States); Oliker, Leonid [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Vuduc, Richard [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Shalf, John [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Yelick, Katherine [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States); Demmel, James [Univ. of California, Berkeley, CA (United States)
2007-01-01
We are witnessing a dramatic change in computer architecture due to the multicore paradigm shift, as every electronic device from cell phones to supercomputers confronts parallelism of unprecedented scale. To fully unleash the potential of these systems, the HPC community must develop multicore specific optimization methodologies for important scientific computations. In this work, we examine sparse matrix-vector multiply (SpMV) - one of the most heavily used kernels in scientific computing - across a broad spectrum of multicore designs. Our experimental platform includes the homogeneous AMD dual-core and Intel quad-core designs, the heterogeneous STI Cell, as well as the first scientific study of the highly multithreaded Sun Niagara2. We present several optimization strategies especially effective for the multicore environment, and demonstrate significant performance improvements compared to existing state-of-the-art serial and parallel SpMV implementations. Additionally, we present key insights into the architectural tradeoffs of leading multicore design strategies, in the context of demanding memory-bound numerical algorithms.
Optimization of sparse matrix-vector multiplication on emerging multicore platforms
Energy Technology Data Exchange (ETDEWEB)
Williams, S; Oliker, L; Vuduc, R; Shalf, J; Yelick, K; Demmel, J
2007-04-16
We are witnessing a dramatic change in computer architecture due to the multicore paradigm shift, as every electronic device from cell phones to supercomputers confronts parallelism of unprecedented scale. To fully unleash the potential of these systems, the HPC community must develop multicore specific optimization methodologies for important scientific computations. In this work, we examine sparse matrix-vector multiply (SpMV)--one of the most heavily used kernels in scientific computing--across a broad spectrum of multicore designs. Our experimental platform includes the homogeneous AMD dual-core and Intel quad-core designs, as well as the highly multithreaded Sun Niagara and heterogeneous STI Cell. We present several optimization strategies especially effective for the multicore environment, and demonstrate significant performance improvements compared to existing state-of-the-art serial and parallel SpMV implementations. Additionally, we present key insights into the architectural tradeoffs of leading multicore design strategies, in the context of demanding memory-bound numerical algorithms.
Energy Technology Data Exchange (ETDEWEB)
Cwik, T.; Jamnejad, V.; Zuffada, C. [California Institute of Technology, Pasadena, CA (United States)
1994-12-31
The usefulness of finite element modeling follows from the ability to accurately simulate the geometry and three-dimensional fields on the scale of a fraction of a wavelength. To make this modeling practical for engineering design, it is necessary to integrate the stages of geometry modeling and mesh generation, numerical solution of the fields-a stage heavily dependent on the efficient use of a sparse matrix equation solver, and display of field information. The stages of geometry modeling, mesh generation, and field display are commonly completed using commercially available software packages. Algorithms for the numerical solution of the fields need to be written for the specific class of problems considered. Interior problems, i.e. simulating fields in waveguides and cavities, have been successfully solved using finite element methods. Exterior problems, i.e. simulating fields scattered or radiated from structures, are more difficult to model because of the need to numerically truncate the finite element mesh. To practically compute a solution to exterior problems, the domain must be truncated at some finite surface where the Sommerfeld radiation condition is enforced, either approximately or exactly. Approximate methods attempt to truncate the mesh using only local field information at each grid point, whereas exact methods are global, needing information from the entire mesh boundary. In this work, a method that couples three-dimensional finite element (FE) solutions interior to the bounding surface, with an efficient integral equation (IE) solution that exactly enforces the Sommerfeld radiation condition is developed. The bounding surface is taken to be a surface of revolution (SOR) to greatly reduce computational expense in the IE portion of the modeling.
DEFF Research Database (Denmark)
Park, Kiwoo; Lee, Kyo Beum; Blaabjerg, Frede
2011-01-01
In this paper, we present a novel Z-source sparse matrix converter (ZSMC) and a compensation method based on a fuzzy logic controller to compensate unbalanced input voltages. The ZSMC is developed based on the structure of an SMC to reduce the number of unipolar power semiconductor switches...... and employs a Z-source network to overcome the inherent limitation of the low voltage transfer ratio of conventional matrix converters. Although the ZSMC is a two-stage converter, it directly connects between a source and a load through a Z-source network, which is designed to have smaller passive components...
Parallel Symmetric Eigenvalue Problem Solvers
2015-05-01
smallest eigenvalues . . . . . . . . . 59 6.1.4 Computing the Fiedler vector . . . . . . . . . . . . . . . . . 59 6.1.5 Computing interior eigenpairs...comparison of several methods of computing the Fiedler vector for cage15 101 xi Figure Page 8.11 Ratio of running times for computing the Fiedler vector...useful in constructing banded preconditioners, because they bring the large elements of a matrix toward the diagonal. To compute the Fiedler vector for
Generalized eigenvalue based spectrum sensing
Shakir, Muhammad
2012-01-01
Spectrum sensing is one of the fundamental components in cognitive radio networks. In this chapter, a generalized spectrum sensing framework which is referred to as Generalized Mean Detector (GMD) has been introduced. In this context, we generalize the detectors based on the eigenvalues of the received signal covariance matrix and transform the eigenvalue based spectrum sensing detectors namely: (i) the Eigenvalue Ratio Detector (ERD) and two newly proposed detectors which are referred to as (ii) the GEometric Mean Detector (GEMD) and (iii) the ARithmetic Mean Detector (ARMD) into an unified framework of generalize spectrum sensing. The foundation of the proposed framework is based on the calculation of exact analytical moments of the random variables of the decision threshold of the respective detectors. The decision threshold has been calculated in a closed form which is based on the approximation of Cumulative Distribution Functions (CDFs) of the respective test statistics. In this context, we exchange the analytical moments of the two random variables of the respective test statistics with the moments of the Gaussian (or Gamma) distribution function. The performance of the eigenvalue based detectors is compared with the several traditional detectors including the energy detector (ED) to validate the importance of the eigenvalue based detectors and the performance of the GEMD and the ARMD particularly in realistic wireless cognitive radio network. Analytical and simulation results show that the newly proposed detectors yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, the presented results based on proposed approximation approaches are in perfect agreement with the empirical results. © 2012 Springer Science+Business Media Dordrecht.
Analysis of multitone holographic interference filters by use of a sparse Hill matrix method
Diehl, D.W.; George, N.
2004-01-01
A theory is presented for the application of Hill's matrix method to the calculation of the reflection and transmission spectra of multitone holographic interference filters in which the permittivity is modulated by a sum of repeating functions of arbitrary period. Such filters are important because
Sparse Non-negative Matrix Factor 2-D Deconvolution for Automatic Transcription of Polyphonic Music
DEFF Research Database (Denmark)
Schmidt, Mikkel N.; Mørup, Morten
2006-01-01
We present a novel method for automatic transcription of polyphonic music based on a recently published algorithm for non-negative matrix factor 2-D deconvolution. The method works by simultaneously estimating a time-frequency model for an instrument and a pattern corresponding to the notes which...
Parity effects in eigenvalue correlators, parametric and crossover ...
Indian Academy of Sciences (India)
This paper summarizes some work that I have been doing on eigenvalue correlators of random matrix models which show some interesting behavior. First we consider matrix models with gaps in their spectrum or density of eigenvalues. The density–density correlators of these models depend on whether , where is the ...
A simple method for complex eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Killingbeck, John P [Mathematics Department, University of Hull, Hull HU6 7RX (United Kingdom); Grosjean, Alain [Laboratoire d' Astrophysique de l' Observatoire de Besancon, CNRS, UMR 6091, 41 bis Avenue de l' Observatoire, BP1615, 25010 Besancon Cedex (France); Jolicard, Georges [Laboratoire d' Astrophysique de l' Observatoire de Besancon, CNRS, UMR 6091, 41 bis Avenue de l' Observatoire, BP1615, 25010 Besancon Cedex (France)
2004-11-05
A simple iterative method is described for finding the eigenvalues of a general square complex matrix. Several numerical examples involving complex symmetric matrices are treated. In particular, it is found that a naive matrix calculation without complex rotation produces resonant state energies in accord with those given by the recently introduced naive complex hypervirial perturbation theory. (letter to the editor)
Effects of Ordering Strategies and Programming Paradigms on Sparse Matrix Computations
Oliker, Leonid; Li, Xiaoye; Husbands, Parry; Biswas, Rupak; Biegel, Bryan (Technical Monitor)
2002-01-01
The Conjugate Gradient (CG) algorithm is perhaps the best-known iterative technique to solve sparse linear systems that are symmetric and positive definite. For systems that are ill-conditioned, it is often necessary to use a preconditioning technique. In this paper, we investigate the effects of various ordering and partitioning strategies on the performance of parallel CG and ILU(O) preconditioned CG (PCG) using different programming paradigms and architectures. Results show that for this class of applications: ordering significantly improves overall performance on both distributed and distributed shared-memory systems, that cache reuse may be more important than reducing communication, that it is possible to achieve message-passing performance using shared-memory constructs through careful data ordering and distribution, and that a hybrid MPI+OpenMP paradigm increases programming complexity with little performance gains. A implementation of CG on the Cray MTA does not require special ordering or partitioning to obtain high efficiency and scalability, giving it a distinct advantage for adaptive applications; however, it shows limited scalability for PCG due to a lack of thread level parallelism.
Eigenvalue distributions of Wilson loops
Energy Technology Data Exchange (ETDEWEB)
Lohmayer, Robert
2010-07-01
In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the
Hine, N D M; Haynes, P D; Mostofi, A A; Payne, M C
2010-09-21
We present calculations of formation energies of defects in an ionic solid (Al(2)O(3)) extrapolated to the dilute limit, corresponding to a simulation cell of infinite size. The large-scale calculations required for this extrapolation are enabled by developments in the approach to parallel sparse matrix algebra operations, which are central to linear-scaling density-functional theory calculations. The computational cost of manipulating sparse matrices, whose sizes are determined by the large number of basis functions present, is greatly improved with this new approach. We present details of the sparse algebra scheme implemented in the ONETEP code using hierarchical sparsity patterns, and demonstrate its use in calculations on a wide range of systems, involving thousands of atoms on hundreds to thousands of parallel processes.
A filtering method for the interval eigenvalue problem
DEFF Research Database (Denmark)
Hladik, Milan; Daney, David; Tsigaridas, Elias
2011-01-01
We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method...
Generalization of Samuelson's inequality and location of eigenvalues
Indian Academy of Sciences (India)
Bounds for the eigenvalues are obtained when a given complex × matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations. Author Affiliations. R Sharma1 R Saini1. Department of Mathematics, H. P. University, Shimla 171 005, India. Dates. Manuscript received: 7 September 2013 ...
Eigenvalues and expansion of bipartite graphs
DEFF Research Database (Denmark)
Høholdt, Tom; Janwa, Heeralal
2012-01-01
We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by ...
Numerical pole assignment by eigenvalue Jacobian inversion
Sevaston, George E.
1986-01-01
A numerical procedure for solving the linear pole placement problem is developed which operates by the inversion of an analytically determined eigenvalue Jacobian matrix. Attention is given to convergence characteristics and pathological situations. It is not concluded that the algorithm developed is suitable for computer-aided control system design with particular reference to the scan platform pointing control system for the Galileo spacecraft.
Eigenvalue ratio detection based on exact moments of smallest and largest eigenvalues
Shakir, Muhammad
2011-01-01
Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes always depend on asymptotic assumptions since the close-formed expression of exact eigenvalues ratio distribution is exceptionally complex to compute in practice. In this paper, non-asymptotic spectrum sensing approach to approximate the extreme eigenvalues is introduced. In this context, the Gaussian approximation approach based on exact analytical moments of extreme eigenvalues is presented. In this approach, the extreme eigenvalues are considered as dependent Gaussian random variables such that the joint probability density function (PDF) is approximated by bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. In this context, the definition of Copula is cited to analyze the extent of the dependency between the extreme eigenvalues. Later, the decision threshold based on the ratio of dependent Gaussian extreme eigenvalues is derived. The performance analysis of our newly proposed approach is compared with the already published asymptotic Tracy-Widom approximation approach. © 2011 ICST.
Shakir, Muhammad
2011-12-01
In this paper, we introduce a new detector referred to as Geometric mean detector (GEMD) which is based on the ratio of the largest eigenvalue to the Geometric mean of the eigenvalues for collaborative spectrum sensing. The decision threshold has been derived by employing Gaussian approximation approach. In this approach, the two random variables, i.e. The largest eigenvalue and the Geometric mean of the eigenvalues are considered as independent Gaussian random variables such that their cumulative distribution functions (CDFs) are approximated by a univariate Gaussian distribution function for any number of cooperating secondary users and received samples. The approximation approach is based on the calculation of exact analytical moments of the largest eigenvalue and the Geometric mean of the eigenvalues of the received covariance matrix. The decision threshold has been calculated by exploiting the CDF of the ratio of two Gaussian distributed random variables. In this context, we exchange the analytical moments of the two random variables with the moments of the Gaussian distribution function. The performance of the detector is compared with the performance of the energy detector and eigenvalue ratio detector. Analytical and simulation results show that our newly proposed detector yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, our results based on proposed approximation approach are in perfect agreement with the empirical results. © 2011 IEEE.
On Selberg's small eigenvalue conjecture and residual eigenvalues
DEFF Research Database (Denmark)
Risager, Morten S.
2011-01-01
We show that Selberg’s eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbat......We show that Selberg’s eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic...
DEFF Research Database (Denmark)
Lindberg, Erik
1997-01-01
In order to obtain insight in the nature of nonlinear oscillators the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator are found and displayed as functions of time. A number of oscillators are studied including Dewey's oscillator (piecewise linear wit...... with negative resistance), Kennedy's Colpitts-oscillator (with and without chaos) and a new 4'th order oscillator with hyper-chaos.......In order to obtain insight in the nature of nonlinear oscillators the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator are found and displayed as functions of time. A number of oscillators are studied including Dewey's oscillator (piecewise linear...
Indian Academy of Sciences (India)
The vibrating string problem is the source of much mathematicsand physics. This article describes Lagrange's formulationof a discretised version of the problem and its solution.This is also the first instance of an eigenvalue problem. Author Affiliations. Rajendra Bhatia1. Ashoka University, Rai, Haryana 131 029, India.
Random eigenvalue problems revisited
Indian Academy of Sciences (India)
The description of real-life engineering structural systems is associated with some amount of uncertainty in specifying material properties, geometric parameters, boundary conditions and applied loads. In the context of structural dynamics it is necessary to consider random eigenvalue problems in order to account for these ...
Random eigenvalue problems revisited
Indian Academy of Sciences (India)
Several studies have been conducted on this topic since the mid-sixties. The. A list of .... Random eigenvalue problems revisited. 297 and various elements of Hij ,i ≤ j are statistically independent and Gaussian. The pdf of H can be expressed as, ...... Generality of this result however remains to be verified in future studies.
Wu, Sheng-Jhih; Chu, Moody T.
2017-08-01
An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.
Extreme eigenvalues of sample covariance and correlation matrices
DEFF Research Database (Denmark)
Heiny, Johannes
2017-01-01
This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... that the extreme eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables. The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high...... dimension of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals...
Extreme eigenvalues of sample covariance and correlation matrices
DEFF Research Database (Denmark)
Heiny, Johannes
This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables. The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension...... of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals...
Xie, Jianwen; Douglas, Pamela K; Wu, Ying Nian; Brody, Arthur L; Anderson, Ariana E
2017-04-15
Brain networks in fMRI are typically identified using spatial independent component analysis (ICA), yet other mathematical constraints provide alternate biologically-plausible frameworks for generating brain networks. Non-negative matrix factorization (NMF) would suppress negative BOLD signal by enforcing positivity. Spatial sparse coding algorithms (L1 Regularized Learning and K-SVD) would impose local specialization and a discouragement of multitasking, where the total observed activity in a single voxel originates from a restricted number of possible brain networks. The assumptions of independence, positivity, and sparsity to encode task-related brain networks are compared; the resulting brain networks within scan for different constraints are used as basis functions to encode observed functional activity. These encodings are then decoded using machine learning, by using the time series weights to predict within scan whether a subject is viewing a video, listening to an audio cue, or at rest, in 304 fMRI scans from 51 subjects. The sparse coding algorithm of L1 Regularized Learning outperformed 4 variations of ICA (pICA and sparse coding algorithms. Holding constant the effect of the extraction algorithm, encodings using sparser spatial networks (containing more zero-valued voxels) had higher classification accuracy (pICA. Negative BOLD signal may capture task-related activations. Copyright © 2017 Elsevier B.V. All rights reserved.
The Schroedinger eigenvalue march
Energy Technology Data Exchange (ETDEWEB)
Tannous, C; Langlois, J, E-mail: tannous@univ-brest.fr [Laboratoire de Magnetisme de Bretagne, CNRS-FRE 3117, Universite de Bretagne Occidentale, BP: 809 Brest CEDEX 29285 (France)
2011-11-15
A simple numerical method for the determination of Schroedinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric double-well (ammonia inversion problem) and Johnson asymmetric potentials, 3D hydrogen atom and Morse potential. Band structure calculation can equally be tackled by extending marching to the complex plane as illustrated with the Kronig-Penney problem.
Bogiatzis, Petros; Ishii, Miaki; Davis, Timothy A.
2016-05-01
For more than two decades, the number of data and model parameters in seismic tomography problems has exceeded the available computational resources required for application of direct computational methods, leaving iterative solvers the only option. One disadvantage of the iterative techniques is that the inverse of the matrix that defines the system is not explicitly formed, and as a consequence, the model resolution and covariance matrices cannot be computed. Despite the significant effort in finding computationally affordable approximations of these matrices, challenges remain, and methods such as the checkerboard resolution tests continue to be used. Based upon recent developments in sparse algorithms and high-performance computing resources, we show that direct methods are becoming feasible for large seismic tomography problems. We demonstrate the application of QR factorization in solving the regional P-wave structure and computing the full resolution matrix with 267 520 model parameters.
DEFF Research Database (Denmark)
Montoya-Martinez, Jair; Artes-Rodriguez, Antonio; Pontil, Massimiliano
2014-01-01
We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured...... matrix and the squared Frobenius norm of the latent source matrix. We develop an alternating optimization algorithm to solve the resulting nonsmooth-nonconvex minimization problem. We analyze the convergence of the optimization procedure, and we compare, under different synthetic scenarios...
Bodewig, E
1959-01-01
Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements. The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant. The text ponders on approximate solutions, as well
Maximal imaginery eigenvalues in optimal systems
Directory of Open Access Journals (Sweden)
David Di Ruscio
1991-07-01
Full Text Available In this note we present equations that uniquely determine the maximum possible imaginary value of the closed loop eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen, provided a real symmetric solution of the algebraic Riccati equation exists. In addition, the corresponding state weight matrix and the solution to the algebraic Riccati equation are derived for a class of linear systems. A fundamental lemma for the existence of a real symmetric solution to the algebraic Riccati equation is derived for this class of linear systems.
On-The-Fly Computation of Frontal Orbitals in Density Matrix Expansions.
Kruchinina, Anastasia; Rudberg, Elias; Rubensson, Emanuel H
2017-12-01
We propose a method for computation of frontal (homo and lumo) orbitals in recursive polynomial expansion algorithms for the density matrix. Such algorithms give a computational cost that increases only linearly with system size for sufficiently sparse systems but a drawback compared to the traditional diagonalization approach is that molecular orbitals are not readily available. Our method is based on the idea to use the polynomial of the density matrix expansion as an eigenvalue filter giving large separation between eigenvalues around homo and lumo [J. Chem. Phys. 128, 176101 (2008)]. This filter is combined with a shift-and-square (folded spectrum) method to move the desired eigenvalue to the end of the spectrum. In this work we propose a transparent way to select recursive expansion iteration and shift for the eigenvector computation that results in a sharp eigenvalue filter. The filter is obtained as a by-product of the density matrix expansion and there is no significant additional cost associated neither with its construction or with its application. This gives a clear-cut and efficient eigenvalue solver that can be used to compute homo and lumo orbitals with sufficient accuracy in a small fraction of the total recursive expansion time. Our algorithms make use of recent homo and lumo eigenvalue estimates that can be obtained at negligible cost [SIAM J. Sci. Comput. 36, B147 (2014)]. We illustrate our method by performing self-consistent field calculations for large scale systems.
Matrix with Prescribed Eigenvectors
Ahmad, Faiz
2011-01-01
It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…
On computation of real eigenvalues of matrices via the Adomian decomposition
Directory of Open Access Journals (Sweden)
Hooman Fatoorehchi
2014-04-01
Full Text Available The problem of matrix eigenvalues is encountered in various fields of engineering endeavor. In this paper, a new approach based on the Adomian decomposition method and the Faddeev-Leverrier’s algorithm is presented for finding real eigenvalues of any desired real matrices. The method features accuracy and simplicity. In contrast to many previous techniques which merely afford one specific eigenvalue of a matrix, the method has the potential to provide all real eigenvalues. Also, the method does not require any initial guesses in its starting point unlike most of iterative techniques. For the sake of illustration, several numerical examples are included.
Shakir, Muhammad Zeeshan
2013-03-01
Eigenvalue Ratio (ER) detector based on the two extreme eigenvalues of the received signal covariance matrix is currently one of the most effective solution for spectrum sensing. However, the analytical results of such scheme often depend on asymptotic assumptions since the distribution of the ratio of two extreme eigenvalues is exceptionally complex to compute. In this paper, a non-asymptotic spectrum sensing approach for ER detector is introduced to approximate the marginal and joint distributions of the two extreme eigenvalues. The two extreme eigenvalues are considered as dependent Gaussian random variables such that their joint probability density function (PDF) is approximated by a bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. The PDF approximation approach is based on the moment matching method where we calculate the exact analytical moments of joint and marginal distributions of the two extreme eigenvalues. The decision threshold is calculated by exploiting the statistical mean and the variance of each of the two extreme eigenvalues and the correlation coefficient between them. The performance analysis of our newly proposed approximation approach is compared with the already published asymptotic Tracy-Widom approximation approach. It has been shown that our results are in perfect agreement with the simulation results for any number of secondary users and received samples. © 2002-2012 IEEE.
Hou, Gary Y; Provost, Jean; Grondin, Julien; Wang, Shutao; Marquet, Fabrice; Bunting, Ethan; Konofagou, Elisa E
2014-11-01
Harmonic motion imaging for focused ultrasound (HMIFU) utilizes an amplitude-modulated HIFU beam to induce a localized focal oscillatory motion simultaneously estimated. The objective of this study is to develop and show the feasibility of a novel fast beamforming algorithm for image reconstruction using GPU-based sparse-matrix operation with real-time feedback. In this study, the algorithm was implemented onto a fully integrated, clinically relevant HMIFU system. A single divergent transmit beam was used while fast beamforming was implemented using a GPU-based delay-and-sum method and a sparse-matrix operation. Axial HMI displacements were then estimated from the RF signals using a 1-D normalized cross-correlation method and streamed to a graphic user interface with frame rates up to 15 Hz, a 100-fold increase compared to conventional CPU-based processing. The real-time feedback rate does not require interrupting the HIFU treatment. Results in phantom experiments showed reproducible HMI images and monitoring of 22 in vitro HIFU treatments using the new 2-D system demonstrated reproducible displacement imaging, and monitoring of 22 in vitro HIFU treatments using the new 2-D system showed a consistent average focal displacement decrease of 46.7 ±14.6% during lesion formation. Complementary focal temperature monitoring also indicated an average rate of displacement increase and decrease with focal temperature at 0.84±1.15%/(°)C, and 2.03±0.93%/(°)C , respectively. These results reinforce the HMIFU capability of estimating and monitoring stiffness related changes in real time. Current ongoing studies include clinical translation of the presented system for monitoring of HIFU treatment for breast and pancreatic tumor applications.
Eigenvalue Conditions for Induced Subgraphs
Directory of Open Access Journals (Sweden)
Harant Jochen
2015-05-01
Full Text Available Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.
Round, Simon; Schafmeister, Frank; Heldwein, Marcelo; Pereira, Eduardo; Serpa, Leonardo; Kolar, Johann
This paper undertakes a comparison of the very sparse matrix converter (VSMC) and the back-to-back voltage DC-link converter (BBC) for a permanent magnetic synchronous motor drive application. The VSMC has the same functionality as the conventional matrix converter but a reduced number of switches and lower control complexity. The two converters are designed, using the same IGBT power modules, with a switching frequency of 40kHz and a thermal rating of 6.8kW at an ambient temperature of 45°C. From the design, the volume of the VSMC is 2.3 liters and is half that of the BBC. The efficiency for the VSMC at full load is 94.5% compared to 92% for the BBC. At very low output frequencies the output current of the VSMC can be increased by 25% above nominal compared to a 54% decrease for the BBC. Overall the VSMC offers advantages in volume and efficiency for motor drive applications requiring switching frequencies above 10kHz.
Extreme eigenvalues of sample covariance and correlation matrices
DEFF Research Database (Denmark)
Heiny, Johannes
of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals......This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index
Extreme eigenvalues of sample covariance and correlation matrices
DEFF Research Database (Denmark)
Heiny, Johannes
2017-01-01
dimension of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals......This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... matrix of a $p$-dimensional heavy-tailed time series when $p$ converges to infinity together with the sample size $n$. We generalize the growth rates of $p$ existing in the literature. Assuming a regular variation condition with tail index $\\alpha
On solving energy-dependent partitioned eigenvalue problem by ...
Indian Academy of Sciences (India)
Abstract. An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of genetic algorithm. The proposed method is tested with matrices of different sizes (30 × 30 to 1000 × 1000). Com- parison is made with Löwdin's strategy for solving the ...
On solving energy-dependent partitioned eigenvalue problem by ...
Indian Academy of Sciences (India)
An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of genetic algorithm. The proposed method is tested with matrices of different sizes (30 × 30 to 1000 × 1000). Comparison is made with Löwdin's strategy for solving the problem.
Basic sparse computations on data parallel computers
Energy Technology Data Exchange (ETDEWEB)
Ferng, W.R. [National Chiao-Tung Univ. (Taiwan, Province of China); Wu, K. [Univ. of Minnesota, Minneapolis, MN (United States); Petiton, S.
1993-12-31
This paper presents a preliminary experimental study of the performance of basic sparse matrix computations on the CM-200 and the CM-5 massively parallel computers. We concentrate on examining various ways of performing general sparse matrix-vector operations and the basic primitives on which there are based in SIMD data parallel mode. Various data structures for storing sparse matrices and their corresponding matrix-vector multiplications are compared.
Perturbation Theory of Embedded Eigenvalues
DEFF Research Database (Denmark)
Engelmann, Matthias
project gives a general and systematic approach to analytic perturbation theory of embedded eigenvalues. The spectral deformation technique originally developed in the theory of dilation analytic potentials in the context of Schrödinger operators is systematized by the use of Mourre theory. The group......We study problems connected to perturbation theory of embedded eigenvalues in two different setups. The first part deals with second order perturbation theory of mass shells in massive translation invariant Nelson type models. To this end an expansion of the eigenvalues w.r.t. fiber parameter up...... of dilations is thereby replaced by the unitary group generated y the conjugate operator. This then allows to treat the perturbation problem with the usual Kato theory....
Directory of Open Access Journals (Sweden)
Seyed Sina Sebtahmadi
2016-11-01
Full Text Available A rotational d-q current control scheme based on a Particle Swarm Optimization- Proportional-Integral (PSO-PI controller, is used to drive an induction motor (IM through an Ultra Sparse Z-source Matrix Converter (USZSMC. To minimize the overall size of the system, the lowest feasible values of Z-source elements are calculated by considering the both timing and aspects of the circuit. A meta-heuristic method is integrated to the control system in order to find optimal coefficient values in a single multimodal problem. Henceforth, the effect of all coefficients in minimizing the total harmonic distortion (THD and balancing the stator current are considered simultaneously. Through changing the reference point of magnitude or frequency, the modulation index can be automatically adjusted and respond to changes without heavy computational cost. The focus of this research is on a reliable and lightweight system with low computational resources. The proposed scheme is validated through both simulation and experimental results.
Zhang, Shao-Liang; Imamura, Toshiyuki; Yamamoto, Yusaku; Kuramashi, Yoshinobu; Hoshi, Takeo
2017-01-01
This book provides state-of-the-art and interdisciplinary topics on solving matrix eigenvalue problems, particularly by using recent petascale and upcoming post-petascale supercomputers. It gathers selected topics presented at the International Workshops on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2014 and EPASA2015), which brought together leading researchers working on the numerical solution of matrix eigenvalue problems to discuss and exchange ideas – and in so doing helped to create a community for researchers in eigenvalue problems. The topics presented in the book, including novel numerical algorithms, high-performance implementation techniques, software developments and sample applications, will contribute to various fields that involve solving large-scale eigenvalue problems.
Solving the RPA eigenvalue equation in real-space
Muta, A; Hashimoto, Y; Yabana, K
2002-01-01
We present a computational method to solve the RPA eigenvalue equation employing a uniform grid representation in three-dimensional Cartesian coordinates. The conjugate gradient method is used for this purpose as an interactive method for a generalized eigenvalue problem. No construction of unoccupied orbitals is required in the procedure. We expect this method to be useful for systems lacking spatial symmetry to calculate accurate eigenvalues and transition matrix elements of a few low-lying excitations. Some applications are presented to demonstrate the feasibility of the method, considering the simplified mean-field model as an example of a nuclear physics system and the electronic excitations in molecules with time-dependent density functional theory as an example of an electronic system. (author)
Fast Solution in Sparse LDA for Binary Classification
Moghaddam, Baback
2010-01-01
An algorithm that performs sparse linear discriminant analysis (Sparse-LDA) finds near-optimal solutions in far less time than the prior art when specialized to binary classification (of 2 classes). Sparse-LDA is a type of feature- or variable- selection problem with numerous applications in statistics, machine learning, computer vision, computational finance, operations research, and bio-informatics. Because of its combinatorial nature, feature- or variable-selection problems are NP-hard or computationally intractable in cases involving more than 30 variables or features. Therefore, one typically seeks approximate solutions by means of greedy search algorithms. The prior Sparse-LDA algorithm was a greedy algorithm that considered the best variable or feature to add/ delete to/ from its subsets in order to maximally discriminate between multiple classes of data. The present algorithm is designed for the special but prevalent case of 2-class or binary classification (e.g. 1 vs. 0, functioning vs. malfunctioning, or change versus no change). The present algorithm provides near-optimal solutions on large real-world datasets having hundreds or even thousands of variables or features (e.g. selecting the fewest wavelength bands in a hyperspectral sensor to do terrain classification) and does so in typical computation times of minutes as compared to days or weeks as taken by the prior art. Sparse LDA requires solving generalized eigenvalue problems for a large number of variable subsets (represented by the submatrices of the input within-class and between-class covariance matrices). In the general (fullrank) case, the amount of computation scales at least cubically with the number of variables and thus the size of the problems that can be solved is limited accordingly. However, in binary classification, the principal eigenvalues can be found using a special analytic formula, without resorting to costly iterative techniques. The present algorithm exploits this analytic
Novel Approach for Calculation and Analysis of Eigenvalues and Eigenvectors in Microgrids: Preprint
Energy Technology Data Exchange (ETDEWEB)
Li, Y.; Gao, W.; Muljadi, E.; Jiang, J.
2014-02-01
This paper proposes a novel approach based on matrix perturbation theory to calculate and analyze eigenvalues and eigenvectors in a microgrid system. Rigorous theoretical analysis to solve eigenvalues and the corresponding eigenvectors for a system under various perturbations caused by fluctuations of irradiance, wind speed, or loads is presented. A computational flowchart is proposed for the unified solution of eigenvalues and eigenvectors in microgrids, and the effectiveness of the matrix perturbation-based approach in microgrids is verified by numerical examples on a typical low-voltage microgrid network.
Directory of Open Access Journals (Sweden)
Maria Malejki
2007-01-01
Full Text Available We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space \\(l^2(\\mathbb{N}\\ by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with \\(n\\to \\infty\\ of the joint error of approximation for the first \\(n\\ eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order \\(n \\times n\\. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004, 39-54]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.
Eigenvalue spacings for quantized cat maps
Gamburd, A; Rockmore, D
2003-01-01
According to one of the basic conjectures in quantum chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms - 'cat maps'. Our numerical experiments indicate that for 'generic' choices of cat maps, the unfolded consecutive spacing distribution in the irreducible components of the Nth quantization (given by the N-dimensional Weil representation) approaches the GOE/GSE law of random matrix theory. For certain special 'arithmetic' transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.
Highly indefinite multigrid for eigenvalue problems
Energy Technology Data Exchange (ETDEWEB)
Borges, L.; Oliveira, S.
1996-12-31
Eigenvalue problems are extremely important in understanding dynamic processes such as vibrations and control systems. Large scale eigenvalue problems can be very difficult to solve, especially if a large number of eigenvalues and the corresponding eigenvectors need to be computed. For solving this problem a multigrid preconditioned algorithm is presented in {open_quotes}The Davidson Algorithm, preconditioning and misconvergence{close_quotes}. Another approach for solving eigenvalue problems is by developing efficient solutions for highly indefinite problems. In this paper we concentrate on the use of new highly indefinite multigrid algorithms for the eigenvalue problem.
Calculating alpha Eigenvalues in a Continuous-Energy Infinite Medium with Monte Carlo
Energy Technology Data Exchange (ETDEWEB)
Betzler, Benjamin R. [Los Alamos National Laboratory; Kiedrowski, Brian C. [Los Alamos National Laboratory; Brown, Forrest B. [Los Alamos National Laboratory; Martin, William R. [Los Alamos National Laboratory
2012-09-04
The {alpha} eigenvalue has implications for time-dependent problems where the system is sub- or supercritical. We present methods and results from calculating the {alpha}-eigenvalue spectrum for a continuous-energy infinite medium with a simplified Monte Carlo transport code. We formulate the {alpha}-eigenvalue problem, detail the Monte Carlo code physics, and provide verification and results. We have a method for calculating the {alpha}-eigenvalue spectrum in a continuous-energy infinite-medium. The continuous-time Markov process described by the transition rate matrix provides a way of obtaining the {alpha}-eigenvalue spectrum and kinetic modes. These are useful for the approximation of the time dependence of the system.
Generalization of Samuelson's inequality and location of eigenvalues
Indian Academy of Sciences (India)
Refinements, extensions and applications of such inequalities have been studied exten- sively in the literature (see [2, ... have observed that if the eigenvalues of an n × n complex matrix are real, as in the case of Hermitian ... k = 1, 2, ··· . (2.3). Applying (2.3) to n − 1 positive real numbers (xi − ¯x)2 ,i = 1, 2,...,n and i = j, we get.
Characterizing and approximating eigenvalue sets of symmetric interval matrices
DEFF Research Database (Denmark)
Hladík, Milan; Daney, David; Tsigaridas, Elias
2011-01-01
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed, with perturbations belonging to some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorith......, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illustrate our approach by several examples and numerical experiments....
Modified Hybrid Freeman/Eigenvalue Decomposition for Polarimetric SAR Data
Directory of Open Access Journals (Sweden)
Shuang Zhang
2015-01-01
Full Text Available Because of the rapid advancement of the airborne sensors and spaceborne sensors, large volumes of fully polarimetric synthetic aperture radar (PolSAR data are available, but they are too complex to interpret difficultly. In this paper, a modified hybrid Freeman/eigenvalue decomposition method for the coherency matrix derived from the fully PolSAR sensors is proposed. The proposed modified hybrid Freeman/eigenvalue decomposition uses a real unitary transformation on the coherency matrix to release correlations between the copolarized term and cross polarized term, and the scattering models are derived from eigenvectors of the coherency matrix with reflection symmetry condition. The anisotropy and entropy are used to determine whether the volume scattering component is derived from the man-made structures or not. Moreover, the scattering powers from the proposed hybrid Freeman/eigenvalue decomposition are all nonnegative values. Fully PolSAR data on San Francisco acquired by AIRSAR sensor are used in the experiments to prove the efficacy of the proposed decomposition.
Current Developments of Sparse Microwave Imaging
Directory of Open Access Journals (Sweden)
Wu Yi-rong
2014-08-01
Full Text Available The sparse microwave imaging combines the sparse signal processing theory with radar imaging to obtain new theory, new system, and new methodology of microwave imaging. In this paper, a brief review of fundamental issues in applying sparse signal processing to radar imaging is provided, including sparse representation, measurement matrix construction, unambiguity reconstruction, and so on. The developments of sparse signal processing in microwave imaging are discussed, and the initial airborne experiments on the prototype Synthetic Aperture Radar (SAR framework with sparse constraints are introduced. The results demonstrate the feasibility and effectiveness of the principle and methodology of sparse microwave imaging. Besides, we also provide an overview of sparse signal processing in various radar applications, including Tomographic SAR (TomoSAR, Inverse SAR (ISAR, Ground Penetrating Radar (GPR as well.
On the limit of extreme eigenvalues of large dimensional random quaternion matrices
Energy Technology Data Exchange (ETDEWEB)
Yin, Yanqing, E-mail: yinyq799@nenu.edu.cn; Bai, Zhidong, E-mail: baizd@nenu.edu.cn; Hu, Jiang, E-mail: huj156@nenu.edu.cn
2014-03-01
Since E.P. Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case.
Arbitrary eigenvalue assignments for linear time-varying multivariable control systems
Nguyen, Charles C.
1987-01-01
The problem of eigenvalue assignments for a class of linear time-varying multivariable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is 'lexicography-fixedly controllable' can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.
DEFF Research Database (Denmark)
Han, Xixuan; Clemmensen, Line Katrine Harder
2015-01-01
We propose a general technique for obtaining sparse solutions to generalized eigenvalue problems, and call it Regularized Generalized Eigen-Decomposition (RGED). For decades, Fisher's discriminant criterion has been applied in supervised feature extraction and discriminant analysis...... techniques, for instance, 2D-Linear Discriminant Analysis (2D-LDA). Furthermore, an iterative algorithm based on the alternating direction method of multipliers is developed. The algorithm approximately solves RGED with monotonically decreasing convergence and at an acceptable speed for results of modest...... accuracy. Numerical experiments based on four data sets of different types of images show that RGED has competitive classification performance with existing multidimensional and sparse techniques of discriminant analysis....
Parallel transposition of sparse data structures
DEFF Research Database (Denmark)
Wang, Hao; Liu, Weifeng; Hou, Kaixi
2016-01-01
Many applications in computational sciences and social sciences exploit sparsity and connectivity of acquired data. Even though many parallel sparse primitives such as sparse matrix-vector (SpMV) multiplication have been extensively studied, some other important building blocks, e.g., parallel...
Andric, I; Jurman, D; Nielsen, H B
2007-01-01
We discuss a dynamical matrix model by which probability distribution is associated with Gaussian ensembles from random matrix theory. We interpret the matrix M as a Hamiltonian representing interaction of a bosonic system with a single fermion. We show that a system of second-quantized fermions influences the ground state of the whole system by producing a gap between the highest occupied eigenvalue and the lowest unoccupied eigenvalue.
Sparse PCA with Oracle Property.
Gu, Quanquan; Wang, Zhaoran; Liu, Han
In this paper, we study the estimation of the k-dimensional sparse principal subspace of covariance matrix Σ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-k, and attains a [Formula: see text] statistical rate of convergence with s being the subspace sparsity level and n the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.
Matrix theory selected topics and useful results
Mehta, Madan Lal
1989-01-01
Matrices and operations on matrices ; determinants ; elementary operations on matrices (continued) ; eigenvalues and eigenvectors, diagonalization of normal matrices ; functions of a matrix ; positive definiteness, various polar forms of a matrix ; special matrices ; matrices with quaternion elements ; inequalities ; generalised inverse of a matrix ; domain of values of a matrix, location and dispersion of eigenvalues ; symmetric functions ; integration over matrix variables ; permanents of doubly stochastic matrices ; infinite matrices ; Alexander matrices, knot polynomials, torsion numbers.
Sparse Frequency Waveform Design for Radar-Embedded Communication
Directory of Open Access Journals (Sweden)
Chaoyun Mai
2016-01-01
Full Text Available According to the Tag application with function of covert communication, a method for sparse frequency waveform design based on radar-embedded communication is proposed. Firstly, sparse frequency waveforms are designed based on power spectral density fitting and quasi-Newton method. Secondly, the eigenvalue decomposition of the sparse frequency waveform sequence is used to get the dominant space. Finally the communication waveforms are designed through the projection of orthogonal pseudorandom vectors in the vertical subspace. Compared with the linear frequency modulation waveform, the sparse frequency waveform can further improve the bandwidth occupation of communication signals, thus achieving higher communication rate. A certain correlation exists between the reciprocally orthogonal communication signals samples and the sparse frequency waveform, which guarantees the low SER (signal error rate and LPI (low probability of intercept. The simulation results verify the effectiveness of this method.
Frequency response as a surrogate eigenvalue problem in topology optimization
DEFF Research Database (Denmark)
Andreassen, Erik; Ferrari, Federico; Sigmund, Ole
2017-01-01
This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity...... associated with computation of eigenvalues in very large problems....
A spectral algorithm for envelope reduction of sparse matrices
Energy Technology Data Exchange (ETDEWEB)
Barnard, S.T.; Simon, H.D. [NASA Ames Research Center, Moffett Field, CA (United States). Cray Research Inc.; Pothen, A. [Univ. of Waterloo, Ontario (Canada). Computer Science Dept.
1993-12-31
A new algorithm for reducing the envelope of a sparse matrix is presented. This algorithm is based on the computation of eigenvectors of the Laplacian matrix associated with the graph of the sparse matrix. A reordering of the sparse matrix is determined based on the numerical values of the entries of an eigenvector of the Laplacian matrix. Numerical results show that the new reordering algorithm can in some cases reduce the envelope by more than a factor of two over the current standard algorithms such as Gibbs-Poole-Stockmeyer (GPS) or SPARSPAK`s reverse Cuthill-McKee (RCM).
Directory of Open Access Journals (Sweden)
Yuuki Kanemiyo
2012-01-01
Full Text Available This paper described a spatial correlation and eigenvalue in a multiple-input multiple-output (MIMO channel. A MIMO channel model with a multipath propagation mechanism was proposed and showed the channel matrix. The spatial correlation coefficient formula −,′−′( between MIMO channel matrix elements was derived for the model and was expressed as a directive wave term added to the product of mobile site correlation −′( and base site correlation −′( without LOS path, which are calculated independently of each other. By using −,′−′(, it is possible to create the channel matrix element with a fixed correlation value estimated by −,′−′( for a given multipath condition and a given antenna configuration. Furthermore, the correlation and the channel matrix eigenvalue were simulated, and the simulated and theoretical correlation values agreed well. The simulated eigenvalue showed that the average of the first eigenvalue λ1 hardly depends on the correlation −,′−′(, but the others do depend on −,′−′( and approach 1 as −,′−′( decreases. Moreover, as the path moves into LOS, the 1 state with mobile movement becomes more stable than the 1 of NLOS path.
Sorting waves and associated eigenvalues
Carbonari, Costanza; Colombini, Marco; Solari, Luca
2017-04-01
The presence of mixed sediment always characterizes gravel bed rivers. Sorting processes take place during bed load transport of heterogeneous sediment mixtures. The two main elements necessary to the occurrence of sorting are the heterogeneous character of sediments and the presence of an active sediment transport. When these two key ingredients are simultaneously present, the segregation of bed material is consistently detected both in the field [7] and in laboratory [3] observations. In heterogeneous sediment transport, bed altimetric variations and sorting always coexist and both mechanisms are independently capable of driving the formation of morphological patterns. Indeed, consistent patterns of longitudinal and transverse sorting are identified almost ubiquitously. In some cases, such as bar formation [2] and channel bends [5], sorting acts as a stabilizing effect and therefore the dominant mechanism driving pattern formation is associated with bed altimetric variations. In other cases, such as longitudinal streaks, sorting enhances system instability and can therefore be considered the prevailing mechanism. Bedload sheets, first observed by Khunle and Southard [1], represent another classic example of a morphological pattern essentially triggered by sorting, as theoretical [4] and experimental [3] results suggested. These sorting waves cause strong spatial and temporal fluctuations of bedload transport rate typical observed in gravel bed rivers. The problem of bed load transport of a sediment mixture is formulated in the framework of a 1D linear stability analysis. The base state consists of a uniform flow in an infinitely wide channel with active bed load transport. The behaviour of the eigenvalues associated with fluid motion, bed evolution and sorting processes in the space of the significant flow and sediment parameters is analysed. A comparison is attempted with the results of the theoretical analysis of Seminara Colombini and Parker [4] and Stecca
Directory of Open Access Journals (Sweden)
Philip Wong
Full Text Available Accurate reconstruction of 3D photoacoustic (PA images requires detection of photoacoustic signals from many angles. Several groups have adopted staring ultrasound arrays, but assessment of array performance has been limited. We previously reported on a method to calibrate a 3D PA tomography (PAT staring array system and analyze system performance using singular value decomposition (SVD. The developed SVD metric, however, was impractical for large system matrices, which are typical of 3D PAT problems. The present study consisted of two main objectives. The first objective aimed to introduce the crosstalk matrix concept to the field of PAT for system design. Figures-of-merit utilized in this study were root mean square error, peak signal-to-noise ratio, mean absolute error, and a three dimensional structural similarity index, which were derived between the normalized spatial crosstalk matrix and the identity matrix. The applicability of this approach for 3D PAT was validated by observing the response of the figures-of-merit in relation to well-understood PAT sampling characteristics (i.e. spatial and temporal sampling rate. The second objective aimed to utilize the figures-of-merit to characterize and improve the performance of a near-spherical staring array design. Transducer arrangement, array radius, and array angular coverage were the design parameters examined. We observed that the performance of a 129-element staring transducer array for 3D PAT could be improved by selection of optimal values of the design parameters. The results suggested that this formulation could be used to objectively characterize 3D PAT system performance and would enable the development of efficient strategies for system design optimization.
Wong, Philip; Kosik, Ivan; Raess, Avery; Carson, Jeffrey J L
2015-01-01
Accurate reconstruction of 3D photoacoustic (PA) images requires detection of photoacoustic signals from many angles. Several groups have adopted staring ultrasound arrays, but assessment of array performance has been limited. We previously reported on a method to calibrate a 3D PA tomography (PAT) staring array system and analyze system performance using singular value decomposition (SVD). The developed SVD metric, however, was impractical for large system matrices, which are typical of 3D PAT problems. The present study consisted of two main objectives. The first objective aimed to introduce the crosstalk matrix concept to the field of PAT for system design. Figures-of-merit utilized in this study were root mean square error, peak signal-to-noise ratio, mean absolute error, and a three dimensional structural similarity index, which were derived between the normalized spatial crosstalk matrix and the identity matrix. The applicability of this approach for 3D PAT was validated by observing the response of the figures-of-merit in relation to well-understood PAT sampling characteristics (i.e. spatial and temporal sampling rate). The second objective aimed to utilize the figures-of-merit to characterize and improve the performance of a near-spherical staring array design. Transducer arrangement, array radius, and array angular coverage were the design parameters examined. We observed that the performance of a 129-element staring transducer array for 3D PAT could be improved by selection of optimal values of the design parameters. The results suggested that this formulation could be used to objectively characterize 3D PAT system performance and would enable the development of efficient strategies for system design optimization.
Gasbarra, Dario; Pajevic, Sinisa; Basser, Peter J.
2017-01-01
Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄. When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model. PMID:28989561
The eigenvalue problem in phase space.
Cohen, Leon
2017-07-27
We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Approximate inverse preconditioners for general sparse matrices
Energy Technology Data Exchange (ETDEWEB)
Chow, E.; Saad, Y. [Univ. of Minnesota, Minneapolis, MN (United States)
1994-12-31
Preconditioned Krylov subspace methods are often very efficient in solving sparse linear matrices that arise from the discretization of elliptic partial differential equations. However, for general sparse indifinite matrices, the usual ILU preconditioners fail, often because of the fact that the resulting factors L and U give rise to unstable forward and backward sweeps. In such cases, alternative preconditioners based on approximate inverses may be attractive. We are currently developing a number of such preconditioners based on iterating on each column to get the approximate inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., we must use sparse-matrix by sparse-vector type operatoins. We will discuss a few options and compare their performance on standard problems from the Harwell-Boeing collection.
Eigenvalues of Covariance Matrix for Two-Source Array Processing
1990-11-01
the need to perform eigen-decompositions. The works by Munier and Delisle [33], Reilly, Chen, and Wong [43], and Friedlander [19] all appeared in 1988...Architectures for Signal Processing III, SPIE vol. 975, Aug 1988, pp. 1 0 1 -1 0 7 . [33] Munier , J. G.Y. Delisle. "A New Algorithm for the
First Bloch eigenvalue in high contrast media
Energy Technology Data Exchange (ETDEWEB)
Briane, Marc, E-mail: mbriane@insa-rennes.fr [Institut de Recherche Mathématique de Rennes, INSA de Rennes (France); Vanninathan, Muthusamy, E-mail: vanni@math.tifrbng.res.in [TIFR-CAM, Bangalore (India)
2014-01-15
This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast εY-periodic conductivity. When the conductivity is bounded in L{sup 1} and the constant of the Poincaré-Wirtinger weighted by the conductivity is very small with respect to ε{sup −2}, the first Bloch eigenvalue converges as ε → 0 to a limit which preserves the second-order expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourth-order under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a L{sup 1}-unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given low-contrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch parameter around zero.
Analysis of eigenvalue correction applied to biometrics
Hendrikse, A.J.; Veldhuis, Raymond N.J.; Spreeuwers, Lieuwe Jan; Bazen, A.M.
Eigenvalue estimation plays an important role in biometrics. However, if the number of samples is limited, estimates are significantly biased. In this article we analyse the influence of this bias on the error rates of PCA/LDA based verification systems, using both synthetic data with realistic
Multiplicities of an Eigenvalue: Some Observations
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 7; Issue 12. Multiplicities of an Eigenvalue: Some Observations. M Thamban Nair. General Article Volume 7 ... Author Affiliations. M Thamban Nair1. Associate Professor Department of Mathematics, Indian Institute of Technology, Chennai 600 036, India.
Eigenvalues of the -Laplacian and disconjugacy criteria
Directory of Open Access Journals (Sweden)
Pinasco Juan P
2006-01-01
Full Text Available We derive oscillation and nonoscillation criteria for the one-dimensional -Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.
A note on quasilinear elliptic eigenvalue problems
Directory of Open Access Journals (Sweden)
Gianni Arioli
1999-11-01
Full Text Available We study an eigenvalue problem by a non-smooth critical point theory. Under general assumptions, we prove the existence of at least one solution as a minimum of a constrained energy functional. We apply some results on critical point theory with symmetry to provide a multiplicity result.
On the sensitivities of multiple eigenvalues
DEFF Research Database (Denmark)
Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang Manh
2011-01-01
polynomials of a number of eigenvalues, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding...
Energy Technology Data Exchange (ETDEWEB)
Basermann, A. [Central Institute for Applied Mathematics, Juelich (Germany)
1994-12-31
For the solution of discretized ordinary or partial differential equations it is necessary to solve systems of equations or eigenproblems with coefficient matrices of different sparsity pattern, depending on the discretization method; using the finite element method (FE) results in largely unstructured systems of equations. Sparse eigenproblems play particularly important roles in the analysis of elastic solids and structures. In the corresponding FE models, the natural frequencies and mode shapes of free vibration are determined as are buckling loads and modes. Another class of problems is related to stability analysis, e.g. of electrical networks. Moreover, approximations of extreme eigenvalues are useful for solving sets of linear equations, e.g. for determining condition numbers of symmetric positive definite matrices or for conjugate gradients methods with polynomial preconditioning. Iterative methods for solving linear systems and eigenproblems mainly consist of matrix-vector products and vector-vector operations; the main work in each iteration is usually the computation of matrix-vector products. Therein, accessing the vector is determined by the sparsity pattern and the storage scheme of the matrix.
On perturbation of eigenvalues embedded at thresholds in a two ...
Indian Academy of Sciences (India)
Abstract. We present some results on the perturbation of eigenvalues embedded at thresholds in a two channel model Hamiltonian with a small off-diagonal perturbation. Examples are given of the various types of behavior of the eigenvalue under perturbation.
Eigenvalue inequalities for the Laplacian with mixed boundary conditions
Lotoreichik, Vladimir; Rohleder, Jonathan
2017-07-01
Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.
Super-quantum curves from super-eigenvalue models
Energy Technology Data Exchange (ETDEWEB)
Ciosmak, Paweł [Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,ul. Banacha 2, 02-097 Warsaw (Poland); Hadasz, Leszek [M. Smoluchowski Institute of Physics, Jagiellonian University,ul. Łojasiewicza 11, 30-348 Kraków (Poland); Manabe, Masahide [Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw (Poland); Sułkowski, Piotr [Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw (Poland); Walter Burke Institute for Theoretical Physics, California Institute of Technology,1200 E. California Blvd, Pasadena, CA 91125 (United States)
2016-10-10
In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce β-deformed version of those models, and derive differential equations for associated α/β-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.
Eigenvalues of large sample covariance matrices of spiked population models
Baik, Jinho; Silverstein, Jack W.
2006-01-01
We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples.
On eigenvalue problems in quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Saha, Aparna; Talukdar, B [Department of Physics, Visva-Bharati University, Santiniketan 731235 (India); Das, Umapada, E-mail: binoy123@bsnl.in [Department of Physics, Abhedananda College, Sainthia 731234 (India)
2011-06-01
To solve quantum mechanical eigenvalue problems using the algorithmic methods recently derived by Nikiforov and Uvarov (1988 Special Functions of Mathematical Physics (Basel: Birkhaeuser)) and Ciftci et al (2003 J. Phys. A: Math. Gen. 36 11807), one needs to first convert the associated wave equation into hypergeometric or closely related forms. We point out that once such forms are obtained, the eigenvalue problem can be satisfactorily solved by only imposing the condition that the regular infinite series solutions of the equations should become polynomials, and one need not take recourse to the use of the algorithmic methods. We first demonstrate the directness and simplicity of our approach by dealing with a few case studies and then present new results for the Woods-Saxon potential.
Sparse Exploratory Factor Analysis.
Trendafilov, Nickolay T; Fontanella, Sara; Adachi, Kohei
2017-07-13
Sparse principal component analysis is a very active research area in the last decade. It produces component loadings with many zero entries which facilitates their interpretation and helps avoid redundant variables. The classic factor analysis is another popular dimension reduction technique which shares similar interpretation problems and could greatly benefit from sparse solutions. Unfortunately, there are very few works considering sparse versions of the classic factor analysis. Our goal is to contribute further in this direction. We revisit the most popular procedures for exploratory factor analysis, maximum likelihood and least squares. Sparse factor loadings are obtained for them by, first, adopting a special reparameterization and, second, by introducing additional [Formula: see text]-norm penalties into the standard factor analysis problems. As a result, we propose sparse versions of the major factor analysis procedures. We illustrate the developed algorithms on well-known psychometric problems. Our sparse solutions are critically compared to ones obtained by other existing methods.
Solving Large-scale Eigenvalue Problems in SciDACApplications
Energy Technology Data Exchange (ETDEWEB)
Yang, Chao
2005-06-29
Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progression using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.
Biclustering via Sparse Singular Value Decomposition
Lee, Mihee
2010-02-16
Sparse singular value decomposition (SSVD) is proposed as a new exploratory analysis tool for biclustering or identifying interpretable row-column associations within high-dimensional data matrices. SSVD seeks a low-rank, checkerboard structured matrix approximation to data matrices. The desired checkerboard structure is achieved by forcing both the left- and right-singular vectors to be sparse, that is, having many zero entries. By interpreting singular vectors as regression coefficient vectors for certain linear regressions, sparsity-inducing regularization penalties are imposed to the least squares regression to produce sparse singular vectors. An efficient iterative algorithm is proposed for computing the sparse singular vectors, along with some discussion of penalty parameter selection. A lung cancer microarray dataset and a food nutrition dataset are used to illustrate SSVD as a biclustering method. SSVD is also compared with some existing biclustering methods using simulated datasets. © 2010, The International Biometric Society.
Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series
DEFF Research Database (Denmark)
Davis, Richard A.; Mikosch, Thomas Valentin; Pfaffel, Olivier
2016-01-01
In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index α∈(0,4) in...... particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the sample covariance matrix and show point process convergence of the normalized eigenvalues. The limiting process has an explicit form involving points of a Poisson process and eigenvalues...... of a non-negative definite matrix. Based on this convergence we derive limit theory for a host of other continuous functionals of the eigenvalues, including the joint convergence of the largest eigenvalues, the joint convergence of the largest eigenvalue and the trace of the sample covariance matrix...
Fast wavelet based sparse approximate inverse preconditioner
Energy Technology Data Exchange (ETDEWEB)
Wan, W.L. [Univ. of California, Los Angeles, CA (United States)
1996-12-31
Incomplete LU factorization is a robust preconditioner for both general and PDE problems but unfortunately not easy to parallelize. Recent study of Huckle and Grote and Chow and Saad showed that sparse approximate inverse could be a potential alternative while readily parallelizable. However, for special class of matrix A that comes from elliptic PDE problems, their preconditioners are not optimal in the sense that independent of mesh size. A reason may be that no good sparse approximate inverse exists for the dense inverse matrix. Our observation is that for this kind of matrices, its inverse entries typically have piecewise smooth changes. We can take advantage of this fact and use wavelet compression techniques to construct a better sparse approximate inverse preconditioner. We shall show numerically that our approach is effective for this kind of matrices.
Haidar, Azzam
2012-01-01
Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel\\'s Math Kernel Library.
Rotational image deblurring with sparse matrices
DEFF Research Database (Denmark)
Hansen, Per Christian; Nagy, James G.; Tigkos, Konstantinos
2014-01-01
We describe iterative deblurring algorithms that can handle blur caused by a rotation along an arbitrary axis (including the common case of pure rotation). Our algorithms use a sparse-matrix representation of the blurring operation, which allows us to easily handle several different boundary cond...
Directory of Open Access Journals (Sweden)
D. K. K. Vamsi
2012-03-01
Full Text Available In this article, we define the Green's matrix for a nonlinear Sturm Liouville problem associated with a pair of dynamic equations on time scales with a singularity at the point of interface. Then using iterative techniques, we obtain eigenvalue intervals for which there exist positive solutions. Then we present iterative schemes for approximating the solutions, and discus an example that illustrates the the results obtained.
Ltaief, Hatem
2012-01-01
This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric eigenvalue problem by computing the Cholesky factorization of the right hand side symmetric definite positive matrix (first stage), and applying the inverse of the freshly computed triangular Cholesky factors to the original dense symmetric matrix of the problem (second stage). Calculating the eigenpairs of the resulting problem is then equivalent to the eigenpairs of the original problem. The computation proceeds by reducing the updated dense symmetric matrix to symmetric band form (third stage). The band structure is further reduced by applying a bulge chasing procedure, which annihilates the extra off-diagonal entries using orthogonal transformations (fourth stage). More details on the third and fourth stage can be found in Haidar et al. [Accepted at SC\\'11, November 2011]. The eigenvalues are then calculated from the tridiagonal form using the standard LAPACK QR algorithm (i.e., DTSEQR routine), while the complex and challenging eigenvector computations will be addressed in a companion paper. The tasks from the various stages can concurrently run in an out-of-order fashion. The data dependencies are cautiously tracked by the dynamic runtime system environment QUARK, which ensures the dependencies are not violated for numerical correctness purposes. The obtained tile four-stage generalized symmetric eigenvalue solver significantly outperforms the state-of-the-art numerical libraries (up to 21-fold speed up against multithreaded LAPACK with optimized multithreaded MKL BLAS and up to 4-fold speed up against the corresponding routine from the commercial numerical software Intel MKL) on four sockets twelve cores AMD system with a 24000×24000 matrix size. © 2012 The authors and IOS Press. All rights reserved.
Directory of Open Access Journals (Sweden)
T. A. Akunov
2014-03-01
Full Text Available The paper deals with steady aperiodic continuous system, state matrix of which has a real spectrum of the eigenvalues which absolute value is less than unity. The latest authors’ works show that for such absolute values and multiple structure of eigenvalues on the free motion trajectories of the system by norm of the state vector the significant overshoot is detected, alternated by monotonous motion toward a state of rest. In order to minimize the overshoot value, it is proposed to modify the structure of the eigenvalues, transforming it into a simple one. The result of structure modification is the following: initial eigenvalue and shifted along the real axis of the complex plane to the left by a fixed value relative to the adjacent eigenvalues; each of them has unit multiplicity. Such modification gives the possibility to form the estimation of the proximity degree of eigenvalues simple structure to the multiple one. Moreover, it can be defined in a relative form, which guarantees the reduction of the above overshoot for the free motion trajectory. Results of computer experiments illustrate the issues of the paper.
Parallel and Scalable Sparse Basic Linear Algebra Subprograms
DEFF Research Database (Denmark)
Liu, Weifeng
matrix-vector multiplication (SpMV) on homogeneous processors such as CPUs, GPUs and Xeon Phi, (3) a new CSR-based SpMV algorithm for a variety of tightly coupled CPU-GPU heterogeneous processors, and (4) a new framework and associated algorithms for sparse matrix-matrix multiplication (SpGEMM) on GPUs...
Zhang, Tianzhu
2015-06-01
Sparse representation has been applied to visual tracking by finding the best target candidate with minimal reconstruction error by use of target templates. However, most sparse representation based trackers only consider holistic or local representations and do not make full use of the intrinsic structure among and inside target candidates, thereby making the representation less effective when similar objects appear or under occlusion. In this paper, we propose a novel Structural Sparse Tracking (SST) algorithm, which not only exploits the intrinsic relationship among target candidates and their local patches to learn their sparse representations jointly, but also preserves the spatial layout structure among the local patches inside each target candidate. We show that our SST algorithm accommodates most existing sparse trackers with the respective merits. Both qualitative and quantitative evaluations on challenging benchmark image sequences demonstrate that the proposed SST algorithm performs favorably against several state-of-the-art methods.
Whence the eigenstate-eigenvalue link?
Gilton, Marian J. R.
2016-08-01
David Wallace has recently argued that the eigenstate-eigenvalue (E-E) link has no place in serious discussions of quantum mechanics on the grounds that, as he claims, the E-E link is an invention of philosophers rather than the community of practicing physicists. This raises an historical question regarding the origin of the link. This paper aims to answer this question by tracing the historical development of the link through six key textbooks of quantum mechanics. In light of the historical evidence from these textbooks, it is argued that Wallace provides insufficient grounds for dismissing the E-E link from discussions of quantum mechanics.
Random matrices, Frobenius eigenvalues, and monodromy
Katz, Nicholas M
1998-01-01
The main topic of this book is the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and L-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinit
Eigenvalue statistics in quantum ideal gases
Eckhardt, B
1998-01-01
The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer $\\alpha>1$ one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function.
Parallel Sparse Linear System and Eigenvalue Problem Solvers: From Multicore to Petascale Computing
2015-06-01
of the Harwell Subroutine Library (HSL), while our symmetric reordering is based on the Fiedler vector of the corresponding...and extended through this grant) for obtaining the Fiedler vector rather than the multilevel eigensolver used in MC73 which
Nonlinear eigenvalue problems and PT-symmetric quantum mechanics
Bender, Carl M.
2017-07-01
Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schrödinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differential equations. The role of an eigenfunction is now played by a separatrix curve, and the special initial condition that gives rise to the separatrix curve is the eigenvalue. The Painlevé transcendents are examples of nonlinear eigenvalue problems, and semiclassical techniques are devised to calculate the behavior of the large eigenvalues. This behavior is found by reducing the Painlevé equation to the linear Schrödinger equation associated with a non-Hermitian PT-symmetric Hamiltonian. The concept of a nonlinear eigenvalue problem extends far beyond the Painlevé equations to huge classes of nonlinear differential equations.
Negre, Christian F A; Mniszewski, Susan M; Cawkwell, Marc J; Bock, Nicolas; Wall, Michael E; Niklasson, Anders M N
2016-07-12
We present a reduced complexity algorithm to compute the inverse overlap factors required to solve the generalized eigenvalue problem in a quantum-based molecular dynamics (MD) simulation. Our method is based on the recursive, iterative refinement of an initial guess of Z (inverse square root of the overlap matrix S). The initial guess of Z is obtained beforehand by using either an approximate divide-and-conquer technique or dynamical methods, propagated within an extended Lagrangian dynamics from previous MD time steps. With this formulation, we achieve long-term stability and energy conservation even under the incomplete, approximate, iterative refinement of Z. Linear-scaling performance is obtained using numerically thresholded sparse matrix algebra based on the ELLPACK-R sparse matrix data format, which also enables efficient shared-memory parallelization. As we show in this article using self-consistent density-functional-based tight-binding MD, our approach is faster than conventional methods based on the diagonalization of overlap matrix S for systems as small as a few hundred atoms, substantially accelerating quantum-based simulations even for molecular structures of intermediate size. For a 4158-atom water-solvated polyalanine system, we find an average speedup factor of 122 for the computation of Z in each MD step.
Polarimetric SAR Image Supervised Classification Method Integrating Eigenvalues
Xing Yanxiao; Zhang Yi; Li Ning; Wang Yu; Hu Guixiang
2016-01-01
Since classification methods based on H/α space have the drawback of yielding poor classification results for terrains with similar scattering features, in this study, we propose a polarimetric Synthetic Aperture Radar (SAR) image classification method based on eigenvalues. First, we extract eigenvalues and fit their distribution with an adaptive Gaussian mixture model. Then, using the naive Bayesian classifier, we obtain preliminary classification results. The distribution of eigenvalues in ...
Finite n Largest Eigenvalue Probability Distribution Function of Gaussian Ensembles
Choup, Leonard N.
2011-01-01
In this paper we focus on the finite n probability distribution function of the largest eigenvalue in the classical Gaussian Ensemble of n by n matrices (GEn). We derive the finite n largest eigenvalue probability distribution function for the Gaussian Orthogonal and Symplectic Ensembles and also prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of Gaussian Symplectic Ensemble. The correction terms to the limiting probability distribution are express...
Energy Technology Data Exchange (ETDEWEB)
Cummings, A. [Physics Department, University College Dublin, Dublin (Ireland); Department of Mathematical Physics, National University of Ireland Maynooth, Kildare (Ireland); O' Sullivan, G. [Physics Department, University College Dublin, Dublin (Ireland); Heffernan, D.M. [Department of Mathematical Physics, National University of Ireland Maynooth, Kildare (Ireland); School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin (Ireland)
2001-09-14
Using the relativistic configuration interaction Hartree-Fock method the Hamiltonian matrices of Ce I, J=4{sup {+-}}, and Pr I, J=11/2{sup {+-}}, are studied. These matrices can be characterized as sparse, banded matrices, with a leading diagonal. Diagonalization of the Hamiltonian results in a set of energy eigenvalues and corresponding eigenvectors and the purpose of this investigation will be to characterize the Hamiltonian matrices and coupling matrices of Ce I and Pr I, for both ls and jj coupling representations, using various statistical predictions of Random Matrix Theory. (author)
Sparse CSEM inversion driven by seismic coherence
Guo, Zhenwei; Dong, Hefeng; Kristensen, Åge
2016-12-01
Marine controlled source electromagnetic (CSEM) data inversion for hydrocarbon exploration is often challenging due to high computational cost, physical memory requirement and low resolution of the obtained resistivity map. This paper aims to enhance both the speed and resolution of CSEM inversion by introducing structural geological information in the inversion algorithm. A coarse mesh is generated for Occam’s inversion, where the parameters are fewer than in the fine regular mesh. This sparse mesh is defined as a coherence-based irregular (IC) sparse mesh, which is based on vertices extracted from available geological information. Inversion results on synthetic data illustrate that the IC sparse mesh has a smaller inversion computational cost compared to the regular dense (RD) mesh. It also has a higher resolution than with a regular sparse (RS) mesh for the same number of estimated parameters. In order to study how the IC sparse mesh reduces the computational time, four different meshes are generated for Occam’s inversion. As a result, an IC sparse mesh can reduce the computational cost while it keeps the resolution as good as a fine regular mesh. The IC sparse mesh reduces the computational cost of the matrix operation for model updates. When the number of estimated parameters reduces to a limited value, the computational cost is independent of the number of parameters. For a testing model with two resistive layers, the inversion result using an IC sparse mesh has higher resolution in both horizontal and vertical directions. Overall, the model representing significant geological information in the IC mesh can improve the resolution of the resistivity models obtained from inversion of CSEM data.
Structure-based bayesian sparse reconstruction
Quadeer, Ahmed Abdul
2012-12-01
Sparse signal reconstruction algorithms have attracted research attention due to their wide applications in various fields. In this paper, we present a simple Bayesian approach that utilizes the sparsity constraint and a priori statistical information (Gaussian or otherwise) to obtain near optimal estimates. In addition, we make use of the rich structure of the sensing matrix encountered in many signal processing applications to develop a fast sparse recovery algorithm. The computational complexity of the proposed algorithm is very low compared with the widely used convex relaxation methods as well as greedy matching pursuit techniques, especially at high sparsity. © 1991-2012 IEEE.
DEFF Research Database (Denmark)
Clemmensen, Line Katrine Harder; Hastie, Trevor; Witten, Daniela
2011-01-01
commonplace in biological and medical applications. In this setting, a traditional approach involves performing feature selection before classification. We propose sparse discriminant analysis, a method for performing linear discriminant analysis with a sparseness criterion imposed such that classification......We consider the problem of performing interpretable classification in the high-dimensional setting, in which the number of features is very large and the number of observations is limited. This setting has been studied extensively in the chemometrics literature, and more recently has become...... and feature selection are performed simultaneously. Sparse discriminant analysis is based on the optimal scoring interpretation of linear discriminant analysis, and can be extended to perform sparse discrimination via mixtures of Gaussians if boundaries between classes are nonlinear or if subgroups...
Sparse structure regularized ranking
Wang, Jim Jing-Yan
2014-04-17
Learning ranking scores is critical for the multimedia database retrieval problem. In this paper, we propose a novel ranking score learning algorithm by exploring the sparse structure and using it to regularize ranking scores. To explore the sparse structure, we assume that each multimedia object could be represented as a sparse linear combination of all other objects, and combination coefficients are regarded as a similarity measure between objects and used to regularize their ranking scores. Moreover, we propose to learn the sparse combination coefficients and the ranking scores simultaneously. A unified objective function is constructed with regard to both the combination coefficients and the ranking scores, and is optimized by an iterative algorithm. Experiments on two multimedia database retrieval data sets demonstrate the significant improvements of the propose algorithm over state-of-the-art ranking score learning algorithms.
A sparse effective Nullstellensatz
Sombra, M
1997-01-01
We present bounds for the sparseness and for the degrees of the polynomials in the Nullstellensatz. Our bounds depend mainly on the unmixed volume of the input polynomial system. The degree bounds can substantially improve the known ones when this polynomial system is sparse, and they are, in the worst case, simply exponential in terms of the number of variables and the maximum degree of the input polynomials.
Parallel sparse direct solver for integrated circuit simulation
Chen, Xiaoming; Yang, Huazhong
2017-01-01
This book describes algorithmic methods and parallelization techniques to design a parallel sparse direct solver which is specifically targeted at integrated circuit simulation problems. The authors describe a complete flow and detailed parallel algorithms of the sparse direct solver. They also show how to improve the performance by simple but effective numerical techniques. The sparse direct solver techniques described can be applied to any SPICE-like integrated circuit simulator and have been proven to be high-performance in actual circuit simulation. Readers will benefit from the state-of-the-art parallel integrated circuit simulation techniques described in this book, especially the latest parallel sparse matrix solution techniques. · Introduces complicated algorithms of sparse linear solvers, using concise principles and simple examples, without complex theory or lengthy derivations; · Describes a parallel sparse direct solver that can be adopted to accelerate any SPICE-like integrated circuit simulato...
Menon, Ravishankar; Gerstoft, Peter; Hodgkiss, William S
2012-11-01
Cross-correlations of diffuse noise fields can be used to extract environmental information. The influence of directional sources (usually ships) often results in a bias of the travel time estimates obtained from the cross-correlations. Using an array of sensors, insights from random matrix theory on the behavior of the eigenvalues of the sample covariance matrix (SCM) in an isotropic noise field are used to isolate the diffuse noise component from the directional sources. A sequential hypothesis testing of the eigenvalues of the SCM reveals eigenvalues dominated by loud sources that are statistical outliers for the assumed diffuse noise model. Travel times obtained from cross-correlations using only the diffuse noise component (i.e., by discarding or attenuating the outliers) converge to the predicted travel times based on the known array sensor spacing and measured sound speed at the site and are stable temporally (i.e., unbiased estimates). Data from the Shallow Water 2006 experiment demonstrates the effectiveness of this approach and that the signal-to-noise ratio builds up as the square root of time, as predicted by theory.
Yoon, Gangjoon; Min, Chohong
2017-11-01
The Shortley-Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-3) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-2), but also keeps the sharp second order convergence.
Spectral properties of Google matrix of Wikipedia and other networks
Ermann, Leonardo; Frahm, Klaus M.; Shepelyansky, Dima L.
2013-05-01
We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method, we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.
Eigenvalue Problem of Scalar Fields in BTZ Black Hole Spacetime
Kuwata, Maiko; Kenmoku, Masakatsu; Shigemoto, Kazuyasu
2008-01-01
We studied the eigenvalue problem of scalar fields in the (2+1)-dimensional BTZ black hole spacetime. The Dirichlet boundary condition at infinity and the Dirichlet or the Neumann boundary condition at the horizon are imposed. Eigenvalues for normal modes are characterized by the principal quantum number $(0
Complex eigenvalue analysis of railway wheel/rail squeal
African Journals Online (AJOL)
DR OKE
complex eigenvalue analysis by the finite element method. The positive real parts of the eigenvalues reflect self-exciting instable vibration, which is closely related to the occurrence of squeal noise. The effect of parameters such as friction coefficient, wheel/rail contact position, axle load, etc. on the instable vibration was ...
Lower Bounds for Eigenvalues by Nonconforming FEM on Convex Domain
Andreev, A. B.; Racheva, M. R.
2010-11-01
In this work we analyze the approximations of second order eigenvalue problems (EVPs). The nonconforming piecewise linear finite element with integral degrees of freedom is used. We prove that the eigenvalues computed by means of this element on convex domain are smaller than the exact ones if the mesh size is small enough. Some numerical results are also given.
Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Jing Wu
2012-01-01
Full Text Available We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation = = where , , , , satisfying , is the standard Riemann-Liouville derivative, , and is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.
Graph theory approach to the eigenvalue problem of large space structures
Reddy, A. S. S. R.; Bainum, P. M.
1981-01-01
Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.
Sparse tensor discriminant analysis.
Lai, Zhihui; Xu, Yong; Yang, Jian; Tang, Jinhui; Zhang, David
2013-10-01
The classical linear discriminant analysis has undergone great development and has recently been extended to different cases. In this paper, a novel discriminant subspace learning method called sparse tensor discriminant analysis (STDA) is proposed, which further extends the recently presented multilinear discriminant analysis to a sparse case. Through introducing the L1 and L2 norms into the objective function of STDA, we can obtain multiple interrelated sparse discriminant subspaces for feature extraction. As there are no closed-form solutions, k-mode optimization technique and the L1 norm sparse regression are combined to iteratively learn the optimal sparse discriminant subspace along different modes of the tensors. Moreover, each non-zero element in each subspace is selected from the most important variables/factors, and thus STDA has the potential to perform better than other discriminant subspace methods. Extensive experiments on face databases (Yale, FERET, and CMU PIE face databases) and the Weizmann action database show that the proposed STDA algorithm demonstrates the most competitive performance against the compared tensor-based methods, particularly in small sample sizes.
Overview of the ArbiTER edge plasma eigenvalue code
Baver, Derek; Myra, James; Umansky, Maxim
2011-10-01
The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue solver that is currently under development for plasma physics applications. The ArbiTER code builds on the equation parser framework of the existing 2DX code, extending it to include a topology parser. This will give the code the capability to model problems with complicated geometries (such as multiple X-points and scrape-off layers) or model equations with arbitrary numbers of dimensions (e.g. for kinetic analysis). In the equation parser framework, model equations are not included in the program's source code. Instead, an input file contains instructions for building a matrix from profile functions and elementary differential operators. The program then executes these instructions in a sequential manner. These instructions may also be translated into analytic form, thus giving the code transparency as well as flexibility. We will present an overview of how the ArbiTER code is to work, as well as preliminary results from early versions of this code. Work supported by the U.S. DOE.
Sparse approximation with bases
2015-01-01
This book systematically presents recent fundamental results on greedy approximation with respect to bases. Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications. The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and do...
Efficient convolutional sparse coding
Wohlberg, Brendt
2017-06-20
Computationally efficient algorithms may be applied for fast dictionary learning solving the convolutional sparse coding problem in the Fourier domain. More specifically, efficient convolutional sparse coding may be derived within an alternating direction method of multipliers (ADMM) framework that utilizes fast Fourier transforms (FFT) to solve the main linear system in the frequency domain. Such algorithms may enable a significant reduction in computational cost over conventional approaches by implementing a linear solver for the most critical and computationally expensive component of the conventional iterative algorithm. The theoretical computational cost of the algorithm may be reduced from O(M.sup.3N) to O(MN log N), where N is the dimensionality of the data and M is the number of elements in the dictionary. This significant improvement in efficiency may greatly increase the range of problems that can practically be addressed via convolutional sparse representations.
Sparse time series chain graphical models for reconstructing genetic networks
Abegaz, Fentaw; Wit, Ernst
We propose a sparse high-dimensional time series chain graphical model for reconstructing genetic networks from gene expression data parametrized by a precision matrix and autoregressive coefficient matrix. We consider the time steps as blocks or chains. The proposed approach explores patterns of
Robust recursive absolute value inequalities discriminant analysis with sparseness.
Li, Chun-Na; Zheng, Zeng-Rong; Liu, Ming-Zeng; Shao, Yuan-Hai; Chen, Wei-Jie
2017-09-01
In this paper, we propose a novel absolute value inequalities discriminant analysis (AVIDA) criterion for supervised dimensionality reduction. Compared with the conventional linear discriminant analysis (LDA), the main characteristics of our AVIDA are robustness and sparseness. By reformulating the generalized eigenvalue problem in LDA to a related SVM-type "concave-convex" problem based on absolute value inequalities loss, our AVIDA is not only more robust to outliers and noises, but also avoids the SSS problem. Moreover, the additional L1-norm regularization term in the objective makes sure sparse discriminant vectors are obtained. A successive linear algorithm is employed to solve the proposed optimization problem, where a series of linear programs are solved. The superiority of our AVIDA is supported by experimental results on artificial examples as well as benchmark image databases. Copyright © 2017 Elsevier Ltd. All rights reserved.
Supervised Transfer Sparse Coding
Al-Shedivat, Maruan
2014-07-27
A combination of the sparse coding and transfer learn- ing techniques was shown to be accurate and robust in classification tasks where training and testing objects have a shared feature space but are sampled from differ- ent underlying distributions, i.e., belong to different do- mains. The key assumption in such case is that in spite of the domain disparity, samples from different domains share some common hidden factors. Previous methods often assumed that all the objects in the target domain are unlabeled, and thus the training set solely comprised objects from the source domain. However, in real world applications, the target domain often has some labeled objects, or one can always manually label a small num- ber of them. In this paper, we explore such possibil- ity and show how a small number of labeled data in the target domain can significantly leverage classifica- tion accuracy of the state-of-the-art transfer sparse cod- ing methods. We further propose a unified framework named supervised transfer sparse coding (STSC) which simultaneously optimizes sparse representation, domain transfer and classification. Experimental results on three applications demonstrate that a little manual labeling and then learning the model in a supervised fashion can significantly improve classification accuracy.
1982-10-27
Detailed header comments? Yes User guide or manual, technical reports, papers, books: Proceedings, of the "I Jormadas Latinoamericanas de Matematica ... Aplicada " held in Santiago, Chile, December 1981 14. Approximate cost of obtaining softwaret Free 15. Restrictions on use: None 1G. Distributor: Same as
Genetic Algorithm Applied to the Eigenvalue Equalization Filtered-x LMS Algorithm (EE-FXLMS
Directory of Open Access Journals (Sweden)
Stephan P. Lovstedt
2008-01-01
Full Text Available The FXLMS algorithm, used extensively in active noise control (ANC, exhibits frequency-dependent convergence behavior. This leads to degraded performance for time-varying tonal noise and noise with multiple stationary tones. Previous work by the authors proposed the eigenvalue equalization filtered-x least mean squares (EE-FXLMS algorithm. For that algorithm, magnitude coefficients of the secondary path transfer function are modified to decrease variation in the eigenvalues of the filtered-x autocorrelation matrix, while preserving the phase, giving faster convergence and increasing overall attenuation. This paper revisits the EE-FXLMS algorithm, using a genetic algorithm to find magnitude coefficients that give the least variation in eigenvalues. This method overcomes some of the problems with implementing the EE-FXLMS algorithm arising from finite resolution of sampled systems. Experimental control results using the original secondary path model, and a modified secondary path model for both the previous implementation of EE-FXLMS and the genetic algorithm implementation are compared.
On the resultant property of the Fisher information matrix of a vector ARMA process
Klein, A.; Mélard, G.; Spreij, P.
2004-01-01
A matrix is called a multiple resultant matrix associated to two matrix polynomials when it becomes singular if and only if the two matrix polynomials have at least one common eigenvalue. In this paper a new multiple resultant matrix is introduced. It concerns the Fisher information matrix (FIM) of
Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.
Ipsen, Jesper R; Kieburg, Mario
2014-03-01
We study the joint probability density of the eigenvalues of a product of rectangular real, complex, or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary, or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously has been discussed for infinite matrix size. Moreover, we derive the joint probability densities of the eigenvalues. To illustrate our results, we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights, we show that the product of complex random matrices yields a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.
Cessna Citation X Business Aircraft Eigenvalue Stability – Part2: Flight Envelope Analysis
Directory of Open Access Journals (Sweden)
Yamina BOUGHARI
2017-12-01
Full Text Available Civil aircraft flight control clearance is a time consuming, thus an expensive process in the aerospace industry. This process has to be investigated and proved to be safe for thousands of combinations in terms of speeds, altitudes, gross weights, Xcg / weight configurations and angles of attack. Even in this case, a worst-case condition that could lead to a critical situation might be missed. To address this problem, models that are able to describe an aircraft’s dynamics by taking into account all uncertainties over a region within a flight envelope have been developed using Linear Fractional Representation. In order to investigate the Cessna Citation X aircraft Eigenvalue Stability envelope, the Linear Fractional Representation models are implemented using the speeds and the altitudes as varying parameters. In this paper Part 2, the aircraft longitudinal eigenvalue stability is analyzed in a continuous range of flight envelope with varying parameter of True airspeed and altitude, instead of a single point, like classical methods. This is known as the aeroelastic stability envelope, required for civil aircraft certification as given by the Circular Advisory “Aeroelastic Stability Substantiation of Transport Category Airplanes AC No: 25.629-18”. In this new methodology the analysis is performed in time domain based on Lyapunov stability and solved by convex optimization algorithms by using the linear matrix inequalities to evaluate the eigenvalue stability, which is reduced to search for the negative eigenvalues in a region of flight envelope. It can also be used to study the stability of a system during an arbitrary motion from one point to another in the flight envelope. A whole aircraft analysis results’ for its entire envelope are presented in the form of graphs, thus offering good readability, and making them easily exploitable.
Directory of Open Access Journals (Sweden)
Ibrahim A. Abbas
2016-06-01
Full Text Available In this article, we consider the problem of a thermoelastic infinite body with a spherical cavity in the context of the theory of fractional order thermoelasticity. The inner surface of the cavity is taken traction free and subjected to a thermal shock. The form of a vector–matrix differential equation has been considered for the governing equations in the Laplace transform domain. The analytical solutions are given by the eigenvalue approach. The graphical results indicate that the fractional parameter effect plays a significant role on all the physical quantities.
Directory of Open Access Journals (Sweden)
Erika Strakova
2017-01-01
Full Text Available A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples.
An eigenvalue localization set for tensors and its applications
Directory of Open Access Journals (Sweden)
Jianxing Zhao
2017-03-01
Full Text Available Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015 and Huang et al. (J. Inequal. Appl. 2016:254, 2016. As an application of this set, new bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of N = { 1 , 2 , … , n } $N=\\{1,2,\\ldots,n\\}$ , we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors. Finally, numerical examples are given to verify the theoretical results.
Eigenvalue estimates of positive integral operators with analytic ...
Indian Academy of Sciences (India)
. Eigenvalue problems; special kernels; integral operators; Hardy spaces;. Smirnov classes; closed analytic curves; Ahlfors ... Given a Riemann mapping function (RMF) ϕ for , i.e. a conformal map of onto the unit disc , define a function K on. ×.
An eigenvalue method for solving transient heat conduction problems
Shih, T. M.; Skladany, J. T.
1983-01-01
The eigenvalue method, which has been used by researchers in structure mechanics, is applied to problems in heat conduction. Its formulation is decribed in terms of an examination of transient heat conduction in a square slab. Taking advantage of the availability of the exact solution, we compare the accuracy and other numerical properties of the eigenvalue method with those of existing numerical schemes. The comparsion shows that, overall, the eigenvalue method appears to be fairly attractive. Furthermore, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed and retained to yield reasonably high accuracy. Greater savings are attained in the computation time for a transient problem with long time duration and a large computational domain.
Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems
Directory of Open Access Journals (Sweden)
Mohammed Al-Refai
2017-01-01
Full Text Available We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.
Sparse inpainting and isotropy
Energy Technology Data Exchange (ETDEWEB)
Feeney, Stephen M.; McEwen, Jason D.; Peiris, Hiranya V. [Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT (United Kingdom); Marinucci, Domenico; Cammarota, Valentina [Department of Mathematics, University of Rome Tor Vergata, via della Ricerca Scientifica 1, Roma, 00133 (Italy); Wandelt, Benjamin D., E-mail: s.feeney@imperial.ac.uk, E-mail: marinucc@axp.mat.uniroma2.it, E-mail: jason.mcewen@ucl.ac.uk, E-mail: h.peiris@ucl.ac.uk, E-mail: wandelt@iap.fr, E-mail: cammarot@axp.mat.uniroma2.it [Kavli Institute for Theoretical Physics, Kohn Hall, University of California, 552 University Road, Santa Barbara, CA, 93106 (United States)
2014-01-01
Sparse inpainting techniques are gaining in popularity as a tool for cosmological data analysis, in particular for handling data which present masked regions and missing observations. We investigate here the relationship between sparse inpainting techniques using the spherical harmonic basis as a dictionary and the isotropy properties of cosmological maps, as for instance those arising from cosmic microwave background (CMB) experiments. In particular, we investigate the possibility that inpainted maps may exhibit anisotropies in the behaviour of higher-order angular polyspectra. We provide analytic computations and simulations of inpainted maps for a Gaussian isotropic model of CMB data, suggesting that the resulting angular trispectrum may exhibit small but non-negligible deviations from isotropy.
On the Riccati transfer matrix method for repetitive structures
Stephen, N.G.
2010-01-01
The Riccati transfer matrix method is employed in the elastostatic analysis of a repetitive structure subject to various loadings; the eigenvalues of particular terms featuring in the recursive relationships show why the method is numerically stable
On the Subspace Projected Approximate Matrix method
Brandts, J.H.; Reis da Silva, R.
2015-01-01
We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method of computing a few eigenvalues of a Hermitian matrix A. It falls in the category of inner-outer iteration methods and aims to reduce the costs of
Directory of Open Access Journals (Sweden)
Schweitzer J.-M.
2010-10-01
Full Text Available Processes carrying out exothermic reactions must ensure safe operating conditions to avoid uncontrolled thermal excursion, also known as runaway. Therefore, a thermal stability analysis is necessary to determine the safe and productive range of operating conditions of highly exothermic processes. Hydrotreating gas oil feeds consists mainly of hydrogenation reactions; processing highly unsaturated feeds such as light cycle oils can be highly exothermic. For this reason, a thermal stability study of this complex refining is performed. Perturbations theory has already been applied to carry out a thermal stability study of this process under dynamic conditions. This method consists in the perturbation of the hydrotreating reactor model and solution of the perturbed model in the form of an eigenvalue problem. The stability condition imposes that all perturbations must tend to zero when time tends to infinity. Some methodology and numerical aspects applying this theory and the effect on stability results are tackled in this work. The formalization of the perturbed model solution as a standard eigenvalue problem or as a generalized eigenvalue problem are presented. The computation of the Jacobian by a numerical approach or with the analytical expressions is also carried out. In both cases, results are compared and their influence on the stability/instability results is presented. Les procédés qui mettent en oeuvre des réactions très exothermiques nécessitent une attention particulière afin d’éviter l’augmentation non contrôlée de la température connue comme emballement thermique. Une analyse de stabilité thermique est nécessaire afin d’établir les conditions d’opération sûre et productive des procédés exothermiques. L’hydrotraitement de gazoles met en oeuvre principalement des réactions d’hydrogénation ; l’hydrotraitement de charges très insaturées comme les gazoles light cycle oil peut être fortement exothermique
Performance analysis for sparse support recovery
Tang, Gongguo
2009-01-01
In this paper, the performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space is analyzed. Support recovery is formulated as a multiple-hypothesis testing problem and both upper and lower bounds on the probability of error are derived for general measurement matrices, by using the Chernoff bound and Fano's inequality, respectively. The form of the upper bound shows that the performance is determined by a single quantity that is a measure of the incoherence of the measurement matrix, while the lower bound reveals the importance of the total measurement gain. To demonstrate its immediate applicability, the lower bound is applied to derive the minimal number of samples needed for accurate direction of arrival (DOA) estimation for an algorithm based on sparse representation. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions to guarantee a vanishing probability of error for majority realizations...
Li, Keqiang; Gao, Feng; Li, Shengbo Eben; Zheng, Yang; Gao, Hongbo
2017-12-01
This study presents a distributed H-infinity control method for uncertain platoons with dimensionally and structurally unknown interaction topologies provided that the associated topological eigenvalues are bounded by a predesigned range.With an inverse model to compensate for nonlinear powertrain dynamics, vehicles in a platoon are modeled by third-order uncertain systems with bounded disturbances. On the basis of the eigenvalue decomposition of topological matrices, we convert the platoon system to a norm-bounded uncertain part and a diagonally structured certain part by applying linear transformation. We then use a common Lyapunov method to design a distributed H-infinity controller. Numerically, two linear matrix inequalities corresponding to the minimum and maximum eigenvalues should be solved. The resulting controller can tolerate interaction topologies with eigenvalues located in a certain range. The proposed method can also ensure robustness performance and disturbance attenuation ability for the closed-loop platoon system. Hardware-in-the-loop tests are performed to validate the effectiveness of our method.
Truncated Linear Statistics Associated with the Top Eigenvalues of Random Matrices
Grabsch, Aurélien; Majumdar, Satya N.; Texier, Christophe
2017-04-01
Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues P(λ _1,\\ldots ,λ _N), many important questions have been related to the study of linear statistics of eigenvalues L=\\sum _{i=1}^Nf(λ _i), where f(λ ) is a known function. We study here truncated linear statistics where the sum is restricted to the N_1analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with f(λ )=√{λ }. Using the Coulomb gas technique, we study the N→ ∞ limit with N_1/N fixed. We show that the constraint that \\tilde{L}=\\sum _{i=1}^{N_1}f(λ _i) is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. Assuming that f(λ ) is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal.
A Unified Software Interface to Solve or Circumvent the Kohn-Sham Eigenvalue Problem: ELSI
Yu, Victor; Huhn, William; Lange, Björn; Blum, Volker; Corsetti, Fabiano; Lin, Lin; Lu, Jianfeng; Vazquez-Mayagoita, Alvaro; Yang, Chao
Solving or circumventing a generalized eigenvalue problem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory (KS-DFT). This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to simplify the access to existing strategies to address the KS eigenvalue problem. Supported algorithms include the massively parallel dense eigensolver ELPA (O (N3)), the orbital minimization method in libOMM (O (N3) with a reduced prefactor), and the Pole EXpansion and Selected Inversion (PEXSI) approach with lower computational complexity (at most O (N2)). The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by a) optional automatic selection of the correct solver depending on the specific problem; b) reasonable default parameters for a chosen solver; and c) automatic conversion between input and internal working matrix formats. Benchmarks are shown for all-electron Hamilton and overlap matrices for system sizes up to several thousand atoms. This work is supported by the National Science Foundation under Grant Number 1450280.
Compressed sensing & sparse filtering
Carmi, Avishy Y; Godsill, Simon J
2013-01-01
This book is aimed at presenting concepts, methods and algorithms ableto cope with undersampled and limited data. One such trend that recently gained popularity and to some extent revolutionised signal processing is compressed sensing. Compressed sensing builds upon the observation that many signals in nature are nearly sparse (or compressible, as they are normally referred to) in some domain, and consequently they can be reconstructed to within high accuracy from far fewer observations than traditionally held to be necessary.Â Apart from compressed sensing this book contains other related app
Wang, Jim Jing-Yan
2014-07-06
Sparse coding approximates the data sample as a sparse linear combination of some basic codewords and uses the sparse codes as new presentations. In this paper, we investigate learning discriminative sparse codes by sparse coding in a semi-supervised manner, where only a few training samples are labeled. By using the manifold structure spanned by the data set of both labeled and unlabeled samples and the constraints provided by the labels of the labeled samples, we learn the variable class labels for all the samples. Furthermore, to improve the discriminative ability of the learned sparse codes, we assume that the class labels could be predicted from the sparse codes directly using a linear classifier. By solving the codebook, sparse codes, class labels and classifier parameters simultaneously in a unified objective function, we develop a semi-supervised sparse coding algorithm. Experiments on two real-world pattern recognition problems demonstrate the advantage of the proposed methods over supervised sparse coding methods on partially labeled data sets.
Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices
Dhesi, G. S.; Ausloos, M.
2016-06-01
Nowadays, strict finite size effects must be taken into account in condensed matter problems when treated through models based on lattices or graphs. On the other hand, the cases of directed bonds or links are known to be highly relevant in topics ranging from ferroelectrics to quotation networks. Combining these two points leads us to examine finite size random matrices. To obtain basic materials properties, the Green's function associated with the matrix has to be calculated. To obtain the first finite size correction, a perturbative scheme is hereby developed within the framework of the replica method. The averaged eigenvalue spectrum and the corresponding Green's function of Wigner random sign real symmetric N ×N matrices to order 1 /N are finally obtained analytically. Related simulation results are also presented. The agreement is excellent between the analytical formulas and finite size matrix numerical diagonalization results, confirming the correctness of the first-order finite size expression.
Methods of intermediate problems for eigenvalues
Weinstein, Alexander
1972-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
Visual tracking based on extreme learning machine and sparse representation.
Wang, Baoxian; Tang, Linbo; Yang, Jinglin; Zhao, Baojun; Wang, Shuigen
2015-10-22
The existing sparse representation-based visual trackers mostly suffer from both being time consuming and having poor robustness problems. To address these issues, a novel tracking method is presented via combining sparse representation and an emerging learning technique, namely extreme learning machine (ELM). Specifically, visual tracking can be divided into two consecutive processes. Firstly, ELM is utilized to find the optimal separate hyperplane between the target observations and background ones. Thus, the trained ELM classification function is able to remove most of the candidate samples related to background contents efficiently, thereby reducing the total computational cost of the following sparse representation. Secondly, to further combine ELM and sparse representation, the resultant confidence values (i.e., probabilities to be a target) of samples on the ELM classification function are used to construct a new manifold learning constraint term of the sparse representation framework, which tends to achieve robuster results. Moreover, the accelerated proximal gradient method is used for deriving the optimal solution (in matrix form) of the constrained sparse tracking model. Additionally, the matrix form solution allows the candidate samples to be calculated in parallel, thereby leading to a higher efficiency. Experiments demonstrate the effectiveness of the proposed tracker.
General eigenvalue correlations for the real Ginibre ensemble
Energy Technology Data Exchange (ETDEWEB)
Sommers, Hans-Juergen; Wieczorek, Waldemar [Fachbereich Physik, Universitaet Duisburg-Essen, 47048 Duisburg (Germany)], E-mail: H.J.Sommers@uni-due.de, E-mail: Waldemar.Wieczorek@uni-due.de
2008-10-10
We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian generating functional for n-point densities. This produces a simple free-fermion diagram expansion for the correlations leading to quaternion determinants in each order n. All will explicitly be given with the help of a very simple symplectic kernel for even dimension N. The kernel is valid for both complex and real eigenvalues and describes a deep connection between both. A slight modification by an artificial additional Grassmannian also solves the more complicated odd-N case. As illustration, we present some numerical results in the space C{sup n} of complex eigenvalue n-tuples.
The nonconforming virtual element method for eigenvalue problems
Energy Technology Data Exchange (ETDEWEB)
Gardini, Francesca [Univ. of Pavia (Italy). Dept. of Mathematics; Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Vacca, Giuseppe [Univ. of Milano-Bicocca, Milan (Italy). Dept. of Mathematics and Applications
2018-02-05
We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L^{2}-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.
Eigenvalues of Bethe vectors in the Gaudin model
Molev, A. I.; Mukhin, E. E.
2017-09-01
According to the Feigin-Frenkel-Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical W-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of q-characters and classical W-algebras.
A teaching proposal for the study of Eigenvectors and Eigenvalues
Directory of Open Access Journals (Sweden)
María José Beltrán Meneu
2017-03-01
Full Text Available In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students’ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.
Partitioning Rectangular and Structurally Nonsymmetric Sparse Matrices for Parallel Processing
Energy Technology Data Exchange (ETDEWEB)
B. Hendrickson; T.G. Kolda
1998-09-01
A common operation in scientific computing is the multiplication of a sparse, rectangular or structurally nonsymmetric matrix and a vector. In many applications the matrix- transpose-vector product is also required. This paper addresses the efficient parallelization of these operations. We show that the problem can be expressed in terms of partitioning bipartite graphs. We then introduce several algorithms for this partitioning problem and compare their performance on a set of test matrices.
The equivalent source method as a sparse signal reconstruction
DEFF Research Database (Denmark)
Fernandez Grande, Efren; Xenaki, Angeliki
2015-01-01
sources in the case of this model. Sparse solutions can be achieved by l-1 norm minimization, providing accurate reconstruction and robustness to noise, because favouring sparsity suppresses noisy components. The study addresses the influence of the ill-conditioning of the propagation matrix, which can...
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available In the solution of boundary value problems, usually zero eigenvalue is ignored. This case also happens in calculating the eigenvalues of matrices, so that we would often like to find the nonzero solutions of the linear system A X = λ X when λ ≠ 0 . But λ = 0 implies that det A = 0 for X ≠ 0 and then the rank of matrix A is reduced at least one degree. This comment can similarly be stated for boundary value problems. In other words, if at least one of the eigens of equations related to the main problem is considered zero, then one of the solutions will be specified in advance. By using this note, first we study a class of special functions and then apply it for the potential, heat, and wave equations in spherical coordinate. In this way, some practical examples are also given.
Directory of Open Access Journals (Sweden)
M. Jazaeri
2012-10-01
Full Text Available Recently, the growing integration of wind energy into power networks has had a significant impact on power systemstability. Amongst types of large capacity wind turbines (WTs, doubly-fed induction generator (DFIG wind turbinesrepresent an important percentage. This paper attemps to study the impact of DFIG wind turbines on the powersystem stability and dynamics by modeling all components of a case study system (CSS. Modal analysis is usedfor the study of the dynamic stability of the CSS. The system dynamics are studied by examining the eigenvaluesof the matrix system of the case study and the impact of all parameters of the CSS are studied in normal,subsynchronous and super-synchronous modes. The results of the eigenvalue analysis are verified by usingdynamic simulation software. The results show that each of the electrical and mechanical parameters of the CSSaffect specific eigenvalues.
Two-dimensional cache-oblivious sparse matrix–vector multiplication
Yzelman, A.N.|info:eu-repo/dai/nl/313872643; Bisseling, R.H.|info:eu-repo/dai/nl/304828068
2011-01-01
In earlier work, we presented a one-dimensional cache-oblivious sparse matrix–vector (SpMV) multiplication scheme which has its roots in one-dimensional sparse matrix partitioning. Partitioning is often used in distributed-memory parallel computing for the SpMV multiplication, an important kernel in
Sparse decompositions in 'incoherent' dictionaries
DEFF Research Database (Denmark)
Gribonval, R.; Nielsen, Morten
2003-01-01
a unique sparse representation in such a dictionary. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of a combinatorial optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may...
Distance-regular Cayley graphs with least eigenvalue -2
van Dam, Edwin; Abdollahi, Alireza; Jazaeri, Mojtaba
2017-01-01
We classify the distance-regular Cayley graphs with least eigenvalue −2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible
Plain strain problem of poroelasticity using eigenvalue approach
Indian Academy of Sciences (India)
A plain strain problem of an isotropic elastic liquid-saturated porous medium in poroelasticity has been studied. The eigenvalue approach using the Laplace and Fourier transforms has been employed and these transforms have been inverted by using a numerical technique. An application of infinite space with ...
On perturbation of eigenvalues embedded at thresholds in a two ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
We assume Vab ∈ B(Hb, Ha), the bounded operators, and Vba = V ∗ ab . The direct sum structure of H0 means that one can easily find examples with eigenvalues embedded at thresholds. To explain the type of results obtained, let us assume that the operatorHa is a Schr ödinger operator with absolutely continuous ...
Emotion recognition using eigenvalues and Levenberg–Marquardt ...
Indian Academy of Sciences (India)
In this paper, a simple and computationally efficient approach is proposed for person independent facial emotion recognition. The proposed approach is based on the significant features of an image, i.e., the collection of few largest eigenvalues (LE). Further, a Levenberg–Marquardt algorithm-based neural network (LMNN) ...
Evaluation of eigenvalues of a smooth potential via Schroedinger ...
Indian Academy of Sciences (India)
Evaluation of eigenvalues of a smooth potential via Schroedinger transmission across multi-step potential. BASUDEB SAHU1,∗, BIDHUBHUSAN SAHU1 and SANTOSH K AGARWALLA2. 1Department of Physics, North Orissa University, Baripada 757 003, India. 2Department of Applied Physics and Ballistics, Fakir Mohan ...
Eigenvalue estimates of positive integral operators with analytic ...
Indian Academy of Sciences (India)
In this paper, we exhibit canonical positive definite integral kernels associated with simply connected domains. We give lower bounds for the eigenvalues of the sums of such kernels. Author Affiliations. Yüksel Soykan1. Department of Mathematics, Art and Science Faculty, Zonguldak Karaelmas University, 67100, ...
Polarimetric SAR Image Supervised Classification Method Integrating Eigenvalues
Directory of Open Access Journals (Sweden)
Xing Yanxiao
2016-04-01
Full Text Available Since classification methods based on H/α space have the drawback of yielding poor classification results for terrains with similar scattering features, in this study, we propose a polarimetric Synthetic Aperture Radar (SAR image classification method based on eigenvalues. First, we extract eigenvalues and fit their distribution with an adaptive Gaussian mixture model. Then, using the naive Bayesian classifier, we obtain preliminary classification results. The distribution of eigenvalues in two kinds of terrains may be similar, leading to incorrect classification in the preliminary step. So, we calculate the similarity of every terrain pair, and add them to the similarity table if their similarity is greater than a given threshold. We then apply the Wishart distance-based KNN classifier to these similar pairs to obtain further classification results. We used the proposed method on both airborne and spaceborne SAR datasets, and the results show that our method can overcome the shortcoming of the H/α-based unsupervised classification method for eigenvalues usage, and produces comparable results with the Support Vector Machine (SVM-based classification method.
Eigenvalue estimates for submanifolds with bounded f-mean curvature
Indian Academy of Sciences (India)
GUANGYUE HUANG
Abstract. In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the f -Laplacian on complete noncompact submanifolds of the weighted. Riemannian manifold (Hm(−1), e−f dv) with respect to the f -mean curvature. In par- ticular, our results generalize those of Cheung and Leung in Math. Z. 236 ...
Numerical analysis of Eigenvalue algorithms based on subspace iterations
Smit, P.
1997-01-01
Eigenvalue problems are important in many applications involving mathematical modelling. The development of hardware and software over the years has enlarged the class of problems that can be solved efficiently. In particular, the iterative algorithms that project the problem to low-dimensional
A combined approach of complex eigenvalue analysis and design of ...
African Journals Online (AJOL)
This paper proposes an approach to investigate the influencing factors of the brake pad on the disc brake squeal by integrating finite element simulations with statistical regression techniques. Complex eigenvalue analysis (CEA) has been widely used to predict unstable frequencies in brake systems models. The finite ...
Complex eigenvalue analysis of railway wheel/rail squeal | Goo ...
African Journals Online (AJOL)
Wheel/rail squeal noise of trains is one of the challenging problems of trains, but a successful model for the squeal problem has not been presented until now. In this study, a new approach was presented to investigate the basic mechanism of the wheel/rail squeal noise, using complex eigenvalue analysis by the finite ...
Balanced and sparse Tamo-Barg codes
Halbawi, Wael
2017-08-29
We construct balanced and sparse generator matrices for Tamo and Barg\\'s Locally Recoverable Codes (LRCs). More specifically, for a cyclic Tamo-Barg code of length n, dimension k and locality r, we show how to deterministically construct a generator matrix where the number of nonzeros in any two columns differs by at most one, and where the weight of every row is d + r - 1, where d is the minimum distance of the code. Since LRCs are designed mainly for distributed storage systems, the results presented in this work provide a computationally balanced and efficient encoding scheme for these codes. The balanced property ensures that the computational effort exerted by any storage node is essentially the same, whilst the sparse property ensures that this effort is minimal. The work presented in this paper extends a similar result previously established for Reed-Solomon (RS) codes, where it is now known that any cyclic RS code possesses a generator matrix that is balanced as described, but is sparsest, meaning that each row has d nonzeros.
Pak, Chan-gi; Lung, Shun-fat
2009-01-01
Modern airplane design is a multidisciplinary task which combines several disciplines such as structures, aerodynamics, flight controls, and sometimes heat transfer. Historically, analytical and experimental investigations concerning the interaction of the elastic airframe with aerodynamic and in retia loads have been conducted during the design phase to determine the existence of aeroelastic instabilities, so called flutter .With the advent and increased usage of flight control systems, there is also a likelihood of instabilities caused by the interaction of the flight control system and the aeroelastic response of the airplane, known as aeroservoelastic instabilities. An in -house code MPASES (Ref. 1), modified from PASES (Ref. 2), is a general purpose digital computer program for the analysis of the closed-loop stability problem. This program used subroutines given in the International Mathematical and Statistical Library (IMSL) (Ref. 3) to compute all of the real and/or complex conjugate pairs of eigenvalues of the Hessenberg matrix. For high fidelity configuration, these aeroelastic system matrices are large and compute all eigenvalues will be time consuming. A subspace iteration method (Ref. 4) for complex eigenvalues problems with nonsymmetric matrices has been formulated and incorporated into the modified program for aeroservoelastic stability (MPASES code). Subspace iteration method only solve for the lowest p eigenvalues and corresponding eigenvectors for aeroelastic and aeroservoelastic analysis. In general, the selection of p is ranging from 10 for wing flutter analysis to 50 for an entire aircraft flutter analysis. The application of this newly incorporated code is an experiment known as the Aerostructures Test Wing (ATW) which was designed by the National Aeronautic and Space Administration (NASA) Dryden Flight Research Center, Edwards, California to research aeroelastic instabilities. Specifically, this experiment was used to study an instability
Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.
2017-04-01
In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O (Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O (Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No chain-type molecules, while supporting sufficient accuracy.
Consensus Convolutional Sparse Coding
Choudhury, Biswarup
2017-04-11
Convolutional sparse coding (CSC) is a promising direction for unsupervised learning in computer vision. In contrast to recent supervised methods, CSC allows for convolutional image representations to be learned that are equally useful for high-level vision tasks and low-level image reconstruction and can be applied to a wide range of tasks without problem-specific retraining. Due to their extreme memory requirements, however, existing CSC solvers have so far been limited to low-dimensional problems and datasets using a handful of low-resolution example images at a time. In this paper, we propose a new approach to solving CSC as a consensus optimization problem, which lifts these limitations. By learning CSC features from large-scale image datasets for the first time, we achieve significant quality improvements in a number of imaging tasks. Moreover, the proposed method enables new applications in high dimensional feature learning that has been intractable using existing CSC methods. This is demonstrated for a variety of reconstruction problems across diverse problem domains, including 3D multispectral demosaickingand 4D light field view synthesis.
Consensus Convolutional Sparse Coding
Choudhury, Biswarup
2017-12-01
Convolutional sparse coding (CSC) is a promising direction for unsupervised learning in computer vision. In contrast to recent supervised methods, CSC allows for convolutional image representations to be learned that are equally useful for high-level vision tasks and low-level image reconstruction and can be applied to a wide range of tasks without problem-specific retraining. Due to their extreme memory requirements, however, existing CSC solvers have so far been limited to low-dimensional problems and datasets using a handful of low-resolution example images at a time. In this paper, we propose a new approach to solving CSC as a consensus optimization problem, which lifts these limitations. By learning CSC features from large-scale image datasets for the first time, we achieve significant quality improvements in a number of imaging tasks. Moreover, the proposed method enables new applications in high-dimensional feature learning that has been intractable using existing CSC methods. This is demonstrated for a variety of reconstruction problems across diverse problem domains, including 3D multispectral demosaicing and 4D light field view synthesis.
In Defense of Sparse Tracking: Circulant Sparse Tracker
Zhang, Tianzhu
2016-12-13
Sparse representation has been introduced to visual tracking by finding the best target candidate with minimal reconstruction error within the particle filter framework. However, most sparse representation based trackers have high computational cost, less than promising tracking performance, and limited feature representation. To deal with the above issues, we propose a novel circulant sparse tracker (CST), which exploits circulant target templates. Because of the circulant structure property, CST has the following advantages: (1) It can refine and reduce particles using circular shifts of target templates. (2) The optimization can be efficiently solved entirely in the Fourier domain. (3) High dimensional features can be embedded into CST to significantly improve tracking performance without sacrificing much computation time. Both qualitative and quantitative evaluations on challenging benchmark sequences demonstrate that CST performs better than all other sparse trackers and favorably against state-of-the-art methods.
Pérez López, César
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Matrix Algebra introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. Starting with a look at symbolic and numeric variables, with an emphasis on vector and matrix variables, you will go on to examine functions and operations that support vectors and matrices as arguments, including those based on analytic parent functions. Computational methods for finding eigenvalues and eigenvectors of matrices are detailed, leading to various matrix decompositions. Applications such as change of bases, the classification of quadratic forms and ...
Staggered chiral random matrix theory
Osborn, James C.
2010-01-01
We present a random matrix theory (RMT) for the staggered lattice QCD Dirac operator. The staggered RMT is equivalent to the zero-momentum limit of the staggered chiral Lagrangian and includes all taste breaking terms at their leading order. This is an extension of previous work which only included some of the taste breaking terms. We will also present some results for the taste breaking contributions to the partition function and the Dirac eigenvalues.
Two new eigenvalue localization sets for tensors and theirs applications
Directory of Open Access Journals (Sweden)
Zhao Jianxing
2017-10-01
Full Text Available A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324 and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50. As an application, a weaker checkable sufficient condition for the positive (semi-definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1, 187-198. As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.
Finite element method for eigenvalue problems in electromagnetics
Reddy, C. J.; Deshpande, Manohar D.; Cockrell, C. R.; Beck, Fred B.
1994-01-01
Finite element method (FEM) has been a very powerful tool to solve many complex problems in electromagnetics. The goal of the current research at the Langley Research Center is to develop a combined FEM/method of moments approach to three-dimensional scattering/radiation problem for objects with arbitrary shape and filled with complex materials. As a first step toward that goal, an exercise is taken to establish the power of FEM, through closed boundary problems. This paper demonstrates the developed of FEM tools for two- and three-dimensional eigenvalue problems in electromagnetics. In section 2, both the scalar and vector finite elements have been used for various waveguide problems to demonstrate the flexibility of FEM. In section 3, vector finite element method has been extended to three-dimensional eigenvalue problems.
Comparing lattice Dirac operators with Random Matrix Theory
Energy Technology Data Exchange (ETDEWEB)
Farchioni, F.; Hipt, I.; Lang, C.B
2000-03-01
We study the eigenvalue spectrum of different lattice Dirac operators (staggered, fixed point, overlap) and discuss their dependence on the topological sectors. Although the model is 2D (the Schwinger model with massless fermions) our observations indicate possible problems in 4D applications. In particular misidentification of the smallest eigenvalues due to non-identification of the topological sector may hinder successful comparison with Random Matrix Theory (RMT)
Random matrix theory and the spectra of overlap fermions
Energy Technology Data Exchange (ETDEWEB)
Shcheredin, S.; Bietenholz, W.; Chiarappa, T.; Jansen, K.; Nagai, K.-I
2004-03-01
The application of Random Matrix Theory to the Dirac operator of QCD yields predictions for the probability distributions of the lowest eigenvalues. We measured Dirac operator spectra using massless overlap fermions in quenched QCD at topological charge {nu} = 0, {+-} 1 and {+-}2, and found agreement with those predictions -- at least for the first non-zero eigenvalue -- if the volume exceeds about (1.2 fm){sup 4}.
Sensorimotor transformation via sparse coding
Takiyama, Ken
2015-01-01
Sensorimotor transformation is indispensable to the accurate motion of the human body in daily life. For instance, when we grasp an object, the distance from our hands to an object needs to be calculated by integrating multisensory inputs, and our motor system needs to appropriately activate the arm and hand muscles to minimize the distance. The sensorimotor transformation is implemented in our neural systems, and recent advances in measurement techniques have revealed an important property of neural systems: a small percentage of neurons exhibits extensive activity while a large percentage shows little activity, i.e., sparse coding. However, we do not yet know the functional role of sparse coding in sensorimotor transformation. In this paper, I show that sparse coding enables complete and robust learning in sensorimotor transformation. In general, if a neural network is trained to maximize the performance on training data, the network shows poor performance on test data. Nevertheless, sparse coding renders compatible the performance of the network on both training and test data. Furthermore, sparse coding can reproduce reported neural activities. Thus, I conclude that sparse coding is necessary and a biologically plausible factor in sensorimotor transformation. PMID:25923980
On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue
Berg, M. van den; Ferone, V.; Nitsch, C.; Trombetti, C.
2016-01-01
Let $\\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\\Omega|$. We obtain some properties of the set function $F:\\Omega\\mapsto \\R^+$ defined by $$ F(\\Omega)=\\frac{T(\\Omega)\\lambda_1(\\Omega)}{|\\Omega|} ,$$ where $T(\\Omega)$ and $\\lambda_1(\\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\\'olya bound $F(\\Omega)\\le 1,$ and show that $$F(\\Omega)\\le 1- \
A nonlinear eigenvalue problem arising in a nanostructured quantum dot
Mohammadi, Abbasali; Bahrami, Fariba
2014-09-01
In this paper we investigate a minimization problem related to the principal eigenvalue of the s-wave Schrödinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the optimization problem and the uniqueness will be addressed when the domain is a ball. The optimized solution can be applied to design new electronic and photonic devices based on the quantum dots.
Detection of Wind Turbine Power Performance Abnormalities Using Eigenvalue Analysis
2014-12-23
Detection of Wind Turbine Power Performance Abnormalities Using Eigenvalue Analysis Georgios Alexandros Skrimpas1, Christian Walsted Sweeney2, Kun S...the following equation: P = 0.5ρACp(λ, β)u 3 (1) where P is the power captured by the wind turbine rotor, ρ is the air density, A is the swept rotor...may complicate the identification of abnor- malities. In addition to the above, the wind turbine power production can be affected by external factors
Eigenvalues from power-series expansions: an alternative approach
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima, Colima (Mexico); Fernandez, Francisco M [INIFTA (UNLP, CCT La Plata-CONICET), Division Quimica Teorica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina)], E-mail: paolo.amore@gmail.com, E-mail: fernande@quimica.unlp.edu.ar
2009-02-20
An appropriate rational approximation to the eigenfunction of the Schroedinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of Hankel determinants. The convergence rate of this approach is greater than that for a well-established method based on power-series expansions weighted by a Gaussian factor with an adjustable parameter (the so-called Hill-determinant method)
Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates
Energy Technology Data Exchange (ETDEWEB)
Perfetti, Christopher M [ORNL; Rearden, Bradley T [ORNL
2015-01-01
This study explored the impact of undersampling on the accuracy of tally estimates in Monte Carlo (MC) calculations. Steady-state MC simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity, and the impact of undersampling on eigenvalue and fuel pin flux/fission estimates was examined. This study observed biases in MC eigenvalue estimates as large as several percent and biases in fuel pin flux/fission tally estimates that exceeded tens, and in some cases hundreds, of percent. This study also investigated five statistical metrics for predicting the occurrence of undersampling biases in MC simulations. Three of the metrics (the Heidelberger-Welch RHW, the Geweke Z-Score, and the Gelman-Rubin diagnostics) are commonly used for diagnosing the convergence of Markov chains, and two of the methods (the Contributing Particles per Generation and Tally Entropy) are new convergence metrics developed in the course of this study. These metrics were implemented in the KENO MC code within the SCALE code system and were evaluated for their reliability at predicting the onset and magnitude of undersampling biases in MC eigenvalue and flux tally estimates in two of the critical models. Of the five methods investigated, the Heidelberger-Welch RHW, the Gelman-Rubin diagnostics, and Tally Entropy produced test metrics that correlated strongly to the size of the observed undersampling biases, indicating their potential to effectively predict the size and prevalence of undersampling biases in MC simulations.
Fast alternating projected gradient descent algorithms for recovering spectrally sparse signals
Cho, Myung
2016-06-24
We propose fast algorithms that speed up or improve the performance of recovering spectrally sparse signals from un-derdetermined measurements. Our algorithms are based on a non-convex approach of using alternating projected gradient descent for structured matrix recovery. We apply this approach to two formulations of structured matrix recovery: Hankel and Toeplitz mosaic structured matrix, and Hankel structured matrix. Our methods provide better recovery performance, and faster signal recovery than existing algorithms, including atomic norm minimization.
Sparse Representation Denoising for Radar High Resolution Range Profiling
Directory of Open Access Journals (Sweden)
Min Li
2014-01-01
Full Text Available Radar high resolution range profile has attracted considerable attention in radar automatic target recognition. In practice, radar return is usually contaminated by noise, which results in profile distortion and recognition performance degradation. To deal with this problem, in this paper, a novel denoising method based on sparse representation is proposed to remove the Gaussian white additive noise. The return is sparsely described in the Fourier redundant dictionary and the denoising problem is described as a sparse representation model. Noise level of the return, which is crucial to the denoising performance but often unknown, is estimated by performing subspace method on the sliding subsequence correlation matrix. Sliding window process enables noise level estimation using only one observation sequence, not only guaranteeing estimation efficiency but also avoiding the influence of profile time-shift sensitivity. Experimental results show that the proposed method can effectively improve the signal-to-noise ratio of the return, leading to a high-quality profile.
Doppler Ambiguity Resolution Based on Random Sparse Probing Pulses
Directory of Open Access Journals (Sweden)
Yunjian Zhang
2015-01-01
Full Text Available A novel method for solving Doppler ambiguous problem based on compressed sensing (CS theory is proposed in this paper. A pulse train with the random and sparse transmitting time is transmitted. The received signals after matched filtering can be viewed as randomly sparse sampling from the traditional fixed-pulse repetition frequency (PRF echo signals. The whole target echo could be reconstructed via CS recovery algorithms. Through refining the sensing matrix, which is equivalent to increase the sampling frequency of target characteristic, the Doppler unambiguous range is enlarged. In particular, Complex Approximate Message Passing (CAMP algorithm is developed to estimate the unambiguity Doppler frequency. Cramer-Rao lower bound expressions are derived for the frequency. Numerical simulations validate the effectiveness of the proposed method. Finally, compared with traditional methods, the proposed method only requires transmitting a few sparse probing pulses to achieve a larger Doppler frequency unambiguous range and can also reduce the consumption of the radar time resources.
On spectral properties of periodic polyharmonic matrix operators
Indian Academy of Sciences (India)
We consider a matrix operator = (-) + in , where ≥ 2, ≥ 1, 4 > + 1, and is the operator of multiplication by a periodic in matrix (). We study spectral properties of in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed.
Matrix Depot: an extensible test matrix collection for Julia
Directory of Open Access Journals (Sweden)
Weijian Zhang
2016-04-01
Full Text Available Matrix Depot is a Julia software package that provides easy access to a large and diverse collection of test matrices. Its novelty is threefold. First, it is extensible by the user, and so can be adapted to include the user’s own test problems. In doing so, it facilitates experimentation and makes it easier to carry out reproducible research. Second, it amalgamates in a single framework two different types of existing matrix collections, comprising parametrized test matrices (including Hansen’s set of regularization test problems and Higham’s Test Matrix Toolbox and real-life sparse matrix data (giving access to the University of Florida sparse matrix collection. Third, it fully exploits the Julia language. It uses multiple dispatch to help provide a simple interface and, in particular, to allow matrices to be generated in any of the numeric data types supported by the language.
OFDM pilot allocation for sparse channel estimation
Pakrooh, Pooria; Marvasti, Farrokh
2011-01-01
In communication systems, efficient use of the spectrum is an indispensable concern. Recently the use of compressed sensing for the purpose of estimating Orthogonal Frequency Division Multiplexing (OFDM) sparse multipath channels has been proposed to decrease the transmitted overhead in form of the pilot subcarriers which are essential for channel estimation. In this paper, we investigate the problem of deterministic pilot allocation in OFDM systems. The method is based on minimizing the coherence of the submatrix of the unitary Discrete Fourier Transform (DFT) matrix associated with the pilot subcarriers. Unlike the usual case of equidistant pilot subcarriers, we show that non-uniform patterns based on cyclic difference sets are optimal. In cases where there are no difference sets, we perform a greedy search method for finding a suboptimal solution. We also investigate the performance of the recovery methods such as Orthogonal Matching Pursuit (OMP) and Iterative Method with Adaptive Thresholding (IMAT) for ...
OFDM pilot allocation for sparse channel estimation
Pakrooh, Pooria; Amini, Arash; Marvasti, Farokh
2012-12-01
In communication systems, efficient use of the spectrum is an indispensable concern. Recently the use of compressed sensing for the purpose of estimating orthogonal frequency division multiplexing (OFDM) sparse multipath channels has been proposed to decrease the transmitted overhead in form of the pilot subcarriers which are essential for channel estimation. In this article, we investigate the problem of deterministic pilot allocation in OFDM systems. The method is based on minimizing the coherence of the submatrix of the unitary discrete fourier transform (DFT) matrix associated with the pilot subcarriers. Unlike the usual case of equidistant pilot subcarriers, we show that non-uniform patterns based on cyclic difference sets are optimal. In cases where there are no difference sets, we perform a greedy method for finding a suboptimal solution. We also investigate the performance of the recovery methods such as orthogonal matching pursuit (OMP) and iterative method with adaptive thresholding (IMAT) for estimation of the channel taps.
Heavy-tailed chiral random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Kanazawa, Takuya [iTHES Research Group and Quantum Hadron Physics Laboratory, RIKEN,Wako, Saitama, 351-0198 (Japan)
2016-05-27
We study an unconventional chiral random matrix model with a heavy-tailed probabilistic weight. The model is shown to exhibit chiral symmetry breaking with no bilinear condensate, in analogy to the Stern phase of QCD. We solve the model analytically and obtain the microscopic spectral density and the smallest eigenvalue distribution for an arbitrary number of flavors and arbitrary quark masses. Exotic behaviors such as non-decoupling of heavy flavors and a power-law tail of the smallest eigenvalue distribution are illustrated.
Heavy-tailed chiral random matrix theory
Kanazawa, Takuya
2016-05-01
We study an unconventional chiral random matrix model with a heavy-tailed probabilistic weight. The model is shown to exhibit chiral symmetry breaking with no bilinear condensate, in analogy to the Stern phase of QCD. We solve the model analytically and obtain the microscopic spectral density and the smallest eigenvalue distribution for an arbitrary number of flavors and arbitrary quark masses. Exotic behaviors such as non-decoupling of heavy flavors and a power-law tail of the smallest eigenvalue distribution are illustrated.
Si, Weijian; Zhao, Pinjiao; Qu, Zhiyu
2016-05-31
This paper presents an L-shaped sparsely-distributed vector sensor (SD-VS) array with four different antenna compositions. With the proposed SD-VS array, a novel two-dimensional (2-D) direction of arrival (DOA) and polarization estimation method is proposed to handle the scenario where uncorrelated and coherent sources coexist. The uncorrelated and coherent sources are separated based on the moduli of the eigenvalues. For the uncorrelated sources, coarse estimates are acquired by extracting the DOA information embedded in the steering vectors from estimated array response matrix of the uncorrelated sources, and they serve as coarse references to disambiguate fine estimates with cyclical ambiguity obtained from the spatial phase factors. For the coherent sources, four Hankel matrices are constructed, with which the coherent sources are resolved in a similar way as for the uncorrelated sources. The proposed SD-VS array requires only two collocated antennas for each vector sensor, thus the mutual coupling effects across the collocated antennas are reduced greatly. Moreover, the inter-sensor spacings are allowed beyond a half-wavelength, which results in an extended array aperture. Simulation results demonstrate the effectiveness and favorable performance of the proposed method.
Towards Google matrix of brain
Energy Technology Data Exchange (ETDEWEB)
Shepelyansky, D.L., E-mail: dima@irsamc.ups-tlse.f [Laboratoire de Physique Theorique (IRSAMC), Universite de Toulouse, UPS, F-31062 Toulouse (France); LPT - IRSAMC, CNRS, F-31062 Toulouse (France); Zhirov, O.V. [Budker Institute of Nuclear Physics, 630090 Novosibirsk (Russian Federation)
2010-07-12
We apply the approach of the Google matrix, used in computer science and World Wide Web, to description of properties of neuronal networks. The Google matrix G is constructed on the basis of neuronal network of a brain model discussed in PNAS 105 (2008) 3593. We show that the spectrum of eigenvalues of G has a gapless structure with long living relaxation modes. The PageRank of the network becomes delocalized for certain values of the Google damping factor {alpha}. The properties of other eigenstates are also analyzed. We discuss further parallels and similarities between the World Wide Web and neuronal networks.
Shi, Zhongrong; Liu, Chuancai
2017-07-01
The objective function of sparse representation to be minimized mainly consists of two parts: reconstruction error and sparsity-inducing regularization (e.g., ℓ0 norm or ℓ1 norm). A regularization parameter allows a trade-off between the two parts. In practical applications, solutions for sparse representation models are highly sensitive to the regularization parameter. To alleviate this phenomenon and improve the discrimination of sparse representation, a reweighted ℓ1 norm minimization-based sparse representation, named class-specific adaptive sparse representation (CSASR), is proposed. In CSASR, a class-specific weight matrix derived from observations of a class is applied to substitute the regularization parameter. The CSASR model mainly has the following two advantages: (i) the class-specific weight matrix can be calculated adaptively from observations; therefore, it is a parameter-free model and (ii) each class has a unique class-specific weight matrix, which enhances the discrimination of sparse representation. The experimental results show that the proposed algorithm achieves superior classification performance compared to traditional sparse representation-based classification algorithms.
Sparse Representations of Hyperspectral Images
Swanson, Robin J.
2015-11-23
Hyperspectral image data has long been an important tool for many areas of sci- ence. The addition of spectral data yields significant improvements in areas such as object and image classification, chemical and mineral composition detection, and astronomy. Traditional capture methods for hyperspectral data often require each wavelength to be captured individually, or by sacrificing spatial resolution. Recently there have been significant improvements in snapshot hyperspectral captures using, in particular, compressed sensing methods. As we move to a compressed sensing image formation model the need for strong image priors to shape our reconstruction, as well as sparse basis become more important. Here we compare several several methods for representing hyperspectral images including learned three dimensional dictionaries, sparse convolutional coding, and decomposable nonlocal tensor dictionaries. Addi- tionally, we further explore their parameter space to identify which parameters provide the most faithful and sparse representations.
Non-convex Statistical Optimization for Sparse Tensor Graphical Model.
Sun, Wei; Wang, Zhaoran; Liu, Han; Cheng, Guang
2015-01-01
We consider the estimation of sparse graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. The penalized maximum likelihood estimation of this model involves minimizing a non-convex objective function. In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with the optimal statistical rate of convergence as well as consistent graph recovery. Notably, such an estimator achieves estimation consistency with only one tensor sample, which is unobserved in previous work. Our theoretical results are backed by thorough numerical studies.
A New Sparse Quasi-Newton Update Method
Directory of Open Access Journals (Sweden)
Minghou Cheng
2012-04-01
Full Text Available Based on the idea of maximum determinant positive definite matrix completion, Yamashita proposed a sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. Such an MCQN update keeps the sparsity structure of the Hessian while relaxing the secant condition. In this paper, we propose an alternative to the MCQN update, in which the quasi-Newton matrix satisfies the secant condition, but does not have the same sparsity structure as the Hessian in general. Our numerical results demonstrate the usefulness of the new MCQN update with the BFGS formula for a collection of test problems. A local and superlinear convergence analysis is also provided for the new MCQN update with the DFP formula.
Image understanding using sparse representations
Thiagarajan, Jayaraman J; Turaga, Pavan; Spanias, Andreas
2014-01-01
Image understanding has been playing an increasingly crucial role in several inverse problems and computer vision. Sparse models form an important component in image understanding, since they emulate the activity of neural receptors in the primary visual cortex of the human brain. Sparse methods have been utilized in several learning problems because of their ability to provide parsimonious, interpretable, and efficient models. Exploiting the sparsity of natural signals has led to advances in several application areas including image compression, denoising, inpainting, compressed sensing, blin
On the adiabatic theorem when eigenvalues dive into the continuum
DEFF Research Database (Denmark)
Cornean, Decebal Horia; Jensen, Arne; Knörr, Hans Konrad
For a Wigner-Weisskopf model of an atom consisting of a quantum dot coupled to an energy reservoir described by a three-dimensional Laplacian we study the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discre...... eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit....
Asymptotic shape of solutions to nonlinear eigenvalue problems
Directory of Open Access Journals (Sweden)
Tetsutaro Shibata
2005-03-01
Full Text Available We consider the nonlinear eigenvalue problem $$ -u''(t = f(lambda, u(t, quad u mbox{greater than} 0, quad u(0 = u(1 = 0, $$ where $lambda > 0$ is a parameter. It is known that under some conditions on $f(lambda, u$, the shape of the solutions associated with $lambda$ is almost `box' when $lambda gg 1$. The purpose of this paper is to study precisely the asymptotic shape of the solutions as $lambda o infty$ from a standpoint of $L^1$-framework. To do this, we establish the asymptotic formulas for $L^1$-norm of the solutions as $lambda o infty$.
p-Norm SDD tensors and eigenvalue localization
Directory of Open Access Journals (Sweden)
Qilong Liu
2016-07-01
Full Text Available Abstract We present a new class of nonsingular tensors (p-norm strictly diagonally dominant tensors, which is a subclass of strong H $\\mathcal{H}$ -tensors. As applications of the results, we give a new eigenvalue inclusion set, which is tighter than those provided by Li et al. (Linear Multilinear Algebra 64:727-736, 2016 in some case. Based on this set, we give a checkable sufficient condition for the positive (semidefiniteness of an even-order symmetric tensor.
Dimensionality of social networks using motifs and eigenvalues.
Directory of Open Access Journals (Sweden)
Anthony Bonato
Full Text Available We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an m-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when m scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.
A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems
Directory of Open Access Journals (Sweden)
Fatemeh Mohammad
2014-05-01
Full Text Available In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem $Ax = \\lambda Bx$[Q.~Ye and P.~Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011 1697-1715]. In particular, the linear convergence property of the inverse subspace iteration is preserved.
A FPC-ROOT Algorithm for 2D-DOA Estimation in Sparse Array
Directory of Open Access Journals (Sweden)
Wenhao Zeng
2016-01-01
Full Text Available To improve the performance of two-dimensional direction-of-arrival (2D DOA estimation in sparse array, this paper presents a Fixed Point Continuation Polynomial Roots (FPC-ROOT algorithm. Firstly, a signal model for DOA estimation is established based on matrix completion and it can be proved that the proposed model meets Null Space Property (NSP. Secondly, left and right singular vectors of received signals matrix are achieved using the matrix completion algorithm. Finally, 2D DOA estimation can be acquired through solving the polynomial roots. The proposed algorithm can achieve high accuracy of 2D DOA estimation in sparse array, without solving autocorrelation matrix of received signals and scanning of two-dimensional spectral peak. Besides, it decreases the number of antennas and lowers computational complexity and meanwhile avoids the angle ambiguity problem. Computer simulations demonstrate that the proposed FPC-ROOT algorithm can obtain the 2D DOA estimation precisely in sparse array.
High-SNR spectrum measurement based on Hadamard encoding and sparse reconstruction
Wang, Zhaoxin; Yue, Jiang; Han, Jing; Li, Long; Jin, Yong; Gao, Yuan; Li, Baoming
2017-12-01
The denoising capabilities of the H-matrix and cyclic S-matrix based on the sparse reconstruction, employed in the Pixel of Focal Plane Coded Visible Spectrometer for spectrum measurement are investigated, where the spectrum is sparse in a known basis. In the measurement process, the digital micromirror device plays an important role, which implements the Hadamard coding. In contrast with Hadamard transform spectrometry, based on the shift invariability, this spectrometer may have the advantage of a high efficiency. Simulations and experiments show that the nonlinear solution with a sparse reconstruction has a better signal-to-noise ratio than the linear solution and the H-matrix outperforms the cyclic S-matrix whether the reconstruction method is nonlinear or linear.
Bostic, Susan W.; Fulton, Robert E.
1987-01-01
Eigenvalue analyses of complex structures is a computationally intensive task which can benefit significantly from new and impending parallel computers. This study reports on a parallel computer implementation of the Lanczos method for free vibration analysis. The approach used here subdivides the major Lanczos calculation tasks into subtasks and introduces parallelism down to the subtask levels such as matrix decomposition and forward/backward substitution. The method was implemented on a commercial parallel computer and results were obtained for a long flexible space structure. While parallel computing efficiency is problem and computer dependent, the efficiency for the Lanczos method was good for a moderate number of processors for the test problem. The greatest reduction in time was realized for the decomposition of the stiffness matrix, a calculation which took 70 percent of the time in the sequential program and which took 25 percent of the time on eight processors. For a sample calculation of the twenty lowest frequencies of a 486 degree of freedom problem, the total sequential computing time was reduced by almost a factor of ten using 16 processors.
Eigenvalue estimation of differential operators with a quantum algorithm
Szkopek, Thomas; Roychowdhury, Vwani; Yablonovitch, Eli; Abrams, Daniel S.
2005-12-01
We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S , acting on functions ψ(x1,x2,…,xD) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy Θ(1/N2) is Θ((2(S+1)(1+1/ν)+D)lnN) qubits and O(N2(S+1)(1+1/ν)lncND) gate operations, where N is the number of points to which each argument is discretized, ν and c are implementation dependent constants of O(1) . Optimal classical methods require Θ(ND) bits and Ω(ND) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D>2(S+1)(1+1/ν) . In the case of Schrödinger’s equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.
Audibert, Lorenzo; Cakoni, Fioralba; Haddar, Houssem
2017-12-01
In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data.
Efficient compressive sampling of spatially sparse fields in wireless sensor networks
Colonnese, Stefania; Cusani, Roberto; Rinauro, Stefano; Ruggiero, Giorgia; Scarano, Gaetano
2013-12-01
Wireless sensor networks (WSNs), i.e., networks of autonomous, wireless sensing nodes spatially deployed over a geographical area, are often faced with acquisition of spatially sparse fields. In this paper, we present a novel bandwidth/energy-efficient compressive sampling (CS) scheme for the acquisition of spatially sparse fields in a WSN. The paper contribution is twofold. Firstly, we introduce a sparse, structured CS matrix and analytically show that it allows accurate reconstruction of bidimensional spatially sparse signals, such as those occurring in several surveillance application. Secondly, we analytically evaluate the energy and bandwidth consumption of our CS scheme when it is applied to data acquisition in a WSN. Numerical results demonstrate that our CS scheme achieves significant energy and bandwidth savings with respect to state-of-the-art approaches when employed for sensing a spatially sparse field by means of a WSN.
Efficient Robust Matrix Factorization with Nonconvex Penalties
Yao, Quanming
2017-01-01
Robust matrix factorization (RMF) is a fundamental tool with lots of applications. The state-of-art is robust matrix factorization by majorization and minimization (RMF-MM) algorithm. It iteratively constructs and minimizes a novel surrogate function. Besides, it is also the only RMF algorithm with convergence guarantee. However, it can only deal with the convex $\\ell_1$-loss and does not utilize sparsity when matrix is sparsely observed. In this paper, we proposed robust matrix factorization...
Sparse Regression by Projection and Sparse Discriminant Analysis
Qi, Xin
2015-04-03
© 2015, © American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. Recent years have seen active developments of various penalized regression methods, such as LASSO and elastic net, to analyze high-dimensional data. In these approaches, the direction and length of the regression coefficients are determined simultaneously. Due to the introduction of penalties, the length of the estimates can be far from being optimal for accurate predictions. We introduce a new framework, regression by projection, and its sparse version to analyze high-dimensional data. The unique nature of this framework is that the directions of the regression coefficients are inferred first, and the lengths and the tuning parameters are determined by a cross-validation procedure to achieve the largest prediction accuracy. We provide a theoretical result for simultaneous model selection consistency and parameter estimation consistency of our method in high dimension. This new framework is then generalized such that it can be applied to principal components analysis, partial least squares, and canonical correlation analysis. We also adapt this framework for discriminant analysis. Compared with the existing methods, where there is relatively little control of the dependency among the sparse components, our method can control the relationships among the components. We present efficient algorithms and related theory for solving the sparse regression by projection problem. Based on extensive simulations and real data analysis, we demonstrate that our method achieves good predictive performance and variable selection in the regression setting, and the ability to control relationships between the sparse components leads to more accurate classification. In supplementary materials available online, the details of the algorithms and theoretical proofs, and R codes for all simulation studies are provided.
Directory of Open Access Journals (Sweden)
Khalil Ben Haddouch
2016-04-01
Full Text Available In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x$-growth conditions $\\Delta^2_{p(x} u=\\lambda |u|^{p(x-2} u$, under Neumann boundary conditions, where $p(x$ is a continuous function defined on the bounded domain with $p(x>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\\sup \\Lambda =+\\infty$, where $\\Lambda$ is the set of all eigenvalues.
Sparse Matrix Motivated Reconstruction of Far-Field Radiation Patterns
2015-03-01
root mean square error (RMSE). In addition, the proposed reconstruction algorithm was evaluated using measured data obtained in an anechoic chamber ...proposed approach for radiation pattern reconstruction, the time required to take measurements in an anechoic chamber can be reduced up to 87%, therefore...13 Fig. 13 2-D radiation pattern of the pyramidal horn ( anechoic chamber ) antenna and measurements used for reconstruction
A FPC-ROOT Algorithm for 2D-DOA Estimation in Sparse Array
Wenhao Zeng; Hongtao Li; Xiaohua Zhu; Chaoyu Wang
2016-01-01
To improve the performance of two-dimensional direction-of-arrival (2D DOA) estimation in sparse array, this paper presents a Fixed Point Continuation Polynomial Roots (FPC-ROOT) algorithm. Firstly, a signal model for DOA estimation is established based on matrix completion and it can be proved that the proposed model meets Null Space Property (NSP). Secondly, left and right singular vectors of received signals matrix are achieved using the matrix completion algorithm. Finally, 2D DOA estimat...
Random matrix theory and three-dimensional QCD
Energy Technology Data Exchange (ETDEWEB)
Verbaarschot, J.J.M.; Zahed, I. (Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794 (United States))
1994-10-24
We suggest that the spectral properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model. The exact spectral density is derived for this family of random matrix models for both even and odd number of fermions. New sum rules for the inverse powers of the eigenvalues of the Dirac operator are obtained. The issue of anomalies in random matrix theories is discussed.
Discrete Painlev\\'e equations and random matrix averages
Forrester, P. J.; Witte, N. S.
2003-01-01
The $\\tau$-function theory of Painlev\\'e systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlev\\'e equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as $...
Color normalization of histology slides using graph regularized sparse NMF
Sha, Lingdao; Schonfeld, Dan; Sethi, Amit
2017-03-01
Computer based automatic medical image processing and quantification are becoming popular in digital pathology. However, preparation of histology slides can vary widely due to differences in staining equipment, procedures and reagents, which can reduce the accuracy of algorithms that analyze their color and texture information. To re- duce the unwanted color variations, various supervised and unsupervised color normalization methods have been proposed. Compared with supervised color normalization methods, unsupervised color normalization methods have advantages of time and cost efficient and universal applicability. Most of the unsupervised color normaliza- tion methods for histology are based on stain separation. Based on the fact that stain concentration cannot be negative and different parts of the tissue absorb different stains, nonnegative matrix factorization (NMF), and particular its sparse version (SNMF), are good candidates for stain separation. However, most of the existing unsupervised color normalization method like PCA, ICA, NMF and SNMF fail to consider important information about sparse manifolds that its pixels occupy, which could potentially result in loss of texture information during color normalization. Manifold learning methods like Graph Laplacian have proven to be very effective in interpreting high-dimensional data. In this paper, we propose a novel unsupervised stain separation method called graph regularized sparse nonnegative matrix factorization (GSNMF). By considering the sparse prior of stain concentration together with manifold information from high-dimensional image data, our method shows better performance in stain color deconvolution than existing unsupervised color deconvolution methods, especially in keeping connected texture information. To utilized the texture information, we construct a nearest neighbor graph between pixels within a spatial area of an image based on their distances using heat kernal in lαβ space. The
Regression with Sparse Approximations of Data
DEFF Research Database (Denmark)
Noorzad, Pardis; Sturm, Bob L.
2012-01-01
We propose sparse approximation weighted regression (SPARROW), a method for local estimation of the regression function that uses sparse approximation with a dictionary of measurements. SPARROW estimates the regression function at a point with a linear combination of a few regressands selected by...... on the sparse approximation process. Our experimental results show the locally constant form of SPARROW performs competitively....
Continuous speech recognition with sparse coding
CSIR Research Space (South Africa)
Smit, WJ
2009-04-01
Full Text Available Sparse coding is an efficient way of coding information. In a sparse code most of the code elements are zero; very few are active. Sparse codes are intended to correspond to the spike trains with which biological neurons communicate. In this article...
Shearlets and Optimally Sparse Approximations
DEFF Research Database (Denmark)
Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q
2012-01-01
Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations...... of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide...... optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction...
Structured sparse models for classification
Castrodad, Alexey
The main focus of this thesis is the modeling and classification of high dimensional data using structured sparsity. Sparse models, where data is assumed to be well represented as a linear combination of a few elements from a dictionary, have gained considerable attention in recent years, and its use has led to state-of-the-art results in many signal and image processing tasks. The success of sparse modeling is highly due to its ability to efficiently use the redundancy of the data and find its underlying structure. On a classification setting, we capitalize on this advantage to properly model and separate the structure of the classes. We design and validate modeling solutions to challenging problems arising in computer vision and remote sensing. We propose both supervised and unsupervised schemes for the modeling of human actions from motion imagery under a wide variety of acquisition condi- tions. In the supervised case, the main goal is to classify the human actions in the video given a predefined set of actions to learn from. In the unsupervised case, the main goal is to an- alyze the spatio-temporal dynamics of the individuals in the scene without having any prior information on the actions themselves. We also propose a model for remotely sensed hysper- spectral imagery, where the main goal is to perform automatic spectral source separation and mapping at the subpixel level. Finally, we present a sparse model for sensor fusion to exploit the common structure and enforce collaboration of hyperspectral with LiDAR data for better mapping capabilities. In all these scenarios, we demonstrate that these data can be expressed as a combination of atoms from a class-structured dictionary. These data representation becomes essentially a "mixture of classes," and by directly exploiting the sparse codes, one can attain highly accurate classification performance with relatively unsophisticated classifiers.
Low-rank sparse learning for robust visual tracking
Zhang, Tianzhu
2012-01-01
In this paper, we propose a new particle-filter based tracking algorithm that exploits the relationship between particles (candidate targets). By representing particles as sparse linear combinations of dictionary templates, this algorithm capitalizes on the inherent low-rank structure of particle representations that are learned jointly. As such, it casts the tracking problem as a low-rank matrix learning problem. This low-rank sparse tracker (LRST) has a number of attractive properties. (1) Since LRST adaptively updates dictionary templates, it can handle significant changes in appearance due to variations in illumination, pose, scale, etc. (2) The linear representation in LRST explicitly incorporates background templates in the dictionary and a sparse error term, which enables LRST to address the tracking drift problem and to be robust against occlusion respectively. (3) LRST is computationally attractive, since the low-rank learning problem can be efficiently solved as a sequence of closed form update operations, which yield a time complexity that is linear in the number of particles and the template size. We evaluate the performance of LRST by applying it to a set of challenging video sequences and comparing it to 6 popular tracking methods. Our experiments show that by representing particles jointly, LRST not only outperforms the state-of-the-art in tracking accuracy but also significantly improves the time complexity of methods that use a similar sparse linear representation model for particles [1]. © 2012 Springer-Verlag.
Efficient Solution of the Electronic Eigenvalue Problem Using Wavepacket Propagation.
Neville, Simon P; Schuurman, Michael S
2018-02-15
We report how imaginary time wavepacket propagation may be used to efficiently calculate the lowest-lying eigenstates of the electronic Hamiltonian. This approach, known as the relaxation method in the quantum dynamics community, represents a fundamentally different approach to the solution of the electronic eigenvalue problem in comparison to traditional iterative subspace diagonalization schemes such as the Davidson and Lanczos methods. In order to render the relaxation method computationally competitive with existing iterative subspace methods, an extended short iterative Lanczos wavepacket propagation scheme is proposed and implemented. In the examples presented here, we show that by using an efficient wavepacket propagation algorithm the relaxation method is, at worst, as computationally expensive as the commonly used block Davidson-Liu algorithm, and in certain cases, significantly less so.
Detection of Wind Turbine Power Performance Abnormalities Using Eigenvalue Analysis
DEFF Research Database (Denmark)
Skrimpas, Georgios Alexandros; Sweeney, Christian Walsted; Marhadi, Kun Saptohartyadi
2014-01-01
, are commercially available and functioning successfully in fixed speed and vari- able speed turbines. Power performance analysis is a method specifically applicable to wind turbines for the detection of power generation changes due to external factors, such as ic- ing, internal factors, such as controller...... malfunction, or delib- erate actions, such as power de-rating. In this paper, power performance analysis is performed by sliding a time-power window and calculating the two eigenvalues corresponding to the two dimensional wind speed - power generation dis- tribution. The power is classified into five bins......Condition monitoring of wind turbines is a field of continu- ous research and development as new turbine configurations enter into the market and new failure modes appear. Systems utilising well established techniques from the energy and in- dustry sector, such as vibration analysis...
Li, Yuanqing; Cichocki, Andrzej; Amari, Shun-Ichi
2006-03-01
In this paper, we use a two-stage sparse factorization approach for blindly estimating the channel parameters and then estimating source components for electroencephalogram (EEG) signals. EEG signals are assumed to be linear mixtures of source components, artifacts, etc. Therefore, a raw EEG data matrix can be factored into the product of two matrices, one of which represents the mixing matrix and the other the source component matrix. Furthermore, the components are sparse in the time-frequency domain, i.e., the factorization is a sparse factorization in the time frequency domain. It is a challenging task to estimate the mixing matrix. Our extensive analysis and computational results, which were based on many sets of EEG data, not only provide firm evidences supporting the above assumption, but also prompt us to propose a new algorithm for estimating the mixing matrix. After the mixing matrix is estimated, the source components are estimated in the time frequency domain using a linear programming method. In an example of the potential applications of our approach, we analyzed the EEG data that was obtained from a modified Sternberg memory experiment. Two almost uncorrelated components obtained by applying the sparse factorization method were selected for phase synchronization analysis. Several interesting findings were obtained, especially that memory-related synchronization and desynchronization appear in the alpha band, and that the strength of alpha band synchronization is related to memory performance.
Space Group Debris Imaging Based on Sparse Sample
Directory of Open Access Journals (Sweden)
Zhu Jiang
2016-02-01
Full Text Available Space group debris imaging is difficult with sparse data in low Pulse Repetition Frequency (PRF spaceborne radar. To solve this problem in the narrow band system, we propose a method for space group debris imaging based on sparse samples. Due to the diversity of mass, density, and other factors, space group debris typically rotates at a high speed in different ways. We can obtain angular velocity through the autocorrelation function based on the diversity in the angular velocity. The scattering field usually presents strong sparsity, so we can utilize the corresponding measurement matrix to extract the data of different debris and then combine it using the sparse method to reconstruct the image. Furthermore, we can solve the Doppler ambiguity with the measurement matrix in low PRF systems and suppress some energy of other debris. Theoretical analysis confirms the validity of this methodology. Our simulation results demonstrate that the proposed method can achieve high-resolution Inverse Synthetic Aperture Radar (ISAR images of space group debris in low PRF systems.
Multiplicities of an Eigenvalue: Some Observations -R-ES ...
Indian Academy of Sciences (India)
Another title for this article could be ' What makes a ma- trix non-diagonalizable?' Of course, the question can be posed not only for matrices, but also for linear transfor- mations on any linear space. Recall that an n x n matrix A is said to be diagonalizable if there exists an n x n diagonal matrix D and an n x n invertible matrix ...
The Real-Valued Sparse Direction of Arrival (DOA Estimation Based on the Khatri-Rao Product
Directory of Open Access Journals (Sweden)
Tao Chen
2016-05-01
Full Text Available There is a problem that complex operation which leads to a heavy calculation burden is required when the direction of arrival (DOA of a sparse signal is estimated by using the array covariance matrix. The solution of the multiple measurement vectors (MMV model is difficult. In this paper, a real-valued sparse DOA estimation algorithm based on the Khatri-Rao (KR product called the L1-RVSKR is proposed. The proposed algorithm is based on the sparse representation of the array covariance matrix. The array covariance matrix is transformed to a real-valued matrix via a unitary transformation so that a real-valued sparse model is achieved. The real-valued sparse model is vectorized for transforming to a single measurement vector (SMV model, and a new virtual overcomplete dictionary is constructed according to the KR product’s property. Finally, the sparse DOA estimation is solved by utilizing the idea of a sparse representation of array covariance vectors (SRACV. The simulation results demonstrate the superior performance and the low computational complexity of the proposed algorithm.
The Real-Valued Sparse Direction of Arrival (DOA) Estimation Based on the Khatri-Rao Product.
Chen, Tao; Wu, Huanxin; Zhao, Zhongkai
2016-05-14
There is a problem that complex operation which leads to a heavy calculation burden is required when the direction of arrival (DOA) of a sparse signal is estimated by using the array covariance matrix. The solution of the multiple measurement vectors (MMV) model is difficult. In this paper, a real-valued sparse DOA estimation algorithm based on the Khatri-Rao (KR) product called the L₁-RVSKR is proposed. The proposed algorithm is based on the sparse representation of the array covariance matrix. The array covariance matrix is transformed to a real-valued matrix via a unitary transformation so that a real-valued sparse model is achieved. The real-valued sparse model is vectorized for transforming to a single measurement vector (SMV) model, and a new virtual overcomplete dictionary is constructed according to the KR product's property. Finally, the sparse DOA estimation is solved by utilizing the idea of a sparse representation of array covariance vectors (SRACV). The simulation results demonstrate the superior performance and the low computational complexity of the proposed algorithm.
Constraints on spacetime manifold in Euclidean supergravity in terms of Dirac eigenvalues
Ciuhu, C.; Vancea, I. V.
1998-01-01
We generalize previous work on Dirac eigenvalues as dynamical variables of Euclidean supergravity. The most general set of constraints on the curvatures of the tangent bundle and on the spinor bundle of the spacetime manifold under which spacetime admits Dirac eigenvalues as observables, are derived.
Eigenvalue spectra of a PT-symmetric coupled quartic potential in ...
Indian Academy of Sciences (India)
Abstract. The Schrödinger equation was solved for a generalized PT -symmetric quartic potential in two dimensions. It was found that, under a suitable ansatz for the wave function, the system possessed real and discrete energy eigenvalues. Analytic expressions for the energy eigenvalues and the eigenfunctions for the first ...
Li, Tiexiang; Huang, Tsung-Ming; Lin, Wen-Wei; Wang, Jenn-Nan
2017-03-01
We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell’s equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.
Cerbelli, Stefano; Giona, Massimiliano; Gorodetskyi, Olexander; Anderson, Patrick D.
2017-07-01
Enforcing the results developed by Gorodetskyi et al. [O. Gorodetskyi, M. Giona, P. Anderson, Phys. Fluids 24, 073603 (2012)] on the application of the mapping matrix formalism to simulate advective-diffusive transport, we investigate the structure and the properties of strange eigenfunctions and of the associated eigenvalues up to values of the Péclet number Pe 𝒪(108). Attention is focused on the possible occurrence of a singular limit for the second eigenvalue, ν2, of the advection-diffusion propagator as the Péclet number, Pe, tends to infinity, and on the structure of the corresponding eigenfunction. Prototypical time-periodic flows on the two-torus are considered, which give rise to toral twist maps with different hyperbolic character, encompassing Anosov, pseudo-Anosov, and smooth nonuniformly hyperbolic systems possessing a hyperbolic set of full measure. We show that for uniformly hyperbolic systems, a singular limit of the dominant decay exponent occurs, log|ν2| → constant≠0 for Pe → ∞, whereas log |ν2| → 0 according to a power-law in smooth non-uniformly hyperbolic systems that are not uniformly hyperbolic. The mere presence of a nonempty set of nonhyperbolic points (even if of zero Lebesgue measure) is thus found to mark the watershed between regular vs. singular behavior of ν2 with Pe as Pe → ∞.
Airborne Radar STAP using Sparse Recovery of Clutter Spectrum
Sun, Ke; Li, Gang; Meng, Huadong; Wang, Xiqin
2010-01-01
Space-time adaptive processing (STAP) is an effective tool for detecting a moving target in spaceborne or airborne radar systems. Statistical-based STAP methods generally need sufficient statistically independent and identically distributed (IID) training data to estimate the clutter characteristics. However, most actual clutter scenarios appear only locally stationary and lack sufficient IID training data. In this paper, by exploiting the intrinsic sparsity of the clutter distribution in the angle-Doppler domain, a new STAP algorithm called SR-STAP is proposed, which uses the technique of sparse recovery to estimate the clutter space-time spectrum. Joint sparse recovery with several training samples is also used to improve the estimation performance. Finally, an effective clutter covariance matrix (CCM) estimate and the corresponding STAP filter are designed based on the estimated clutter spectrum. Both the Mountaintop data and simulated experiments have illustrated the fast convergence rate of this approach...
A Spectral Algorithm for Envelope Reduction of Sparse Matrices
Barnard, Stephen T.; Pothen, Alex; Simon, Horst D.
1993-01-01
The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelope-reducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. This Laplacian eigenvector solves a continuous relaxation of a discrete problem related to envelope minimization called the minimum 2-sum problem. The permutation vector computed by the spectral algorithm is a closest permutation vector to the specified Laplacian eigenvector. Numerical results show that the new reordering algorithm usually computes smaller envelope sizes than those obtained from the current standard algorithms such as Gibbs-Poole-Stockmeyer (GPS) or SPARSPAK reverse Cuthill-McKee (RCM), in some cases reducing the envelope by more than a factor of two.
Delocalization transition for the Google matrix.
Giraud, Olivier; Georgeot, Bertrand; Shepelyansky, Dima L
2009-08-01
We study the localization properties of eigenvectors of the Google matrix, generated both from the world wide web and from the Albert-Barabási model of networks. We establish the emergence of a delocalization phase for the PageRank vector when network parameters are changed. For networks with localized PageRank, eigenvalues of the matrix in the complex plane with a modulus above a certain threshold correspond to localized eigenfunctions while eigenvalues below this threshold are associated with delocalized relaxation modes. We argue that, for networks with delocalized PageRank, the efficiency of information retrieval by Google-type search is strongly affected since the PageRank values have no clear hierarchical structure in this case.
Sparse Matrices in Frame Theory
DEFF Research Database (Denmark)
Lemvig, Jakob; Krahmer, Felix; Kutyniok, Gitta
2014-01-01
Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames...... yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices...
Dynamic Representations of Sparse Graphs
DEFF Research Database (Denmark)
Brodal, Gerth Stølting; Fagerberg, Rolf
1999-01-01
We present a linear space data structure for maintaining graphs with bounded arboricity—a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth—under edge insertions, edge deletions, and adjacency queries. The data structure supports adjacency queries in worst...... case O(c) time, and edge insertions and edge deletions in amortized O(1) and O(c+log n) time, respectively, where n is the number of nodes in the graph, and c is the bound on the arboricity....
Language Recognition via Sparse Coding
2016-09-08
combination of basis vectors in a dictionary (also learned). Nonzeros in the computed sparse code quantify the presence of specific basis vectors. By...for x using Dt−1a 6: update At := At−1 + yy> and Bt := Bt −1 + xy> 7: update by block-coordinate descent Dta := argminD′ 1 t [ 1 2 Tr(D′>D′At)− Tr(D... Bt )] 8: end 9: return: Dta U"erance(1(from(language(li(( U"erance(2(from(language(li# ( (((((((…( ( U"erance(m(from(language(li( U"erance(1(from
Asteroid Models from Sparse Photometry
Hanuš, Josef
2013-01-01
We investigate the photometric accuracy of the sparse data from astrometric surveys available on AstDyS. We use data from seven surveys with the best accu- racy in combination with relative lightcurves in the lightcurve inversion method to derive ∼300 new asteroid physical models (i.e., convex shapes and rotational states). We introduce several reliability tests that we use on all new asteroid mod- els. We investigate rotational properties of our MBAs sample (∼450 models here or previously de...
LiDAR point classification based on sparse representation
Li, Nan; Pfeifer, Norbert; Liu, Chun
2017-04-01
In order to combine the initial spatial structure and features of LiDAR data for accurate classification. The LiDAR data is represented as a 4-order tensor. Sparse representation for classification(SRC) method is used for LiDAR tensor classification. It turns out SRC need only a few of training samples from each class, meanwhile can achieve good classification result. Multiple features are extracted from raw LiDAR points to generate a high-dimensional vector at each point. Then the LiDAR tensor is built by the spatial distribution and feature vectors of the point neighborhood. The entries of LiDAR tensor are accessed via four indexes. Each index is called mode: three spatial modes in direction X ,Y ,Z and one feature mode. Sparse representation for classification(SRC) method is proposed in this paper. The sparsity algorithm is to find the best represent the test sample by sparse linear combination of training samples from a dictionary. To explore the sparsity of LiDAR tensor, the tucker decomposition is used. It decomposes a tensor into a core tensor multiplied by a matrix along each mode. Those matrices could be considered as the principal components in each mode. The entries of core tensor show the level of interaction between the different components. Therefore, the LiDAR tensor can be approximately represented by a sparse tensor multiplied by a matrix selected from a dictionary along each mode. The matrices decomposed from training samples are arranged as initial elements in the dictionary. By dictionary learning, a reconstructive and discriminative structure dictionary along each mode is built. The overall structure dictionary composes of class-specified sub-dictionaries. Then the sparse core tensor is calculated by tensor OMP(Orthogonal Matching Pursuit) method based on dictionaries along each mode. It is expected that original tensor should be well recovered by sub-dictionary associated with relevant class, while entries in the sparse tensor associated with
Directory of Open Access Journals (Sweden)
Damian Wiśniewski
2016-07-01
Full Text Available We consider the eigenvalue problem for the p(x-Laplace - Beltrami operator on the unit sphere. We prove an integro - differential inequality related to the smallest positive eigenvalue of this problem.
A Matrix Formalism for Landau Damping
Energy Technology Data Exchange (ETDEWEB)
Prabhakar, Shyam
1998-10-20
Existing methods of analyzing the effect of bunch-to-bunch tune shifts on coupled bunch instabilities are applicable to beams with a single unstable mode, or a few non-interacting unstable modes. We present a more general approach that involves computing the eigenvalues of a reduced state matrix. The method is applied to the analysis of PEP-II longitudinal coupled bunch modes, a large number of which are unstable in the absence of feedback.
Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator
Hirota, K.
2017-10-01
We study the eigenvalues of the self-adjoint Zakharov-Shabat operator corresponding to the defocusing nonlinear Schrödinger equation in the inverse scattering method. Real eigenvalues exist when the square of the potential has a simple well. We derive two types of quantization conditions for the eigenvalues by using the exact WKB method and show, moreover, that the eigenvalues stay real for a sufficiently small non-self-adjoint perturbation when the potential has some P T -like symmetry.
A preconditioned inexact newton method for nonlinear sparse electromagnetic imaging
Desmal, Abdulla
2015-03-01
A nonlinear inversion scheme for the electromagnetic microwave imaging of domains with sparse content is proposed. Scattering equations are constructed using a contrast-source (CS) formulation. The proposed method uses an inexact Newton (IN) scheme to tackle the nonlinearity of these equations. At every IN iteration, a system of equations, which involves the Frechet derivative (FD) matrix of the CS operator, is solved for the IN step. A sparsity constraint is enforced on the solution via thresholded Landweber iterations, and the convergence is significantly increased using a preconditioner that levels the FD matrix\\'s singular values associated with contrast and equivalent currents. To increase the accuracy, the weight of the regularization\\'s penalty term is reduced during the IN iterations consistently with the scheme\\'s quadratic convergence. At the end of each IN iteration, an additional thresholding, which removes small \\'ripples\\' that are produced by the IN step, is applied to maintain the solution\\'s sparsity. Numerical results demonstrate the applicability of the proposed method in recovering sparse and discontinuous dielectric profiles with high contrast values.
Permuting sparse rectangular matrices into block-diagonal form
Energy Technology Data Exchange (ETDEWEB)
Aykanat, Cevdet; Pinar, Ali; Catalyurek, Umit V.
2002-12-09
This work investigates the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for the solution of the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Besides proposing the models to represent sparse matrices and investigating related combinatorial problems, we provide a detailed survey of relevant literature to bridge the gap between different societies, investigate existing techniques for partitioning and propose new ones, and finally present a thorough empirical study of these techniques. Our experiments on a wide range of matrices, using state-of-the-art graph and hypergraph partitioning tools MeTiS and PaT oH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and run time.
Matrix models with Penner interaction inspired by interacting ...
Indian Academy of Sciences (India)
tems [5], wireless communication [6], stock and financial market [7] and many more. In many of these applications, universal properties of the eigenvalue statistics of the random matrix models are used like the spacing distributions. More recently, it has been applied to study RNA structure combinatorics which includes the ...
Partial Discharge Detection and Recognition in Random Matrix Theory Paradigm
National Research Council Canada - National Science Library
Luo, Lingen; Han, Bei; Chen, Jingde; Sheng, Gehao; Jiang, Xiuchen
.... Take advantage of the affluent results from random matrix theory (RMT), such as eigenvalue analysis, M-P law, the ring law, and so on, a novel methodology in RMT paradigm is proposed for fast PD pulse detection in this paper...
Bayesian Inference Methods for Sparse Channel Estimation
DEFF Research Database (Denmark)
Pedersen, Niels Lovmand
2013-01-01
This thesis deals with sparse Bayesian learning (SBL) with application to radio channel estimation. As opposed to the classical approach for sparse signal representation, we focus on the problem of inferring complex signals. Our investigations within SBL constitute the basis for the development...... of Bayesian inference algorithms for sparse channel estimation. Sparse inference methods aim at finding the sparse representation of a signal given in some overcomplete dictionary of basis vectors. Within this context, one of our main contributions to the field of SBL is a hierarchical representation...... complex prior representation achieve improved sparsity representations in low signalto- noise ratio as opposed to state-of-the-art sparse estimators. This result is of particular importance for the applicability of the algorithms in the field of channel estimation. We then derive various iterative...
Han, Rui-Qi; Xie, Wen-Jie; Xiong, Xiong; Zhang, Wei; Zhou, Wei-Xing
The correlation structure of a stock market contains important financial contents, which may change remarkably due to the occurrence of financial crisis. We perform a comparative analysis of the Chinese stock market around the occurrence of the 2008 crisis based on the random matrix analysis of high-frequency stock returns of 1228 Chinese stocks. Both raw correlation matrix and partial correlation matrix with respect to the market index in two time periods of one year are investigated. We find that the Chinese stocks have stronger average correlation and partial correlation in 2008 than in 2007 and the average partial correlation is significantly weaker than the average correlation in each period. Accordingly, the largest eigenvalue of the correlation matrix is remarkably greater than that of the partial correlation matrix in each period. Moreover, each largest eigenvalue and its eigenvector reflect an evident market effect, while other deviating eigenvalues do not. We find no evidence that deviating eigenvalues contain industrial sectorial information. Surprisingly, the eigenvectors of the second largest eigenvalues in 2007 and of the third largest eigenvalues in 2008 are able to distinguish the stocks from the two exchanges. We also find that the component magnitudes of the some largest eigenvectors are proportional to the stocks’ capitalizations.
Sparse High Dimensional Models in Economics.
Fan, Jianqing; Lv, Jinchi; Qi, Lei
2011-09-01
This paper reviews the literature on sparse high dimensional models and discusses some applications in economics and finance. Recent developments of theory, methods, and implementations in penalized least squares and penalized likelihood methods are highlighted. These variable selection methods are proved to be effective in high dimensional sparse modeling. The limits of dimensionality that regularization methods can handle, the role of penalty functions, and their statistical properties are detailed. Some recent advances in ultra-high dimensional sparse modeling are also briefly discussed.
Pilipchuk, L. A.; Pilipchuk, A. S.
2016-12-01
For constructing of the solutions of the sparse linear systems we propose effective methods, technologies and their implementation in Wolfram Mathematica. Sparse systems of these types appear in generalized network flow programming problems in the form of restrictions and can be characterized as systems with a large sparse sub-matrix representing the embedded network structure. In addition, such systems arise in estimating traffic in the generalized graph or multigraph on its unobservable part. For computing of each vector of the basis solution space with linear estimate in the worst case we propose effective algorithms and data structures in the case when a support of the multigraph or graph for the sparse systems contains a cycles.
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry
Directory of Open Access Journals (Sweden)
Kwang C. Shin
2010-02-01
Full Text Available We study the eigenvalue problem −u''+V(zu=λu in the complex plane with the boundary condition that u(z decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2, where V(z=−(iz^m−P(iz for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.
On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations
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Heinrich Voss
2016-06-01
Full Text Available Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.
S4 solution of the transport equation for eigenvalues using Legendre polynomials
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Öztürk Hakan
2017-01-01
Full Text Available Numerical solution of the transport equation for monoenergetic neutrons scattered isotropically through the medium of a finite homogeneous slab is studied for the determination of the eigenvalues. After obtaining the discrete ordinates form of the transport equation, separated homogeneous and particular solutions are formed and then the eigenvalues are calculated using the Gauss-Legendre quadrature set. Then, the calculated eigenvalues for various values of the c0, the mean number of secondary neutrons per collision, are given in the tables.
Sparse Reconstruction Schemes for Nonlinear Electromagnetic Imaging
Desmal, Abdulla
2016-03-01
Electromagnetic imaging is the problem of determining material properties from scattered fields measured away from the domain under investigation. Solving this inverse problem is a challenging task because (i) it is ill-posed due to the presence of (smoothing) integral operators used in the representation of scattered fields in terms of material properties, and scattered fields are obtained at a finite set of points through noisy measurements; and (ii) it is nonlinear simply due the fact that scattered fields are nonlinear functions of the material properties. The work described in this thesis tackles the ill-posedness of the electromagnetic imaging problem using sparsity-based regularization techniques, which assume that the scatterer(s) occupy only a small fraction of the investigation domain. More specifically, four novel imaging methods are formulated and implemented. (i) Sparsity-regularized Born iterative method iteratively linearizes the nonlinear inverse scattering problem and each linear problem is regularized using an improved iterative shrinkage algorithm enforcing the sparsity constraint. (ii) Sparsity-regularized nonlinear inexact Newton method calls for the solution of a linear system involving the Frechet derivative matrix of the forward scattering operator at every iteration step. For faster convergence, the solution of this matrix system is regularized under the sparsity constraint and preconditioned by leveling the matrix singular values. (iii) Sparsity-regularized nonlinear Tikhonov method directly solves the nonlinear minimization problem using Landweber iterations, where a thresholding function is applied at every iteration step to enforce the sparsity constraint. (iv) This last scheme is accelerated using a projected steepest descent method when it is applied to three-dimensional investigation domains. Projection replaces the thresholding operation and enforces the sparsity constraint. Numerical experiments, which are carried out using
Programming for Sparse Minimax Optimization
DEFF Research Database (Denmark)
Jonasson, K.; Madsen, Kaj
1994-01-01
We present an algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration, a linear minimax problem is solved for a basic step. If necessary, this is followed...... by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming....... The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ...
Sparse and stable Markowitz portfolios.
Brodie, Joshua; Daubechies, Ingrid; De Mol, Christine; Giannone, Domenico; Loris, Ignace
2009-07-28
We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e., portfolios with only few active positions), and allows accounting for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naïve evenly weighted portfolio.
Haidar, Azzam
2011-01-01
This paper introduces a novel implementation in reducing a symmetric dense matrix to tridiagonal form, which is the preprocessing step toward solving symmetric eigenvalue problems. Based on tile algorithms, the reduction follows a two-stage approach, where the tile matrix is first reduced to symmetric band form prior to the final condensed structure. The challenging trade-off between algorithmic performance and task granularity has been tackled through a grouping technique, which consists of aggregating fine-grained and memory-aware computational tasks during both stages, while sustaining the application\\'s overall high performance. A dynamic runtime environment system then schedules the different tasks in an out-of-order fashion. The performance for the tridiagonal reduction reported in this paper is unprecedented. Our implementation results in up to 50-fold and 12-fold improvement (130 Gflop/s) compared to the equivalent routines from LAPACK V3.2 and Intel MKL V10.3, respectively, on an eight socket hexa-core AMD Opteron multicore shared-memory system with a matrix size of 24000×24000. Copyright 2011 ACM.
Low-Rank Sparse Coding for Image Classification
Zhang, Tianzhu
2013-12-01
In this paper, we propose a low-rank sparse coding (LRSC) method that exploits local structure information among features in an image for the purpose of image-level classification. LRSC represents densely sampled SIFT descriptors, in a spatial neighborhood, collectively as low-rank, sparse linear combinations of code words. As such, it casts the feature coding problem as a low-rank matrix learning problem, which is different from previous methods that encode features independently. This LRSC has a number of attractive properties. (1) It encourages sparsity in feature codes, locality in codebook construction, and low-rankness for spatial consistency. (2) LRSC encodes local features jointly by considering their low-rank structure information, and is computationally attractive. We evaluate the LRSC by comparing its performance on a set of challenging benchmarks with that of 7 popular coding and other state-of-the-art methods. Our experiments show that by representing local features jointly, LRSC not only outperforms the state-of-the-art in classification accuracy but also improves the time complexity of methods that use a similar sparse linear representation model for feature coding.
Efficient MATLAB computations with sparse and factored tensors.
Energy Technology Data Exchange (ETDEWEB)
Bader, Brett William; Kolda, Tamara Gibson (Sandia National Lab, Livermore, CA)
2006-12-01
In this paper, the term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: a Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.
Nonextensive random-matrix theory based on Kaniadakis entropy
Abul-Magd, A. Y.
2006-01-01
The joint eigenvalue distributions of random-matrix ensembles are derived by applying the principle maximum entropy to the Renyi, Abe and Kaniadakis entropies. While the Renyi entropy produces essentially the same matrix-element distributions as the previously obtained expression by using the Tsallis entropy, and the Abe entropy does not lead to a closed form expression, the Kaniadakis entropy leads to a new generalized form of the Wigner surmise that describes a transition of the spacing dis...
Block Hadamard measurement matrix with arbitrary dimension in compressed sensing
Liu, Shaoqiang; Yan, Xiaoyan; Fan, Xiaoping; Li, Fei; Xu, Wen
2017-01-01
As Hadamard measurement matrix cannot be used for compressing signals with dimension of a non-integral power-of-2, this paper proposes a construction method of block Hadamard measurement matrix with arbitrary dimension. According to the dimension N of signals to be measured, firstly, construct a set of Hadamard sub matrixes with different dimensions and make the sum of these dimensions equals to N. Then, arrange the Hadamard sub matrixes in a certain order to form a block diagonal matrix. Finally, take the former M rows of the block diagonal matrix as the measurement matrix. The proposed measurement matrix which retains the orthogonality of Hadamard matrix and sparsity of block diagonal matrix has highly sparse structure, simple hardware implements and general applicability. Simulation results show that the performance of our measurement matrix is better than Gaussian matrix, Logistic chaotic matrix, and Toeplitz matrix.
Two conjectures about spectral density of diluted sparse Bernoulli random matrices
Nechaev, S. K.
2014-01-01
We consider the ensemble of $N\\times N$ ($N\\gg 1$) symmetric random matrices with the bimodal independent distribution of matrix elements: each element could be either "1" with the probability $p$, or "0" otherwise. We pay attention to the "diluted" sparse regime, taking $p=1/N +\\epsilon$, where $0
U(1) staggered Dirac operator and random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Berg, Bernd A.; Markum, Harald; Pullirsch, Rainer; Wettig, Tilo
2000-03-01
We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge theory and its relationship to random matrix theory. In the confined as well as in the Coulomb phase the nearest-neighbor spacing distribution of the unfolded eigenvalues is well described by the chiral unitary ensemble. The same is true for the distribution of the smallest eigenvalue and the microscopic spectral density in the confined phase. The physical origin of the chiral condensate in this phase deserves further study.
Random matrix theory and QCD at nonzero chemical potential
Energy Technology Data Exchange (ETDEWEB)
Verbaarschot, J.J.M. [State Univ. of New York, Stony Brook, NY (United States). Dept. of Physics and Astronomy
1998-11-02
In this lecture we give a brief review of chiral random matrix theory (chRMT) and its applications to QCD at nonzero chemical potential. We present both analytical arguments involving chiral perturbation theory and numerical evidence from lattice QCD simulations showing that correlations of the smallest Dirac eigenvalues are described by chRMT. We discuss the range of validity of chRMT and emphasize the importance of universality. For chRMT`s at {mu}{ne}0 we identify universal properties of the Dirac eigenvalues and study the effect of quenching on the distribution of Yang-Lee zeros. (orig.) 86 refs.
A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential
Directory of Open Access Journals (Sweden)
Craig Haile
2000-01-01
Full Text Available We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^alpha$, $k ge 0$, $k in mathbb{R}, alpha ge 2$ on a bounded domain $Omega$ in $mathbb{R}^n$, $(n ge 2$ with Dirichlet boundary conditions. In particular, we compare the eigenvalue $lambda_2(Omega$ of the operator $-Delta + k |x|^alpha $ on $Omega$ with the eigenvalue $lambda_2(S_1$ of the same operator $-Delta +kr^alpha$ on a ball $S_1$, where $S_1$ has radius such that the first eigenvalues are the same ($lambda_1(Omega = lambda_1(S_1$. The main result is to show $lambda_2(Omega le lambda_2(S_1$. We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain $Omega$ with Dirichlet boundary conditions.
Vannell, Eric C.; Kenny, Sean P.; Maghami, Peiman G.
1995-03-01
The erection and deployment of large flexible structures having thousands of degrees of freedom requires controllers based on new techniques of eigenvalue assignment that are computationally stable and more efficient. Scientists at NASA Langley Research Center have developed a novel and efficient algorithm for the eigenvalue assignment of large, time-invariant systems using full-state and output feedback. The objectives of this research were to improve upon the output feedback version of this algorithm, to produce a toolbox of MATLAB functions based on the efficient eigenvalue assignment algorithm, and to experimentally verify the algorithm and software by implementing controllers designed using the MATLAB toolbox on the phase 2 configuration of NASA Langley's controls-structures interaction evolutionary model, a laboratory model used to study space structures. Results from laboratory tests and computer simulations show that effective controllers can be designed using software based on the efficient eigenvalue assignment algorithm.
Directory of Open Access Journals (Sweden)
Feng Qi
2014-10-01
Full Text Available The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.
Eigenvalues of the Laplacian on Regular Polygons and Polygons Resulting from Their Disection
Cureton, L. M.; Kuttler, J. R.
1999-02-01
The eigenvalue problem is considered for the Laplacian on regular polygons, with either Dirichlet or Neumann boundary conditions, which will be related to the unit circle by a conformal mapping. The polygonal problem is then equivalent to a weighted eigenvalue problem on the circle with the same boundary conditions. Upper bounds are found for the eigenvalues by the Rayleigh-Ritz method, where the trial functions are the eigenfunctions of the unweighted problem on the circle. These are products of Bessel and trigonometrical functions, and so the required integrals simplify greatly, with a new recursion formula used to generate some Bessel function integrals. Numerical results are given for the case of the hexagon with Dirichlet conditions. Consideration of symmetry classes makes computations more efficient, and gives as a byproduct the eigenvalues of a number of polygons, such as trapezoids and diamonds, which result from disecting the hexagon. Comparisons of the hexagon results are made with previous work.
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal Diaz del Castillo 340, Colima, Colima (Mexico)], E-mail: paolo.amore@gmail.com; Fernandez, Francisco M. [INIFTA (Conicet, UNLP), Division Quimica Teorica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina)], E-mail: fernande@quimica.unlp.edu.ar
2008-04-28
We show that the Riccati-Pade method is suitable for the calculation of the complex eigenvalues of the Schroedinger equation with a repulsive exponential potential. The accuracy of the results is remarkable for realistic potential parameters.
Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale
Directory of Open Access Journals (Sweden)
John M. Davis
2000-01-01
. Values of the parameter λ (eigenvalues are determined for which this problem has a positive solution. The methods used here extend recent results by allowing for a broader class of functions for a(t.
Directory of Open Access Journals (Sweden)
Serguei I. Iakovlev
2013-01-01
Full Text Available It is shown that any \\(\\mu \\in \\mathbb{C}\\ is an infinite multiplicity eigenvalue of the Steklov smoothing operator \\(S_h\\ acting on the space \\(L^1_{loc}(\\mathbb{R}\\. For \\(\\mu \
M3: Matrix Multiplication on MapReduce
DEFF Research Database (Denmark)
Silvestri, Francesco; Ceccarello, Matteo
2015-01-01
M3 is an Hadoop library for performing dense and sparse matrix multiplication in MapReduce. The library is based on multi-round algorithms exploiting the 3D decomposition of the problem.......M3 is an Hadoop library for performing dense and sparse matrix multiplication in MapReduce. The library is based on multi-round algorithms exploiting the 3D decomposition of the problem....
Stochastic R matrix for Uq (An(1))
Kuniba, A.; Mangazeev, V. V.; Maruyama, S.; Okado, M.
2016-12-01
We show that the quantum R matrix for symmetric tensor representations of Uq (An(1)) satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the q-Hahn process for n = 1. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of n species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.
Heuveline, V
2003-01-01
Recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators. The standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed. They are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds. Further, the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues. A new method is presented which combines multigrid techniques with the Lanczos process. The resulting scheme requires O(mn) arithmetic operations and O(n) storage requirement, where n is the number of unknowns and m, the number of needed eigenvalues. The discretization of the considered differential operators is realized by means of p-finite elements and is applicable on general geometries. Numerical exp...
Directory of Open Access Journals (Sweden)
Pengzhan Huang
2011-01-01
Full Text Available Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and it is a favorite method among these methods for the Stokes eigenvalue problem.
The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods
Hu, Jun; Huang, Yunqing; Lin, Qun
2011-01-01
The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonco...
Eigenvalue analysis of a cantilever beam-rigid-body MEMS gyroscope
Lajimi, Seyed Amir Mousavi; Abdel-Rahman, Eihab
2014-01-01
The eigenvalues of a new microbeam-rigid-body gyroscope are computed and studied to show the variation of frequencies versus the input spin rate. To this end, assuming the harmonic solution of the dynamic equation of motion the characteristic equation is obtained and solved for the natural frequencies of the system in the rotating frame. It is shown that the difference between the natural frequencies (eigenvalues) proportionally grows with the input angular displacement rate.
A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Zixia Yuan
2007-12-01
Full Text Available Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.
A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Dou Jingbo
2007-01-01
Full Text Available Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the -sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.
Exploiting sparseness in de novo genome assembly.
Ye, Chengxi; Ma, Zhanshan Sam; Cannon, Charles H; Pop, Mihai; Yu, Douglas W
2012-04-19
The very large memory requirements for the construction of assembly graphs for de novo genome assembly limit current algorithms to super-computing environments. In this paper, we demonstrate that constructing a sparse assembly graph which stores only a small fraction of the observed k-mers as nodes and the links between these nodes allows the de novo assembly of even moderately-sized genomes (~500 M) on a typical laptop computer. We implement this sparse graph concept in a proof-of-principle software package, SparseAssembler, utilizing a new sparse k-mer graph structure evolved from the de Bruijn graph. We test our SparseAssembler with both simulated and real data, achieving ~90% memory savings and retaining high assembly accuracy, without sacrificing speed in comparison to existing de novo assemblers.
Fast Estimation of Approximate Matrix Ranks Using Spectral Densities.
Ubaru, Shashanka; Saad, Yousef; Seghouane, Abd-Krim
2017-05-01
Many machine learning and data-related applications require the knowledge of approximate ranks of large data matrices at hand. This letter presents two computationally inexpensive techniques to estimate the approximate ranks of such matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of finding eigenvalues of the matrix at a given point on the real line. Integrating the spectral density over an interval gives the eigenvalue count of the matrix in that interval. Therefore, the rank can be approximated by integrating the spectral density over a carefully selected interval. Two different approaches are discussed to estimate the approximate rank, one based on Chebyshev polynomials and the other based on the Lanczos algorithm. In order to obtain the appropriate interval, it is necessary to locate a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that contribute to the matrix rank. A method for locating this gap and selecting the interval of integration is proposed based on the plot of the spectral density. Numerical experiments illustrate the performance of these techniques on matrices from typical applications.
Parity effects in eigenvalue correlators, parametric and crossover ...
Indian Academy of Sciences (India)
Application to mesoscopic systems ... in the Hamiltonian of the chaotic system or external magnetic field on a sample of material) is varied. ... M4, where M is a random N ¢N matrix. I will present some results for the two-point density– density correlation functions which show interesting parity effects and further characterizes.
On the nonnegative inverse eigenvalue problem of traditional matrices
Directory of Open Access Journals (Sweden)
Alimohammad Nazari
2014-07-01
Full Text Available In this paper, at first for a given set of real or complex numbers $\\sigma$ with nonnegativesummation, we introduce some special conditions that with them there is no nonnegativetridiagonal matrix in which $\\sigma$ is its spectrum. In continue we present some conditions forexistence such nonnegative tridiagonal matrices.
Threshold partitioning of sparse matrices and applications to Markov chains
Energy Technology Data Exchange (ETDEWEB)
Choi, Hwajeong; Szyld, D.B. [Temple Univ., Philadelphia, PA (United States)
1996-12-31
It is well known that the order of the variables and equations of a large, sparse linear system influences the performance of classical iterative methods. In particular if, after a symmetric permutation, the blocks in the diagonal have more nonzeros, classical block methods have a faster asymptotic rate of convergence. In this paper, different ordering and partitioning algorithms for sparse matrices are presented. They are modifications of PABLO. In the new algorithms, in addition to the location of the nonzeros, the values of the entries are taken into account. The matrix resulting after the symmetric permutation has dense blocks along the diagonal, and small entries in the off-diagonal blocks. Parameters can be easily adjusted to obtain, for example, denser blocks, or blocks with elements of larger magnitude. In particular, when the matrices represent Markov chains, the permuted matrices are well suited for block iterative methods that find the corresponding probability distribution. Applications to three types of methods are explored: (1) Classical block methods, such as Block Gauss Seidel. (2) Preconditioned GMRES, where a block diagonal preconditioner is used. (3) Iterative aggregation method (also called aggregation/disaggregation) where the partition obtained from the ordering algorithm with certain parameters is used as an aggregation scheme. In all three cases, experiments are presented which illustrate the performance of the methods with the new orderings. The complexity of the new algorithms is linear in the number of nonzeros and the order of the matrix, and thus adding little computational effort to the overall solution.
Fuzzy Field Theory as a Random Matrix Model
Tekel, Juraj
This dissertation considers the theory of scalar fields on fuzzy spaces from the point of view of random matrices. First we define random matrix ensembles, which are natural description of such theory. These ensembles are new and the novel feature is a presence of kinetic term in the probability measure, which couples the random matrix to a set of external matrices and thus breaks the original symmetry. Considering the case of a free field ensemble, which is generalization of a Gaussian matrix ensemble, we develop a technique to compute expectation values of the observables of the theory based on explicit Wick contractions and we write down recursion rules for these. We show that the eigenvalue distribution of the random matrix follows the Wigner semicircle distribution with a rescaled radius. We also compute distributions of the matrix Laplacian of the random matrix given by the new term and demonstrate that the eigenvalues of these two matrices are correlated. We demonstrate the robustness of the method by computing expectation values and distributions for more complicated observables. We then consider the ensemble corresponding to an interacting field theory, with a quartic interaction. We use the same method to compute the distribution of the eigenvalues and show that the presence of the kinetic terms rescales the distribution given by the original theory, which is a polynomially deformed Wigner semicircle. We compute the eigenvalue distribution of the matrix Laplacian and the joint distribution up to second order in the correlation and we show that the correlation between the two changes from the free field case. Finally, as an application of these results, we compute the phase diagram of the fuzzy scalar field theory, we find multiscaling which stabilizes this diagram in the limit of large matrices and compare it with the results obtained numerically and by considering the kinetic part as a perturbation.
Sparse estimation for structural variability
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Singh Rohit
2011-04-01
Full Text Available Abstract Background Proteins are dynamic molecules that exhibit a wide range of motions; often these conformational changes are important for protein function. Determining biologically relevant conformational changes, or true variability, efficiently is challenging due to the noise present in structure data. Results In this paper we present a novel approach to elucidate conformational variability in structures solved using X-ray crystallography. We first infer an ensemble to represent the experimental data and then formulate the identification of truly variable members of the ensemble (as opposed to those that vary only due to noise as a sparse estimation problem. Our results indicate that the algorithm is able to accurately distinguish genuine conformational changes from variability due to noise. We validate our predictions for structures in the Protein Data Bank by comparing with NMR experiments, as well as on synthetic data. In addition to improved performance over existing methods, the algorithm is robust to the levels of noise present in real data. In the case of Human Ubiquitin-conjugating enzyme Ubc9, variability identified by the algorithm corresponds to functionally important residues implicated by mutagenesis experiments. Our algorithm is also general enough to be integrated into state-of-the-art software tools for structure-inference.
Sparse time-frequency representations.
Gardner, Timothy J; Magnasco, Marcelo O
2006-04-18
Auditory neurons preserve exquisite temporal information about sound features, but we do not know how the brain uses this information to process the rapidly changing sounds of the natural world. Simple arguments for effective use of temporal information led us to consider the reassignment class of time-frequency representations as a model of auditory processing. Reassigned time-frequency representations can track isolated simple signals with accuracy unlimited by the time-frequency uncertainty principle, but lack of a general theory has hampered their application to complex sounds. We describe the reassigned representations for white noise and show that even spectrally dense signals produce sparse reassignments: the representation collapses onto a thin set of lines arranged in a froth-like pattern. Preserving phase information allows reconstruction of the original signal. We define a notion of "consensus," based on stability of reassignment to time-scale changes, which produces sharp spectral estimates for a wide class of complex mixed signals. As the only currently known class of time-frequency representations that is always "in focus" this methodology has general utility in signal analysis. It may also help explain the remarkable acuity of auditory perception. Many details of complex sounds that are virtually undetectable in standard sonograms are readily perceptible and visible in reassignment.
Post-Nonlinear Sparse Component Analysis Using Single-Source Zones and Functional Data Clustering
Puigt, Matthieu; Mouchtaris, Athanasios
2012-01-01
In this paper, we introduce a general extension of linear sparse component analysis (SCA) approaches to postnonlinear (PNL) mixtures. In particular, and contrary to the state-of-art methods, our approaches use a weak sparsity source assumption: we look for tiny temporal zones where only one source is active. We investigate two nonlinear single-source confidence measures, using the mutual information and a local linear tangent space approximation (LTSA). For this latter measure, we derive two extensions of linear single-source measures, respectively based on correlation (LTSA-correlation) and eigenvalues (LTSA-PCA). A second novelty of our approach consists of applying functional data clustering techniques to the scattered observations in the above single-source zones, thus allowing us to accurately estimate them.We first study a classical approach using a B-spline approximation, and then two approaches which locally approximate the nonlinear functions as lines. Finally, we extend our PNL methods to more gener...
Frequency and 2D Angle Estimation Based on a Sparse Uniform Array of Electromagnetic Vector Sensors
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Kwong Sam
2006-01-01
Full Text Available We present an ESPRIT-based algorithm that yields extended-aperture two-dimensional (2D arrival angle and carrier frequency estimates with a sparse uniform array of electromagnetic vector sensors. The ESPRIT-based frequency estimates are first achieved by using the temporal invariance structure out of the two time-delayed sets of data collected from vector sensor array. Each incident source's coarse direction of arrival (DOA estimation is then obtained through the Poynting vector estimates (using a vector cross-product estimator. The frequency and coarse angle estimate results are used jointly to disambiguate the cyclic phase ambiguities in ESPRIT's eigenvalues when the intervector sensor spacing exceeds a half wavelength. Monte Carlo simulation results verified the effectiveness of the proposed method.
Wavelet sparse transform optimization in image reconstruction based on compressed sensing
Ziran, Wei; Huachuang, Wang; Jianlin, Zhang
2017-06-01
The high image sparsity is very important to improve the accuracy of compressed sensing reconstruction image, and the wavelet transform can make the image sparse obviously. This paper is the optimization method based on wavelet sparse transform in image reconstruction based on compressed sensing, and we have designed a restraining matrix to optimize the wavelet sparse transform. Firstly, the wavelet coefficients are obtained by wavelet transform of the original signal data, and the wavelet coefficients have a tendency of decreasing gradually. The restraining matrix is used to restrain the small coefficients and is a part of image sparse transform, so as to make the wavelet coefficients more sparse. When the sampling rate is between 0. 15 and 0. 45, the simulation results show that the quality promotion of the reconstructed image is the best, and the peak signal to noise ratio (PSNR) is increased by about 0.5dB to 1dB. At the same time, it is more obvious to improve the reconstruction accuracy of the fingerprint texture image, which to some extent makes up for the shortcomings that reconstruction of texture image by compressed sensing based on the wavelet transform has the low accuracy.
Directory of Open Access Journals (Sweden)
Yudan Ren
Full Text Available Functional neuroimaging is widely used to examine changes in brain function associated with age, gender or neuropsychiatric conditions. FMRI (functional magnetic resonance imaging studies employ either laboratory-designed tasks that engage the brain with abstracted and repeated stimuli, or resting state paradigms with little behavioral constraint. Recently, novel neuroimaging paradigms using naturalistic stimuli are gaining increasing attraction, as they offer an ecologically-valid condition to approximate brain function in real life. Wider application of naturalistic paradigms in exploring individual differences in brain function, however, awaits further advances in statistical methods for modeling dynamic and complex dataset. Here, we developed a novel data-driven strategy that employs group sparse representation to assess gender differences in brain responses during naturalistic emotional experience. Comparing to independent component analysis (ICA, sparse coding algorithm considers the intrinsic sparsity of neural coding and thus could be more suitable in modeling dynamic whole-brain fMRI signals. An online dictionary learning and sparse coding algorithm was applied to the aggregated fMRI signals from both groups, which was subsequently factorized into a common time series signal dictionary matrix and the associated weight coefficient matrix. Our results demonstrate that group sparse representation can effectively identify gender differences in functional brain network during natural viewing, with improved sensitivity and reliability over ICA-based method. Group sparse representation hence offers a superior data-driven strategy for examining brain function during naturalistic conditions, with great potential for clinical application in neuropsychiatric disorders.
Numerical solution of quadratic matrix equations for free vibration analysis of structures
Gupta, K. K.
1975-01-01
This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.
Sparse Gaussian graphical mixture model
ANANI, Lotsi; WIT, Ernst
2016-01-01
This paper considers the problem of networks reconstruction from heterogeneous data using a Gaussian Graphical Mixture Model (GGMM). It is well known that parameter estimation in this context is challenging due to large numbers of variables coupled with the degenerate nature of the likelihood. We propose as a solution a penalized maximum likelihood technique by imposing an l1 penalty on the precision matrix. Our approach shrinks the parameters thereby resulting in better identifiability and v...
Tang, Xin; Feng, Guo-Can; Li, Xiao-Xin; Cai, Jia-Xin
2015-01-01
Face recognition is challenging especially when the images from different persons are similar to each other due to variations in illumination, expression, and occlusion. If we have sufficient training images of each person which can span the facial variations of that person under testing conditions, sparse representation based classification (SRC) achieves very promising results. However, in many applications, face recognition often encounters the small sample size problem arising from the small number of available training images for each person. In this paper, we present a novel face recognition framework by utilizing low-rank and sparse error matrix decomposition, and sparse coding techniques (LRSE+SC). Firstly, the low-rank matrix recovery technique is applied to decompose the face images per class into a low-rank matrix and a sparse error matrix. The low-rank matrix of each individual is a class-specific dictionary and it captures the discriminative feature of this individual. The sparse error matrix represents the intra-class variations, such as illumination, expression changes. Secondly, we combine the low-rank part (representative basis) of each person into a supervised dictionary and integrate all the sparse error matrix of each individual into a within-individual variant dictionary which can be applied to represent the possible variations between the testing and training images. Then these two dictionaries are used to code the query image. The within-individual variant dictionary can be shared by all the subjects and only contribute to explain the lighting conditions, expressions, and occlusions of the query image rather than discrimination. At last, a reconstruction-based scheme is adopted for face recognition. Since the within-individual dictionary is introduced, LRSE+SC can handle the problem of the corrupted training data and the situation that not all subjects have enough samples for training. Experimental results show that our method achieves the
Directory of Open Access Journals (Sweden)
Xin Tang
Full Text Available Face recognition is challenging especially when the images from different persons are similar to each other due to variations in illumination, expression, and occlusion. If we have sufficient training images of each person which can span the facial variations of that person under testing conditions, sparse representation based classification (SRC achieves very promising results. However, in many applications, face recognition often encounters the small sample size problem arising from the small number of available training images for each person. In this paper, we present a novel face recognition framework by utilizing low-rank and sparse error matrix decomposition, and sparse coding techniques (LRSE+SC. Firstly, the low-rank matrix recovery technique is applied to decompose the face images per class into a low-rank matrix and a sparse error matrix. The low-rank matrix of each individual is a class-specific dictionary and it captures the discriminative feature of this individual. The sparse error matrix represents the intra-class variations, such as illumination, expression changes. Secondly, we combine the low-rank part (representative basis of each person into a supervised dictionary and integrate all the sparse error matrix of each individual into a within-individual variant dictionary which can be applied to represent the possible variations between the testing and training images. Then these two dictionaries are used to code the query image. The within-individual variant dictionary can be shared by all the subjects and only contribute to explain the lighting conditions, expressions, and occlusions of the query image rather than discrimination. At last, a reconstruction-based scheme is adopted for face recognition. Since the within-individual dictionary is introduced, LRSE+SC can handle the problem of the corrupted training data and the situation that not all subjects have enough samples for training. Experimental results show that our
Discriminative sparse coding on multi-manifolds
Wang, J.J.-Y.
2013-09-26
Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics. However, the conventional sparse coding algorithms and their manifold-regularized variants (graph sparse coding and Laplacian sparse coding), learn codebooks and codes in an unsupervised manner and neglect class information that is available in the training set. To address this problem, we propose a novel discriminative sparse coding method based on multi-manifolds, that learns discriminative class-conditioned codebooks and sparse codes from both data feature spaces and class labels. First, the entire training set is partitioned into multiple manifolds according to the class labels. Then, we formulate the sparse coding as a manifold-manifold matching problem and learn class-conditioned codebooks and codes to maximize the manifold margins of different classes. Lastly, we present a data sample-manifold matching-based strategy to classify the unlabeled data samples. Experimental results on somatic mutations identification and breast tumor classification based on ultrasonic images demonstrate the efficacy of the proposed data representation and classification approach. 2013 The Authors. All rights reserved.
Wenner, Michael T.
is simulated multiple times, this methodology often slows down the final keff convergence, but an increase coupling between source zones with important yet low probability interaction is observed. This is an important finding for loosely coupled systems and may be useful in their analysis. The third major focus of this dissertation concerns the use of the standard cumulative fission matrix methodology for high dominance ratio problems which results in high source correlation. Source eigenvector confidence is calculated utilizing a Monte Carlo iterated confidence approach and shown to be superior to the currently used plus and minus fission matrix methodology. Utilizing the fission matrix based approach with appropriately meshing and particle density, it is shown that the fission matrix elements tend to be independent. As a result, the keff and the source eigenvector can be calculated without bias, which is not the case for the standard methodology due to the source correlation. This approach was tested with a 1-D multigroup eigenvalue code developed for this work. A preliminary automatic mesh and particle population diagnostic were formulated to ensure independent and normal fission matrix elements. The algorithm was extended in parallel to show the favorable speedup possible with the fission matrix based approach. (Full text of this dissertation may be available via the University of Florida Libraries web site. Please check http://www.uflib.ufl.edu/etd.html)
Parallel and Scalable Sparse Basic Linear Algebra Subprograms
DEFF Research Database (Denmark)
Liu, Weifeng
Sparse basic linear algebra subprograms (BLAS) are fundamental building blocks for numerous scientific computations and graph applications. Compared with Dense BLAS, parallelization of Sparse BLAS routines entails extra challenges due to the irregularity of sparse data structures. This thesis...
Low-temperature random matrix theory at the soft edge
Energy Technology Data Exchange (ETDEWEB)
Edelman, Alan [Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States); Persson, Per-Olof [Department of Mathematics, University of California, Berkeley, California 94720 (United States); Sutton, Brian D. [Department of Mathematics, Randolph-Macon College, Ashland, Virginia 23005 (United States)
2014-06-15
“Low temperature” random matrix theory is the study of random eigenvalues as energy is removed. In standard notation, β is identified with inverse temperature, and low temperatures are achieved through the limit β → ∞. In this paper, we derive statistics for low-temperature random matrices at the “soft edge,” which describes the extreme eigenvalues for many random matrix distributions. Specifically, new asymptotics are found for the expected value and standard deviation of the general-β Tracy-Widom distribution. The new techniques utilize beta ensembles, stochastic differential operators, and Riccati diffusions. The asymptotics fit known high-temperature statistics curiously well and contribute to the larger program of general-β random matrix theory.
Low-temperature random matrix theory at the soft edge
Edelman, Alan; Persson, Per-Olof; Sutton, Brian D.
2014-06-01
"Low temperature" random matrix theory is the study of random eigenvalues as energy is removed. In standard notation, β is identified with inverse temperature, and low temperatures are achieved through the limit β → ∞. In this paper, we derive statistics for low-temperature random matrices at the "soft edge," which describes the extreme eigenvalues for many random matrix distributions. Specifically, new asymptotics are found for the expected value and standard deviation of the general-β Tracy-Widom distribution. The new techniques utilize beta ensembles, stochastic differential operators, and Riccati diffusions. The asymptotics fit known high-temperature statistics curiously well and contribute to the larger program of general-β random matrix theory.
Probabilistic Recovery Guarantees for Sparsely Corrupted Signals
Pope, Graeme; Studer, Christoph
2012-01-01
We consider the recovery of sparse signals subject to sparse interference, as introduced in Studer et al., IEEE Trans. IT, 2012. We present novel probabilistic recovery guarantees for this framework, covering varying degrees of knowledge of the signal and interference support, which are relevant for a large number of practical applications. Our results assume that the sparsifying dictionaries are solely characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.
Sparse Multivariate Modeling: Priors and Applications
DEFF Research Database (Denmark)
Henao, Ricardo
to use them as hypothesis generating tools. All of our models start from a family of structures, for instance factor models, directed acyclic graphs, classifiers, etc. Then we let them be selectively sparse as a way to provide them with structural fl exibility and interpretability. Finally, we complement...... modeling, a model for peptide-protein/protein-protein interactions called latent protein tree, a framework for sparse Gaussian process classification based on active set selection and a linear multi-category sparse classifier specially targeted to gene expression data. The thesis is organized to provide...
Sparse modeling theory, algorithms, and applications
Rish, Irina
2014-01-01
""A comprehensive, clear, and well-articulated book on sparse modeling. This book will stand as a prime reference to the research community for many years to come.""-Ricardo Vilalta, Department of Computer Science, University of Houston""This book provides a modern introduction to sparse methods for machine learning and signal processing, with a comprehensive treatment of both theory and algorithms. Sparse Modeling is an ideal book for a first-year graduate course.""-Francis Bach, INRIA - École Normale Supřieure, Paris
Subspace Based Blind Sparse Channel Estimation
DEFF Research Database (Denmark)
Hayashi, Kazunori; Matsushima, Hiroki; Sakai, Hideaki
2012-01-01
The paper proposes a subspace based blind sparse channel estimation method using 1–2 optimization by replacing the 2–norm minimization in the conventional subspace based method by the 1–norm minimization problem. Numerical results confirm that the proposed method can significantly improve...... the estimation accuracy for the sparse channel, while achieving the same performance as the conventional subspace method when the channel is dense. Moreover, the proposed method enables us to estimate the channel response with unknown channel order if the channel is sparse enough....
Codesign of Beam Pattern and Sparse Frequency Waveforms for MIMO Radar
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Chaoyun Mai
2015-01-01
Full Text Available Multiple-input multiple-output (MIMO radar takes the advantages of high degrees of freedom for beam pattern design and waveform optimization, because each antenna in centralized MIMO radar system can transmit different signal waveforms. When continuous band is divided into several pieces, sparse frequency radar waveforms play an important role due to the special pattern of the sparse spectrum. In this paper, we start from the covariance matrix of the transmitted waveform and extend the concept of sparse frequency design to the study of MIMO radar beam pattern. With this idea in mind, we first solve the problem of semidefinite constraint by optimization tools and get the desired covariance matrix of the ideal beam pattern. Then, we use the acquired covariance matrix and generalize the objective function by adding the constraint of both constant modulus of the signals and corresponding spectrum. Finally, we solve the objective function by the cyclic algorithm and obtain the sparse frequency MIMO radar waveforms with desired beam pattern. The simulation results verify the effectiveness of this method.
Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations
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Huiping Cao
2014-01-01
Full Text Available Schubert’s method is an extension of Broyden’s method for solving sparse nonlinear equations, which can preserve the zero-nonzero structure defined by the sparse Jacobian matrix and can retain many good properties of Broyden’s method. In particular, Schubert’s method has been proved to be locally and q-superlinearly convergent. In this paper, we globalize Schubert’s method by using a nonmonotone line search. Under appropriate conditions, we show that the proposed algorithm converges globally and superlinearly. Some preliminary numerical experiments are presented, which demonstrate that our algorithm is effective for large-scale problems.
Speech Denoising in White Noise Based on Signal Subspace Low-rank Plus Sparse Decomposition
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yuan Shuai
2017-01-01
Full Text Available In this paper, a new subspace speech enhancement method using low-rank and sparse decomposition is presented. In the proposed method, we firstly structure the corrupted data as a Toeplitz matrix and estimate its effective rank for the underlying human speech signal. Then the low-rank and sparse decomposition is performed with the guidance of speech rank value to remove the noise. Extensive experiments have been carried out in white Gaussian noise condition, and experimental results show the proposed method performs better than conventional speech enhancement methods, in terms of yielding less residual noise and lower speech distortion.
A sparse equivalent source method for near-field acoustic holography
DEFF Research Database (Denmark)
Fernandez Grande, Efren; Xenaki, Angeliki; Gerstoft, Peter
2017-01-01
on the superposition of few waves) that are accurate when the acoustic sources are spatially localized. The importance of obtaining a non-redundant representation, i.e., a sensing matrix with low column coherence, and the inherent ill-conditioning of near-field reconstruction problems is addressed. Numerical......This study examines a near-field acoustic holography method consisting of a sparse formulation of the equivalent source method, based on the compressive sensing (CS) framework. The method, denoted Compressive–Equivalent Source Method (C-ESM), encourages spatially sparse solutions (based...
AKNS eigenvalue spectrum for densely spaced envelope solitary waves
Slunyaev, Alexey; Starobor, Alexey
2010-05-01
The problem of the influence of one envelope soliton to the discrete eigenvalues of the associated scattering problem for the other envelope soliton, which is situated close to the first one, is discussed. Envelope solitons are exact solutions of the integrable nonlinear Schrödinger equation (NLS). Their generalizations (taking into account the background nonlinear waves [1-4] or strongly nonlinear effects [5, 6]) are possible candidates to rogue waves in the ocean. The envelope solitary waves could be in principle detected in the stochastic wave field by approaches based on the Inverse Scattering Technique in terms of ‘unstable modes' (see [1-3]), or envelope solitons [7-8]. However, densely spaced intense groups influence the spectrum of the associated scattering problem, so that the solitary trains cannot be considered alone. Here we solve the initial-value problem exactly for some simplified configurations of the wave field, representing two closely placed intense wave groups, within the frameworks of the NLS equation by virtue of the solution of the AKNS system [9]. We show that the analogues of the level splitting and the tunneling effects, known in quantum physics, exist in the context of the NLS equation, and thus may be observed in application to sea waves [10]. These effects make the detecting of single solitary wave groups surrounded by other nonlinear wave groups difficult. [1]. A.L. Islas, C.M. Schober (2005) Predicting rogue waves in random oceanic sea states. Phys. Fluids 17, 031701-1-4. [2]. A.R. Osborne, M. Onorato, M. Serio (2005) Nonlinear Fourier analysis of deep-water random surface waves: Theoretical formulation and and experimental observations of rogue waves. 14th Aha Huliko's Winter Workshop, Honolulu, Hawaii. [3]. C.M. Schober, A. Calini (2008) Rogue waves in higher order nonlinear Schrödinger models. In: Extreme Waves (Eds.: E. Pelinovsky & C. Kharif), Springer. [4]. N. Akhmediev, A. Ankiewicz, M. Taki (2009) Waves that appear from
Second Workshop on Sparse Grids and Applications
Pflüger, Dirk
2014-01-01
Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the “curse” of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.
Sparse Image Reconstruction in Computed Tomography
DEFF Research Database (Denmark)
Jørgensen, Jakob Sauer
(CS), have shown significant empirical potential for this purpose. For example, total variation regularized image reconstruction has been shown in some cases to allow reducing x-ray exposure by a factor of 10 or more, while maintaining or even improving image quality compared to conventional...... and limitations of sparse reconstruction methods in CT, in particular in a quantitative sense. For example, relations between image properties such as contrast, structure and sparsity, tolerable noise levels, suficient sampling levels, the choice of sparse reconstruction formulation and the achievable image...... applications. This thesis takes a systematic approach toward establishing quantitative understanding of conditions for sparse reconstruction to work well in CT. A general framework for analyzing sparse reconstruction methods in CT is introduced and two sets of computational tools are proposed: 1...
Beam Combination for Sparse Aperture Telescopes Project
National Aeronautics and Space Administration — The Stellar Imager, an ultraviolet, sparse-aperture telescope, was one of the fifteen Vision Missions chosen for a study completed last year. Stellar Imager will...
Sparse representations for radar with Matlab examples
Knee, Peter
2012-01-01
Although the field of sparse representations is relatively new, research activities in academic and industrial research labs are already producing encouraging results. The sparse signal or parameter model motivated several researchers and practitioners to explore high complexity/wide bandwidth applications such as Digital TV, MRI processing, and certain defense applications. The potential signal processing advancements in this area may influence radar technologies. This book presents the basic mathematical concepts along with a number of useful MATLAB® examples to emphasize the practical imple
Gui, Tao; Lu, Chao; Lau, Alan Pak Tao; Wai, P K A
2017-08-21
In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.
Luo, FuSheng; Lin, Qun; Xie, HeHu
2012-05-01
This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.
Visual tracking via robust multitask sparse prototypes
Zhang, Huanlong; Hu, Shiqiang; Yu, Junyang
2015-03-01
Sparse representation has been applied to an online subspace learning-based tracking problem. To handle partial occlusion effectively, some researchers introduce l1 regularization to principal component analysis (PCA) reconstruction. However, in these traditional tracking methods, the representation of each object observation is often viewed as an individual task so the inter-relationship between PCA basis vectors is ignored. We propose a new online visual tracking algorithm with multitask sparse prototypes, which combines multitask sparse learning with PCA-based subspace representation. We first extend a visual tracking algorithm with sparse prototypes in multitask learning framework to mine inter-relations between subtasks. Then, to avoid the problem that enforcing all subtasks to share the same structure may result in degraded tracking results, we impose group sparse constraints on the coefficients of PCA basis vectors and element-wise sparse constraints on the error coefficients, respectively. Finally, we show that the proposed optimization problem can be effectively solved using the accelerated proximal gradient method with the fast convergence. Experimental results compared with the state-of-the-art tracking methods demonstrate that the proposed algorithm achieves favorable performance when the object undergoes partial occlusion, motion blur, and illumination changes.
Efficient, sparse biological network determination
Directory of Open Access Journals (Sweden)
Papachristodoulou Antonis
2009-02-01
Full Text Available Abstract Background Determining the interaction topology of biological systems is a topic that currently attracts significant research interest. Typical models for such systems take the form of differential equations that involve polynomial and rational functions. Such nonlinear models make the problem of determining the connectivity of biochemical networks from time-series experimental data much harder. The use of linear dynamics and linearization techniques that have been proposed in the past can circumvent this, but the general problem of developing efficient algorithms for models that provide more accurate system descriptions remains open. Results We present a network determination algorithm that can treat model descriptions with polynomial and rational functions and which does not make use of linearization. For this purpose, we make use of the observation that biochemical networks are in general 'sparse' and minimize the 1-norm of the decision variables (sum of weighted network connections while constraints keep the error between data and the network dynamics small. The emphasis of our methodology is on determining the interconnection topology rather than the specific reaction constants and it takes into account the necessary properties that a chemical reaction network should have – something that techniques based on linearization can not. The problem can be formulated as a Linear Program, a convex optimization problem, for which efficient algorithms are available that can treat large data sets efficiently and uncertainties in data or model parameters. Conclusion The presented methodology is able to predict with accuracy and efficiency the connectivity structure of a chemical reaction network with mass action kinetics and of a gene regulatory network from simulation data even if the dynamics of these systems are non-polynomial (rational and uncertainties in the data are taken into account. It also produces a network structure that can
Efficient, sparse biological network determination.
August, Elias; Papachristodoulou, Antonis
2009-02-23
Determining the interaction topology of biological systems is a topic that currently attracts significant research interest. Typical models for such systems take the form of differential equations that involve polynomial and rational functions. Such nonlinear models make the problem of determining the connectivity of biochemical networks from time-series experimental data much harder. The use of linear dynamics and linearization techniques that have been proposed in the past can circumvent this, but the general problem of developing efficient algorithms for models that provide more accurate system descriptions remains open. We present a network determination algorithm that can treat model descriptions with polynomial and rational functions and which does not make use of linearization. For this purpose, we make use of the observation that biochemical networks are in general 'sparse' and minimize the 1-norm of the decision variables (sum of weighted network connections) while constraints keep the error between data and the network dynamics small. The emphasis of our methodology is on determining the interconnection topology rather than the specific reaction constants and it takes into account the necessary properties that a chemical reaction network should have - something that techniques based on linearization can not. The problem can be formulated as a Linear Program, a convex optimization problem, for which efficient algorithms are available that can treat large data sets efficiently and uncertainties in data or model parameters. The presented methodology is able to predict with accuracy and efficiency the connectivity structure of a chemical reaction network with mass action kinetics and of a gene regulatory network from simulation data even if the dynamics of these systems are non-polynomial (rational) and uncertainties in the data are taken into account. It also produces a network structure that can explain the real experimental data of L. lactis and is similar
Wu, Xiaohu; Jin, Can; Fu, Ceji
2017-11-01
We conduct a comprehensive analysis of the eigenvalues for propagation of electromagnetic (EM) waves in anisotropic materials. The analysis can be employed to judge the characteristics of the eigenvalues for EM waves in anisotropic materials. Taking the advantage of the non-opposite eigenvalues, we show that unidirectional transmission and unidirectional invisibility can be realized with a slab of natural anisotropic material. Our results in this work may provide important guidelines for design of novel devices of unidirectional transmission or unidirectional invisibility.
On affine non-negative matrix factorization
DEFF Research Database (Denmark)
Laurberg, Hans; Hansen, Lars Kai
2007-01-01
We generalize the non-negative matrix factorization (NMF) generative model to incorporate an explicit offset. Multiplicative estimation algorithms are provided for the resulting sparse affine NMF model. We show that the affine model has improved uniqueness properties and leads to more accurate id...
Dependence of Eigenvalues of a Class of Higher-Order Sturm-Liouville Problems on the Boundary
Directory of Open Access Journals (Sweden)
Qiuxia Yang
2015-01-01
Full Text Available We show that the eigenvalues of a class of higher-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all 2kth-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest eigenvalue.
Eigenvalue-based harmonic stability analysis method in inverter-fed power systems
DEFF Research Database (Denmark)
Wang, Yanbo; Wang, Xiongfei; Blaabjerg, Frede
2015-01-01
This paper presents an eigenvalue-based harmonic stability analysis method for inverter-fed power systems. A full-order small-signal model for a droop-controlled Distributed Generation (DG) inverter is built first, including the time delay of digital control system, inner current and voltage...... resonance and instability in the power system. Eigenvalues associated with time delay of inverter and inner controller parameters is obtained, which shows the time delay has an important effect on harmonic instability of inverter-fed power systems. Simulation results are given for validating the proposed...... control loops, and outer droop-based power control loop. Based on the inverter model, an overall small-signal model of a two-inverter-fed system is then established, and the eigenvalue-based stability analysis is subsequently performed to assess the influence of controller parameters on the harmonic...
New expansions of numerical eigenvalues for -Delta uDλ rho u by nonconforming elements
Lin, Qun; Huang, Hung-Tsai; Li, Zi-Cai
2008-12-01
The paper explores new expansions of the eigenvalues for -Delta uDλ rho u in S with Dirichlet boundary conditions by the bilinear element (denoted Q_1 ) and three nonconforming elements, the rotated bilinear element (denoted Q_1^{rot} ), the extension of Q_1^{rot} (denoted EQ_1^{rot} ) and Wilson's elements. The expansions indicate that Q_1 and Q_1^{rot} provide upper bounds of the eigenvalues, and that EQ_1^{rot} and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the O(h^4) convergence rate can be obtained, where h is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.
Al NMR: a novel NMR data processing program optimized for sparse sampling
Energy Technology Data Exchange (ETDEWEB)
Gledhill, John M.; Wand, A. Joshua, E-mail: wand@mail.med.upenn.edu [University of Pennsylvania, Graduate Group in Biochemistry and Molecular Biophysics, Perelman School of Medicine (United States)
2012-01-15
Sparse sampling in biomolecular multidimensional NMR offers increased acquisition speed and resolution and, if appropriate conditions are met, an increase in sensitivity. Sparse sampling of indirectly detected time domains combined with the direct truly multidimensional Fourier transform has elicited particular attention because of the ability to generate a final spectrum amenable to traditional analysis techniques. A number of sparse sampling schemes have been described including radial sampling, random sampling, concentric sampling and variations thereof. A fundamental feature of these sampling schemes is that the resulting time domain data array is not amenable to traditional Fourier transform based processing and phasing correction techniques. In addition, radial sampling approaches offer a number of advantages and capabilities that are also not accessible using standard NMR processing techniques. These include sensitivity enhancement, sub-matrix processing and determination of minimal sets of sampling angles. Here we describe a new software package (Al NMR) that enables these capabilities in the context of a general NMR data processing environment.
Franklin, Joel N
2003-01-01
Mathematically rigorous introduction covers vector and matrix norms, the condition-number of a matrix, positive and irreducible matrices, much more. Only elementary algebra and calculus required. Includes problem-solving exercises. 1968 edition.
Eigenvalue solution for the convected wave equation in a circular soft wall duct
Alonso, Jose S.; Burdisso, Ricardo A.
2008-09-01
A numerical approach to find the eigenvalues of the wave equation applied in a circular duct with convective flow and soft wall boundary conditions is proposed. The characteristic equation is solved in the frequency domain as a function of a locally reacting acoustic impedance and the flow Mach number. In addition, the presence of the convective flow couples the solution of the eigenvalues with the axial propagation constants. The unknown eigenvalues are also related to these propagation constants by a quadratic expression that leads to two solutions. These two solutions replaced into the characteristic equation generate two separate eigenvalue problems depending on the direction of propagation. Given that the resulting nonlinear complex-valued equations do not provide the solution explicitly, a numerical technique must be used. The proposed approach is based on the minimization of the absolute value of the characteristic equation by the Nelder-Mead simplex method. The main advantage of this method is that it only uses function evaluations, rather than derivatives, and geometric reasoning. The minimization is performed starting from very low frequencies and increasing by small steps to the particular frequency of interest. The initial guess for the first frequency of calculation is provided as the hard wall eigenvalue solution. Then, the solution from the previous step is used as the initial value for the next calculation. This approach was specifically developed for applications with resonator-type liners commonly used in the commercial aviation industry, where the low-frequency behavior resembles that of a hard wall and agrees with the first initial guess for the first frequency of calculation. The numerical technique was found to be very robust in terms of convergence and stability. Also, the method provides a physical meaning for each eigenvalue since the variation as a function of frequency can be clearly followed with respect to the values that are originally
Google matrix and Ulam networks of intermittency maps.
Ermann, L; Shepelyansky, D L
2010-03-01
We study the properties of the Google matrix of an Ulam network generated by intermittency maps. This network is created by the Ulam method which gives a matrix approximant for the Perron-Frobenius operator of dynamical map. The spectral properties of eigenvalues and eigenvectors of this matrix are analyzed. We show that the PageRank of the system is characterized by a power law decay with the exponent beta dependent on map parameters and the Google damping factor alpha . Under certain conditions the PageRank is completely delocalized so that the Google search in such a situation becomes inefficient.
Application of Random Matrix Theory to Complex Networks
Rai, Aparna; Jalan, Sarika
The present article provides an overview of recent developments in spectral analysis of complex networks under random matrix theory framework. Adjacency matrix of unweighted networks, reviewed here, differ drastically from a random matrix, as former have only binary entries. Remarkably, short range correlations in corresponding eigenvalues of such matrices exhibit Gaussian orthogonal statistics of RMT and thus bring them into the universality class. Spectral rigidity of spectra provides measure of randomness in underlying networks. We will consider several examples of model networks vastly studied in last two decades. To the end we would provide potential of RMT framework and obtained results to understand and predict behavior of complex systems with underlying network structure.
Bifurcations of Eigenvalues of Gyroscopic Systems with Parameters Near Stability Boundaries
DEFF Research Database (Denmark)
Seyranian, Alexander P.; Kliem, Wolfhard
1999-01-01
The paper deals with stability problems of linear gyroscopic systems with finite or infinite degrees of freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulae for the behavior of eigenvalues under a change of parameters are obtained. It is sh......The paper deals with stability problems of linear gyroscopic systems with finite or infinite degrees of freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulae for the behavior of eigenvalues under a change of parameters are obtained...
On the Solution of the Eigenvalue Assignment Problem for Discrete-Time Systems
Directory of Open Access Journals (Sweden)
El-Sayed M. E. Mostafa
2017-01-01
Full Text Available The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples.
Directory of Open Access Journals (Sweden)
Winfried Auzinger
2006-01-01
Full Text Available We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.
Real eigenvalue analysis in NASTRAN by the tridiagonal reduction (FEER) method
Newman, M.; Flanagen, P. F.; Rogers, J. L., Jr.
1976-01-01
Implementation of the tridiagonal reduction method for real eigenvalue extraction in structural vibration and buckling problems is described. The basic concepts underlying the method are summarized and special features, such as the computation of error bounds and default modes of operation are discussed. In addition, the new user information and error messages and optional diagnostic output relating to the tridiagonal reduction method are presented. Some numerical results and initial experiences relating to usage in the NASTRAN environment are provided, including comparisons with other existing NASTRAN eigenvalue methods.
AMP-Inspired Deep Networks for Sparse Linear Inverse Problems
Borgerding, Mark; Schniter, Philip; Rangan, Sundeep
2017-08-01
Deep learning has gained great popularity due to its widespread success on many inference problems. We consider the application of deep learning to the sparse linear inverse problem, where one seeks to recover a sparse signal from a few noisy linear measurements. In this paper, we propose two novel neural-network architectures that decouple prediction errors across layers in the same way that the approximate message passing (AMP) algorithms decouple them across iterations: through Onsager correction. First, we propose a "learned AMP" network that significantly improves upon Gregor and LeCun's "learned ISTA." Second, inspired by the recently proposed "vector AMP" (VAMP) algorithm, we propose a "learned VAMP" network that offers increased robustness to deviations in the measurement matrix from i.i.d. Gaussian. In both cases, we jointly learn the linear transforms and scalar nonlinearities of the network. Interestingly, with i.i.d. signals, the linear transforms and scalar nonlinearities prescribed by the VAMP algorithm coincide with the values learned through back-propagation, leading to an intuitive interpretation of learned VAMP. Finally, we apply our methods to two problems from 5G wireless communications: compressive random access and massive-MIMO channel estimation.
Low-Complexity Bayesian Estimation of Cluster-Sparse Channels
Ballal, Tarig
2015-09-18
This paper addresses the problem of channel impulse response estimation for cluster-sparse channels under the Bayesian estimation framework. We develop a novel low-complexity minimum mean squared error (MMSE) estimator by exploiting the sparsity of the received signal profile and the structure of the measurement matrix. It is shown that due to the banded Toeplitz/circulant structure of the measurement matrix, a channel impulse response, such as underwater acoustic channel impulse responses, can be partitioned into a number of orthogonal or approximately orthogonal clusters. The orthogonal clusters, the sparsity of the channel impulse response and the structure of the measurement matrix, all combined, result in a computationally superior realization of the MMSE channel estimator. The MMSE estimator calculations boil down to simpler in-cluster calculations that can be reused in different clusters. The reduction in computational complexity allows for a more accurate implementation of the MMSE estimator. The proposed approach is tested using synthetic Gaussian channels, as well as simulated underwater acoustic channels. Symbol-error-rate performance and computation time confirm the superiority of the proposed method compared to selected benchmark methods in systems with preamble-based training signals transmitted over clustersparse channels.
Highly sparse representations from dictionaries are unique and independent of the sparseness measure
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
2007-01-01
The purpose of this paper is to study sparse representations of signals from a general dictionary in a Banach space. For so-called localized frames in Hilbert spaces, the canonical frame coefficients are shown to provide a near sparsest expansion for several sparseness measures. However, for frames...
When sparse coding meets ranking: a joint framework for learning sparse codes and ranking scores
Wang, Jim Jing-Yan
2017-06-28
Sparse coding, which represents a data point as a sparse reconstruction code with regard to a dictionary, has been a popular data representation method. Meanwhile, in database retrieval problems, learning the ranking scores from data points plays an important role. Up to now, these two problems have always been considered separately, assuming that data coding and ranking are two independent and irrelevant problems. However, is there any internal relationship between sparse coding and ranking score learning? If yes, how to explore and make use of this internal relationship? In this paper, we try to answer these questions by developing the first joint sparse coding and ranking score learning algorithm. To explore the local distribution in the sparse code space, and also to bridge coding and ranking problems, we assume that in the neighborhood of each data point, the ranking scores can be approximated from the corresponding sparse codes by a local linear function. By considering the local approximation error of ranking scores, the reconstruction error and sparsity of sparse coding, and the query information provided by the user, we construct a unified objective function for learning of sparse codes, the dictionary and ranking scores. We further develop an iterative algorithm to solve this optimization problem.
Rank Awareness in Group-Sparse Recovery of Multi-Echo MR Images
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Rabab Ward
2013-03-01
Full Text Available This work addresses the problem of recovering multi-echo T1 or T2 weighted images from their partial K-space scans. Recent studies have shown that the best results are obtained when all the multi-echo images are reconstructed by simultaneously exploiting their intra-image spatial redundancy and inter-echo correlation. The aforesaid studies either stack the vectorised images (formed by row or columns concatenation as columns of a Multiple Measurement Vector (MMV matrix or concatenate them as a long vector. Owing to the inter-image correlation, the thus formed MMV matrix or the long concatenated vector is row-sparse or group-sparse respectively in a transform domain (wavelets. Consequently the reconstruction problem was formulated as a row-sparse MMV recovery or a group-sparse vector recovery. In this work we show that when the multi-echo images are arranged in the MMV form, the thus formed matrix is low-rank. We show that better reconstruction accuracy can be obtained when the information about rank-deficiency is incorporated into the row/group sparse recovery problem. Mathematically, this leads to a constrained optimization problem where the objective function promotes the signal’s groups-sparsity as well as its rank-deficiency; the objective function is minimized subject to data fidelity constraints. The experiments were carried out on ex vivo and in vivo T2 weighted images of a rat's spinal cord. Results show that this method yields considerably superior results than state-of-the-art reconstruction techniques.
Sparse Text Indexing in Small Space
DEFF Research Database (Denmark)
Bille, Philip; Fischer, Johannes; Gørtz, Inge Li
2016-01-01
In this work, we present efficient algorithms for constructing sparse suffix trees, sparse suffix arrays, and sparse position heaps for b arbitrary positions of a text T of length n while using only O(b) words of space during the construction. Attempts at breaking the naïve bound of Ω(nb) time...... for constructing sparse suffix trees in O(b) space can be traced back to the origins of string indexing in 1968. First results were not obtained until 1996, but only for the case in which the b suffixes were evenly spaced in T. In this article, there is no constraint on the locations of the suffixes. Our main...... contribution is to show that the sparse suffix tree (and array) can be constructed in O(nlog 2b) time. To achieve this, we develop a technique that allows one to efficiently answer b longest common prefix queries on suffixes of T, using only O(b) space. We expect that this technique will prove useful in many...
Sparse Learning with Stochastic Composite Optimization.
Zhang, Weizhong; Zhang, Lijun; Jin, Zhongming; Jin, Rong; Cai, Deng; Li, Xuelong; Liang, Ronghua; He, Xiaofei
2017-06-01
In this paper, we study Stochastic Composite Optimization (SCO) for sparse learning that aims to learn a sparse solution from a composite function. Most of the recent SCO algorithms have already reached the optimal expected convergence rate O(1/λT), but they often fail to deliver sparse solutions at the end either due to the limited sparsity regularization during stochastic optimization (SO) or due to the limitation in online-to-batch conversion. Even when the objective function is strongly convex, their high probability bounds can only attain O(√{log(1/δ)/T}) with δ is the failure probability, which is much worse than the expected convergence rate. To address these limitations, we propose a simple yet effective two-phase Stochastic Composite Optimization scheme by adding a novel powerful sparse online-to-batch conversion to the general Stochastic Optimization algorithms. We further develop three concrete algorithms, OptimalSL, LastSL and AverageSL, directly under our scheme to prove the effectiveness of the proposed scheme. Both the theoretical analysis and the experiment results show that our methods can really outperform the existing methods at the ability of sparse learning and at the meantime we can improve the high probability bound to approximately O(log(log(T)/δ)/λT).
Sparse Bayesian learning for DOA estimation with mutual coupling.
Dai, Jisheng; Hu, Nan; Xu, Weichao; Chang, Chunqi
2015-10-16
Sparse Bayesian learning (SBL) has given renewed interest to the problem of direction-of-arrival (DOA) estimation. It is generally assumed that the measurement matrix in SBL is precisely known. Unfortunately, this assumption may be invalid in practice due to the imperfect manifold caused by unknown or misspecified mutual coupling. This paper describes a modified SBL method for joint estimation of DOAs and mutual coupling coefficients with uniform linear arrays (ULAs). Unlike the existing method that only uses stationary priors, our new approach utilizes a hierarchical form of the Student t prior to enforce the sparsity of the unknown signal more heavily. We also provide a distinct Bayesian inference for the expectation-maximization (EM) algorithm, which can update the mutual coupling coefficients more efficiently. Another difference is that our method uses an additional singular value decomposition (SVD) to reduce the computational complexity of the signal reconstruction process and the sensitivity to the measurement noise.
Sparse Bayesian Learning for DOA Estimation with Mutual Coupling
Directory of Open Access Journals (Sweden)
Jisheng Dai
2015-10-01
Full Text Available Sparse Bayesian learning (SBL has given renewed interest to the problem of direction-of-arrival (DOA estimation. It is generally assumed that the measurement matrix in SBL is precisely known. Unfortunately, this assumption may be invalid in practice due to the imperfect manifold caused by unknown or misspecified mutual coupling. This paper describes a modified SBL method for joint estimation of DOAs and mutual coupling coefficients with uniform linear arrays (ULAs. Unlike the existing method that only uses stationary priors, our new approach utilizes a hierarchical form of the Student t prior to enforce the sparsity of the unknown signal more heavily. We also provide a distinct Bayesian inference for the expectation-maximization (EM algorithm, which can update the mutual coupling coefficients more efficiently. Another difference is that our method uses an additional singular value decomposition (SVD to reduce the computational complexity of the signal reconstruction process and the sensitivity to the measurement noise.
Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems
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Eric Bavier
2012-01-01
Full Text Available Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.
A tight and explicit representation of Q in sparse QR factorization
Energy Technology Data Exchange (ETDEWEB)
Ng, E.G.; Peyton, B.W.
1992-05-01
In QR factorization of a sparse m{times}n matrix A (m {ge} n) the orthogonal factor Q is often stored implicitly as a lower trapezoidal matrix H known as the Householder matrix. This paper presents a simple characterization of the row structure of Q, which could be used as the basis for a sparse data structure that can store Q explicitly. The new characterization is a simple extension of a well known row-oriented characterization of the structure of H. Hare, Johnson, Olesky, and van den Driessche have recently provided a complete sparsity analysis of the QR factorization. Let U be the matrix consisting of the first n columns of Q. Using results from, we show that the data structures for H and U resulting from our characterizations are tight when A is a strong Hall matrix. We also show that H and the lower trapezoidal part of U have the same sparsity characterization when A is strong Hall. We then show that this characterization can be extended to any weak Hall matrix that has been permuted into block upper triangular form. Finally, we show that permuting to block triangular form never increases the fill incurred during the factorization.
Effective Lagrangians and chiral random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Halasz, M.A.; Verbaarschot, J.J.M. [Department of Physics, State University of New York, Stony Brook, New York 11794 (United States)
1995-08-15
Recently, sum rules were derived for the inverse eigenvalues of the Dirac operator. They were obtained in two different ways: (i) starting from the low-energy effective Lagrangian and (ii) starting from a random matrix theory with the symmetries of the Dirac operator. This suggests that the effective theory can be obtained directly from the random matrix theory. Previously, this was shown for three or more colors with fundamental fermions. In this paper we construct the effective theory from a random matrix theory for two colors in the fundamental representation and for an arbitrary number of colors in the adjoint representation. We construct a fermionic partition function for Majorana fermions in Euclidean spacetime. Their reality condition is formulated in terms of complex conjugation of the second kind.
SPARSE ELECTROMAGNETIC IMAGING USING NONLINEAR LANDWEBER ITERATIONS
Desmal, Abdulla
2015-07-29
A scheme for efficiently solving the nonlinear electromagnetic inverse scattering problem on sparse investigation domains is described. The proposed scheme reconstructs the (complex) dielectric permittivity of an investigation domain from fields measured away from the domain itself. Least-squares data misfit between the computed scattered fields, which are expressed as a nonlinear function of the permittivity, and the measured fields is constrained by the L0/L1-norm of the solution. The resulting minimization problem is solved using nonlinear Landweber iterations, where at each iteration a thresholding function is applied to enforce the sparseness-promoting L0/L1-norm constraint. The thresholded nonlinear Landweber iterations are applied to several two-dimensional problems, where the ``measured\\'\\' fields are synthetically generated or obtained from actual experiments. These numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed scheme in reconstructing sparse profiles with high permittivity values.
A Fast Gradient Method for Nonnegative Sparse Regression With Self-Dictionary
Gillis, Nicolas; Luce, Robert
2018-01-01
A nonnegative matrix factorization (NMF) can be computed efficiently under the separability assumption, which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these conic basis columns are based on nonnegative sparse regression and self dictionaries, and require the solution of large-scale convex optimization problems. In this paper we study a particular nonnegative sparse regression model with self dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and real-world hyperspectral images.
A unified approach to sparse signal processing
Marvasti, Farokh; Amini, Arash; Haddadi, Farzan; Soltanolkotabi, Mahdi; Khalaj, Babak Hossein; Aldroubi, Akram; Sanei, Saeid; Chambers, Janathon
2012-12-01
A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing, and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The error locator polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method under noisy environments. The iterative methods developed for sampling and coding applications are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. Sparsity in unobservable source signals is also shown to facilitate source separation in sparse component analysis; the algorithms developed in this area such as linear programming and matching pursuit are also widely used in compressed sensing. Finally
Fundamentals of matrix analysis with applications
Saff, Edward Barry
2015-01-01
This book provides comprehensive coverage of matrix theory from a geometric and physical perspective, and the authors address the functionality of matrices and their ability to illustrate and aid in many practical applications. Readers are introduced to inverses and eigenvalues through physical examples such as rotations, reflections, and projections, and only then are computational details described and explored. MATLAB is utilized to aid in reader comprehension, and the authors are careful to address the issue of rank fragility so readers are not flummoxed when MATLAB displays conflict wit
The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces
Ćurgus, Branko; Dijksma, Aad; Read, Tom
2001-01-01
The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert
Yadav, Jayprakash; Rafique, S. M.; Kumari, Shanti
2009-10-01
In the present paper some superconducting (SC) state parameters of metals Ga, Cd and In have been studied through Harrison's First Principle [HFP] pseudopotential technique using McMillan's formalism. The impact of choosing two different sets of core energy eigenvalues viz. Herman-Skillman and Clementi (or Experimental) has been studied.
On the asymptotic of an eigenvalue problem with 2n2n2n interior ...
Indian Academy of Sciences (India)
On the asymptotic of an eigenvalue problem with 2n2n2n interior singularities. A NEAMATY and S HAGHAIEGHY. Department of Mathematics, Faculty of Basic Sciences, Mazandaran University,. Babolsar, Iran. E-mail: namaty@umz.ac.ir; namtyumcc@yahoo.com. MS received 6 January 2008; revised 28 July 2009. Abstract ...
On the extrema of Dirichlet's first eigenvalue of a family of punctured ...
Indian Academy of Sciences (India)
Home; Journals; Proceedings – Mathematical Sciences; Volume 122; Issue 2. On the Extrema of Dirichlet's First Eigenvalue of a Family of Punctured Regular Polygons in Two Dimensional Space Forms. A R Aithal Rajesh Raut. Volume 122 Issue 2 May 2012 pp 257-281 ...
Directory of Open Access Journals (Sweden)
Hai Bi
2012-01-01
Full Text Available This paper discusses efficient numerical methods for the Steklov eigenvalue problem and establishes a new multiscale discretization scheme and an adaptive algorithm based on the Rayleigh quotient iterative method. The efficiency of these schemes is analyzed theoretically, and the constants appeared in the error estimates are also analyzed elaborately. Finally, numerical experiments are provided to support the theory.
Emphasizing Visualization and Physical Applications in the Study of Eigenvectors and Eigenvalues
BeltrÁn-Meneu, María José; Murillo-Arcila, Marina; Albarracín, Lluís
2017-01-01
This article presents a teaching proposal that emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. More concretely, these concepts were introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was designed after observing architecture students…
On the asymptotic of an eigenvalue problem with 2n2n2n interior ...
Indian Academy of Sciences (India)
where satisfies Dirichlet boundary conditions and is a real-valued function which has even number of singularities at c 1 , … , c 2 n ∈ ( a , b ) . We will study the asymptotic eigenvalue near the singularity points. Author Affiliations. A Neamaty1 S Haghaieghy1. Department of Mathematics, Faculty of Basic Sciences, ...
M. Nool (Margreet); A. van der Ploeg (Auke)
1997-01-01
textabstractWe study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form $A x = lambda B x$, in which the complex matrices $A$ and $B$ are block tridiagonal, and $B$ is Hermitian positive
Eigenvalues for Iterative Systems of (n,p-Type Fractional Order Boundary Value Problems
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K. R. Prasad
2014-04-01
Full Text Available In this paper, we determine the eigenvalue intervals of λ1, λ2, ..., λn for which the iterative system of (n,p-type fractional order two-point boundary value problem has a positive solution by an application of Guo-Krasnosel’skii fixed point theorem on a cone.
Bonito, Andrea
2011-01-01
We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H-α with α ∈ (1/2, 1). The method is shown to be convergent and spectrally correct. © 2011 American Mathematical Society.
A multilevel in space and energy solver for multigroup diffusion eigenvalue problems
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Ben C. Yee
2017-09-01
Full Text Available In this paper, we present a new multilevel in space and energy diffusion (MSED method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1 a grey (one-group diffusion equation used to efficiently converge the fission source and eigenvalue, (2 a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3 a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.
Gould, S H
1966-01-01
The first edition of this book gave a systematic exposition of the Weinstein method of calculating lower bounds of eigenvalues by means of intermediate problems. This second edition presents new developments in the framework of the material contained in the first edition, which is retained in somewhat modified form.
Directory of Open Access Journals (Sweden)
Boubakari Ibrahimou
2013-01-01
maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.
Expansion by eigenvectors in case of simple eigenvalues of singular differential operator
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O. V. Makhnei
2011-06-01
Full Text Available The asymptotic formulas with large values of parameter for solutions of singular differential equation allow us to estimate Green's function of the boundary-value problem. With the help of this estimation the expansion of singular dierential operator by eigenvectors in the case of simple eigenvalues is constructed.
The Second Eigenvalue of the p-Laplacian as p Goes to 1
Directory of Open Access Journals (Sweden)
Enea Parini
2010-01-01
Full Text Available The asymptotic behaviour of the second eigenvalue of the p-Laplacian operator as p goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if p is close enough to 1.
Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods
Alonso, Ana; Dello Russo, Anahí
2009-01-01
This paper deals with the nonconforming spectral approximation of variationally posed eigenvalue problems. It is an extension to more general situations of known previous results about nonconforming methods. As an application of the present theory, convergence and optimal order error estimates are proved for the lowest order Crouzeix-Raviart approximation of the eigenpairs of two representative second-order elliptical operators.
The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile ...
Indian Academy of Sciences (India)
In this paper we present a unified and simplified approach to the universal eigenvalue inequalities of Payne–Pólya–Weinberger, Hile–Protter, and Yang. We then generalize these results to inhomogeneous membranes and Schrödinger's equation with a nonnegative potential. We also show that Yang's inequality is always ...
Lq-perturbations of leading coefficients of elliptic operators: Asymptotics of eigenvalues
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Vladimir Kozlov
2006-01-01
Full Text Available We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L∞-norm.
Wunsche, A.
1993-01-01
The eigenvalue problem of the operator a + zeta(boson creation operator) is solved for arbitrarily complex zeta by applying a nonunitary operator to the vacuum state. This nonunitary approach is compared with the unitary approach leading for the absolute value of zeta less than 1 to squeezed coherent states.
Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem
Raman, Aaswath
2010-02-26
We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.
Bounds and extremal domains for Robin eigenvalues with negative boundary parameter
Czech Academy of Sciences Publication Activity Database
Antunes, P. R. S.; Freitas, P.; Krejčiřík, David
2017-01-01
Roč. 10, č. 4 (2017), s. 357-379 ISSN 1864-8258 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : Eigenvalue optimisation * Robin Laplacian * negative boundary parameter * Bareket's conjecture Subject RIV: BA - General Mathematics Impact factor: 1.182, year: 2016
Binary Sparse Phase Retrieval via Simulated Annealing
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Wei Peng
2016-01-01
Full Text Available This paper presents the Simulated Annealing Sparse PhAse Recovery (SASPAR algorithm for reconstructing sparse binary signals from their phaseless magnitudes of the Fourier transform. The greedy strategy version is also proposed for a comparison, which is a parameter-free algorithm. Sufficient numeric simulations indicate that our method is quite effective and suggest the binary model is robust. The SASPAR algorithm seems competitive to the existing methods for its efficiency and high recovery rate even with fewer Fourier measurements.
Supersymmetry in random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Kieburg, Mario
2010-05-04
I study the applications of supersymmetry in random matrix theory. I generalize the supersymmetry method and develop three new approaches to calculate eigenvalue correlation functions. These correlation functions are averages over ratios of characteristic polynomials. In the first part of this thesis, I derive a relation between integrals over anti-commuting variables (Grassmann variables) and differential operators with respect to commuting variables. With this relation I rederive Cauchy- like integral theorems. As a new application I trace the supermatrix Bessel function back to a product of two ordinary matrix Bessel functions. In the second part, I apply the generalized Hubbard-Stratonovich transformation to arbitrary rotation invariant ensembles of real symmetric and Hermitian self-dual matrices. This extends the approach for unitarily rotation invariant matrix ensembles. For the k-point correlation functions I derive supersymmetric integral expressions in a unifying way. I prove the equivalence between the generalized Hubbard-Stratonovich transformation and the superbosonization formula. Moreover, I develop an alternative mapping from ordinary space to superspace. After comparing the results of this approach with the other two supersymmetry methods, I obtain explicit functional expressions for the probability densities in superspace. If the probability density of the matrix ensemble factorizes, then the generating functions exhibit determinantal and Pfaffian structures. For some matrix ensembles this was already shown with help of other approaches. I show that these structures appear by a purely algebraic manipulation. In this new approach I use structures naturally appearing in superspace. I derive determinantal and Pfaffian structures for three types of integrals without actually mapping onto superspace. These three types of integrals are quite general and, thus, they are applicable to a broad class of matrix ensembles. (orig.)
Chiral random matrix theory and effective theories of QCD
Energy Technology Data Exchange (ETDEWEB)
Takahashi, K.; Iida, S
2000-05-08
The correlations of the QCD Dirac eigenvalues are studied with use of an extended chiral random matrix model. The inclusion of spatial dependence which the original model lacks enables us to investigate the effects of diffusion modes. We get analytical expressions of level correlation functions with non-universal behavior caused by diffusion modes which is characterized by Thouless energy. Pion mode is shown to be responsible for these diffusion effects when QCD vacuum is considered a disordered medium.
On spectral properties of periodic polyharmonic matrix operators
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
in L2(Rn) are not degenerated for almost all values of quasimomentum. In the matrix case, when v0s = v0q for at least one pair of q,s,q = s, the scalar scheme does not work, for the reason that the Bloch eigenvalues of the 'unperturbed' operator H0 = (− )l + V0 are degenerated for all values of quasimomentum. Here we will ...
Nobi, Ashadun; Maeng, Seong Eun; Ha, Gyeong Gyun; Lee, Jae Woo
2013-02-01
We analyzed cross-correlations between price fluctuations of global financial indices (20 daily stock indices over the world) and local indices (daily indices of 200 companies in the Korean stock market) by using random matrix theory (RMT). We compared eigenvalues and components of the largest and the second largest eigenvectors of the cross-correlation matrix before, during, and after the global financial the crisis in the year 2008. We find that the majority of its eigenvalues fall within the RMT bounds [ λ -, λ +], where λ - and λ + are the lower and the upper bounds of the eigenvalues of random correlation matrices. The components of the eigenvectors for the largest positive eigenvalues indicate the identical financial market mode dominating the global and local indices. On the other hand, the components of the eigenvector corresponding to the second largest eigenvalue are positive and negative values alternatively. The components before the crisis change sign during the crisis, and those during the crisis change sign after the crisis. The largest inverse participation ratio (IPR) corresponding to the smallest eigenvector is higher after the crisis than during any other periods in the global and local indices. During the global financial the crisis, the correlations among the global indices and among the local stock indices are perturbed significantly. However, the correlations between indices quickly recover the trends before the crisis.
Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States
Ambainis, Andris; Harrow, Aram W.; Hastings, Matthew B.
2012-02-01
We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in {(mathbb {C}^d)^{⊗ k}}, where k and p/ d k are fixed while d → ∞. When k = 1, the Marčenko-Pastur law determines (up to small corrections) not only the largest eigenvalue ({(1+sqrt{p/d^k})^2}) but the smallest eigenvalue {(min(0,1-sqrt{p/d^k})^2)} and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately {(1+sqrt{p/d^k})^2} and the spectral density approaches that of the Marčenko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter. Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.
Directory of Open Access Journals (Sweden)
HongZhong Tang
2016-01-01
Full Text Available Optimizing the mutual coherence of a learned dictionary plays an important role in sparse representation and compressed sensing. In this paper, a efficient framework is developed to learn an incoherent dictionary for sparse representation. In particular, the coherence of a previous dictionary (or Gram matrix is reduced sequentially by finding a new dictionary (or Gram matrix, which is closest to the reference unit norm tight frame of the previous dictionary (or Gram matrix. The optimization problem can be solved by restricting the tightness and coherence alternately at each iteration of the algorithm. The significant and different aspect of our proposed framework is that the learned dictionary can approximate an equiangular tight frame. Furthermore, manifold optimization is used to avoid the degeneracy of sparse representation while only reducing the coherence of the learned dictionary. This can be performed after the dictionary update process rather than during the dictionary update process. Experiments on synthetic and real audio data show that our proposed methods give notable improvements in lower coherence, have faster running times, and are extremely robust compared to several existing methods.
Mini-lecture course: Introduction into hierarchical matrix technique
Litvinenko, Alexander
2017-12-14
The H-matrix format has a log-linear computational cost and storage O(kn log n), where the rank k is a small integer and n is the number of locations (mesh points). The H-matrix technique allows us to work with general class of matrices (not only structured or Toeplits or sparse). H-matrices can keep the H-matrix data format during linear algebra operations (inverse, update, Schur complement).
Google matrix of the citation network of Physical Review.
Frahm, Klaus M; Eom, Young-Ho; Shepelyansky, Dima L
2014-05-01
We study the statistical properties of spectrum and eigenstates of the Google matrix of the citation network of Physical Review for the period 1893-2009. The main fraction of complex eigenvalues with largest modulus is determined numerically by different methods based on high-precision computations with up to p = 16384 binary digits that allow us to resolve hard numerical problems for small eigenvalues. The nearly nilpotent matrix structure allows us to obtain a semianalytical computation of eigenvalues. We find that the spectrum is characterized by the fractal Weyl law with a fractal dimension d(f) ≈ 1. It is found that the majority of eigenvectors are located in a localized phase. The statistical distribution of articles in the PageRank-CheiRank plane is established providing a better understanding of information flows on the network. The concept of ImpactRank is proposed to determine an influence domain of a given article. We also discuss the properties of random matrix models of Perron-Frobenius operators.
A sparse-grid isogeometric solver
Beck, Joakim
2018-02-28
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90’s in the context of the approximation of high-dimensional PDEs.The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.
A sparse version of IGA solvers
Beck, Joakim
2017-07-30
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse grids construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.
Array aperture extrapolation using sparse reconstruction
Anitori, L.; Rossum, W.L. van; Huizing, A.G.
2015-01-01
In this paper we present some preliminary results on antenna array extrapolation for Direction Of Arrival (DOA) estimation using Sparse Reconstruction (SR). The objective of this study is to establish wether it is possible to achieve with an array of a given physical length the performance (in terms
Self-Control in Sparsely Coded Networks
Dominguez, D. R. C.; Bollé, D.
1998-03-01
A complete self-control mechanism is proposed in the dynamics of neural networks through the introduction of a time-dependent threshold, determined in function of both the noise and the pattern activity in the network. Especially for sparsely coded models this mechanism is shown to considerably improve the storage capacity, the basins of attraction, and the mutual information content.
Comparison of sparse point distribution models
DEFF Research Database (Denmark)
Erbou, Søren Gylling Hemmingsen; Vester-Christensen, Martin; Larsen, Rasmus
2010-01-01
This paper compares several methods for obtaining sparse and compact point distribution models suited for data sets containing many variables. These are evaluated on a database consisting of 3D surfaces of a section of the pelvic bone obtained from CT scans of 33 porcine carcasses. The superior m...... in slaughterhouses....
Multisnapshot Sparse Bayesian Learning for DOA
DEFF Research Database (Denmark)
Gerstoft, Peter; Mecklenbrauker, Christoph F.; Xenaki, Angeliki
2016-01-01
The directions of arrival (DOA) of plane waves are estimated from multisnapshot sensor array data using sparse Bayesian learning (SBL). The prior for the source amplitudes is assumed independent zero-mean complex Gaussian distributed with hyperparameters, the unknown variances (i.e., the source p...... is discussed and evaluated competitively against LASSO (l(1)-regularization), conventional beamforming, and MUSIC....
Scalable group level probabilistic sparse factor analysis
DEFF Research Database (Denmark)
Hinrich, Jesper Løve; Nielsen, Søren Føns Vind; Riis, Nicolai Andre Brogaard
2017-01-01
shows that noise is reduced in areas typically associated with activation by the experimental design. The psFA model identifies sparse components and the probabilistic setting provides a natural way to handle parameter uncertainties. The variational Bayesian framework easily extends to more complex...
Directory of Open Access Journals (Sweden)
Zakaria El Allali
2018-01-01
Full Text Available In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent $\\Delta^2_{p(x} u=\\lambda V(x |u|^{q(x-2} u$, in a smooth bounded domain,under Neumann boundary conditions, where $\\lambda$ is a positive real number, $p,q: \\overline{\\Omega} \\rightarrow \\mathbb{R}$, are continuous functions, and $V$ is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.
Bienvenu, François; Akçay, Erol; Legendre, Stéphane; McCandlish, David M
2017-06-01
Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units. Copyright © 2017 Elsevier Inc. All rights reserved.
Yamanaka, Masanori
2013-08-01
We apply the random matrix theory to analyze the molecular dynamics simulation of macromolecules, such as proteins. The eigensystem of the cross-correlation matrix for the time series of the atomic coordinates is analyzed. We study a data set with seven different sampling intervals to observe the characteristic motion at each time scale. In all cases, the unfolded eigenvalue spacings are in agreement with the predictions of random matrix theory. In the short-time scale, the cross-correlation matrix has the universal properties of the Gaussian orthogonal ensemble. The eigenvalue distribution and inverse participation ratio have a crossover behavior between the universal and nonuniversal classes, which is distinct from the known results such as the financial time series. Analyzing the inverse participation ratio, we extract the correlated cluster of atoms and decompose it to subclusters.
The Performance Analysis Based on SAR Sample Covariance Matrix
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Esra Erten
2012-03-01
Full Text Available Multi-channel systems appear in several fields of application in science. In the Synthetic Aperture Radar (SAR context, multi-channel systems may refer to different domains, as multi-polarization, multi-interferometric or multi-temporal data, or even a combination of them. Due to the inherent speckle phenomenon present in SAR images, the statistical description of the data is almost mandatory for its utilization. The complex images acquired over natural media present in general zero-mean circular Gaussian characteristics. In this case, second order statistics as the multi-channel covariance matrix fully describe the data. For practical situations however, the covariance matrix has to be estimated using a limited number of samples, and this sample covariance matrix follow the complex Wishart distribution. In this context, the eigendecomposition of the multi-channel covariance matrix has been shown in different areas of high relevance regarding the physical properties of the imaged scene. Specifically, the maximum eigenvalue of the covariance matrix has been frequently used in different applications as target or change detection, estimation of the dominant scattering mechanism in polarimetric data, moving target indication, etc. In this paper, the statistical behavior of the maximum eigenvalue derived from the eigendecomposition of the sample multi-channel covariance matrix in terms of multi-channel SAR images is simplified for SAR community. Validation is performed against simulated data and examples of estimation and detection problems using the analytical expressions are as well given.
Kumar, Santosh; Sambasivam, Bharath; Anand, Shashank
2017-08-01
The statistical behaviour of the smallest eigenvalue has important implications for systems which can be modeled using a Wishart-Laguerre ensemble, the regular one or the fixed trace one. For example, the density of the smallest eigenvalue of the Wishart-Laguerre ensemble plays a crucial role in characterizing multiple channel telecommunication systems. Similarly, in the quantum entanglement problem, the smallest eigenvalue of the fixed trace ensemble carries information regarding the nature of entanglement. For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme suggested by Edelman to directly obtain the exact expression for the smallest eigenvalue density. In the case of complex Wishart-Laguerre matrices, for finding exact and explicit expressions for the smallest eigenvalue density, existing results based on determinants become impractical when the determinants involve large-size matrices. In this work, we derive a recurrence scheme for the complex case which is analogous to that of Edelman’s for the real case. This is used to obtain exact results for the smallest eigenvalue density for both the regular, and the fixed trace complex Wishart-Laguerre ensembles. We validate our analytical results using Monte Carlo simulations. We also study scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of coupled kicked tops.