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Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 906 (36 self)
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and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A
A Singular Value Thresholding Algorithm for Matrix Completion
, 2008
"... This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of reco ..."
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Cited by 553 (21 self)
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toimplement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative and produces a sequence of matrices {X k, Y k} and at each step, mainly performs a softthresholding operation on the singular values of the matrix Y k. There are two
A multilinear singular value decomposition
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are ..."
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Cited by 470 (22 self)
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Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc
Using Linear Algebra for Intelligent Information Retrieval
 SIAM REVIEW
, 1995
"... Currently, most approaches to retrieving textual materials from scientific databases depend on a lexical match between words in users' requests and those in or assigned to documents in a database. Because of the tremendous diversity in the words people use to describe the same document, lexical ..."
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Cited by 675 (18 self)
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, lexical methods are necessarily incomplete and imprecise. Using the singular value decomposition (SVD), one can take advantage of the implicit higherorder structure in the association of terms with documents by determining the SVD of large sparse term by document matrices. Terms and documents represented
Parallel Numerical Linear Algebra
, 1993
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illust ..."
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Cited by 776 (23 self)
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, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
Indexing by latent semantic analysis
 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE
, 1990
"... A new method for automatic indexing and retrieval is described. The approach is to take advantage of implicit higherorder structure in the association of terms with documents (“semantic structure”) in order to improve the detection of relevant documents on the basis of terms found in queries. The p ..."
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Cited by 3775 (35 self)
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. The particular technique used is singularvalue decomposition, in which a large term by document matrix is decomposed into a set of ca. 100 orthogonal factors from which the original matrix can be approximated by linear combination. Documents are represented by ca. 100 item vectors of factor weights. Queries
Probabilistic Latent Semantic Analysis
 In Proc. of Uncertainty in Artificial Intelligence, UAI’99
, 1999
"... Probabilistic Latent Semantic Analysis is a novel statistical technique for the analysis of twomode and cooccurrence data, which has applications in information retrieval and filtering, natural language processing, machine learning from text, and in related areas. Compared to standard Latent Sema ..."
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Cited by 770 (9 self)
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Semantic Analysis which stems from linear algebra and performs a Singular Value Decomposition of cooccurrence tables, the proposed method is based on a mixture decomposition derived from a latent class model. This results in a more principled approach which has a solid foundation in statistics. In order
Probabilistic Latent Semantic Indexing
, 1999
"... Probabilistic Latent Semantic Indexing is a novel approach to automated document indexing which is based on a statistical latent class model for factor analysis of count data. Fitted from a training corpus of text documents by a generalization of the Expectation Maximization algorithm, the utilized ..."
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Cited by 1227 (10 self)
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model is able to deal with domainspecific synonymy as well as with polysemous words. In contrast to standard Latent Semantic Indexing (LSI) by Singular Value Decomposition, the probabilistic variant has a solid statistical foundation and defines a proper generative data model. Retrieval experiments
Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 722 (18 self)
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processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum
Unsupervised Learning by Probabilistic Latent Semantic Analysis
 Machine Learning
, 2001
"... Abstract. This paper presents a novel statistical method for factor analysis of binary and count data which is closely related to a technique known as Latent Semantic Analysis. In contrast to the latter method which stems from linear algebra and performs a Singular Value Decomposition of cooccurren ..."
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Cited by 617 (4 self)
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Abstract. This paper presents a novel statistical method for factor analysis of binary and count data which is closely related to a technique known as Latent Semantic Analysis. In contrast to the latter method which stems from linear algebra and performs a Singular Value Decomposition of co
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