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41
Numerical methods for computing angles between linear subspaces
 Math. Comp
, 1973
"... Foundation. Assume that two subspaces F and G of a unitary space are defined.. as the ranges(or nullspacd of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles ek(F,G) and orthogonal sets of principal vectors u k 6 F and vk c G, k = 1,2,. ..."
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Cited by 144 (3 self)
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Foundation. Assume that two subspaces F and G of a unitary space are defined.. as the ranges(or nullspacd of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles ek(F,G) and orthogonal sets of principal vectors u k 6 F and vk c G, k = 1,2,..., q = dim(G) 2 dim(F). An important application in statistics is computing the canonical correlations uk = cos 8 k between two sets of variates. A perturbation analysis shows that the condition number for ek essentially is max(K(A),K(B)), where K denotes the condition number of a matrix. The algorithms are based on a preliminary &Rfactorization of A and B (or AH and BH), for which either the method of Householder transformations (HT) or the modified GramSchmidt method (MGS) is used. Then cos Ok and sin 0 k are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives Bk with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not in general true for MGS. Finally the case when A and/or B are rank deficient is discussed..1.
On the Early History of the Singular Value Decomposition
, 1992
"... This paper surveys the contributions of five mathematicians  Eugenio Beltrami (18351899), Camille Jordan (18381921), James Joseph Sylvester (18141897), Erhard Schmidt (18761959), and Hermann Weyl (18851955)  who were responsible for establishing the existence of the singular value de ..."
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Cited by 104 (1 self)
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This paper surveys the contributions of five mathematicians  Eugenio Beltrami (18351899), Camille Jordan (18381921), James Joseph Sylvester (18141897), Erhard Schmidt (18761959), and Hermann Weyl (18851955)  who were responsible for establishing the existence of the singular value decomposition and developing its theory.
Perturbation Theory for the Singular Value Decomposition
 IN SVD AND SIGNAL PROCESSING, II: ALGORITHMS, ANALYSIS AND APPLICATIONS
, 1990
"... The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the effects of the noise on the singular values and singular v ..."
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Cited by 40 (0 self)
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The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the effects of the noise on the singular values and singular vectors  a problem in classical perturbation theory. In this paper we survey the perturbation theory of the singular value decomposition.
Contextual Spelling Correction Using Latent Semantic Analysis
 In Proc. 5th Conference on Applied Natural Language Processing
, 1997
"... Contextual spelling errors are defined as the use of an incorrect, though valid, word in a particular sentence or context. Traditional spelling checkers flag misspelled words, but they do not typically attempt to identify words that are used incorrectly in a sentence. We explore the use of Lat ..."
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Cited by 29 (0 self)
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Contextual spelling errors are defined as the use of an incorrect, though valid, word in a particular sentence or context. Traditional spelling checkers flag misspelled words, but they do not typically attempt to identify words that are used incorrectly in a sentence. We explore the use of Latent Semantic Analysis for correcting these incorrectly used words and the results are compared to earlier work based on a Bayesian classifier.
Perturbation Theory for Homogeneous Polynomial Eigenvalue Problems
, 2001
"... We consider polynomial eigenvalue problems P (A; ; )x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (; ) 2 C 2 . In this framework innite eigenvalues are on the same footing as nite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs ..."
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Cited by 23 (2 self)
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We consider polynomial eigenvalue problems P (A; ; )x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (; ) 2 C 2 . In this framework innite eigenvalues are on the same footing as nite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is wellposed when its eigenvalues are simple. We dene the condition numbers of a simple eigenvalue (; ) and a corresponding eigenvector x and show that the distance to the nearest illposed problem is equal to the reciprocal of the condition number of the eigenvector x. We describe a bihomogeneous Newton method for the solution of the homogeneous polynomial eigenvalue problem. Key words. polynomial eigenvalue problem, matrix polynomial, quadratic eigenvalue problem, condition number AMS subject classications. 65F15, 15A18 1
A Unitary Invariant in Riemannian Geometry
"... We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role ..."
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Cited by 15 (2 self)
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We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi– Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.
Efficient Singular Value Decomposition via Improved Document Sampling
 DEPT. OF COMPUTER SCIENCE, DUKE UNIVERSITY
, 1999
"... Singular value decomposition (SVD) is a generalpurpose mathematical analysis tool that has been used in a variety of informationretrieval applications. As the size and complexity of retrieval collections increase, it is crucial for our analysis tools to scale accordingly. To this end, we have stud ..."
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Cited by 11 (1 self)
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Singular value decomposition (SVD) is a generalpurpose mathematical analysis tool that has been used in a variety of informationretrieval applications. As the size and complexity of retrieval collections increase, it is crucial for our analysis tools to scale accordingly. To this end, we have studied the application of a new theoretically justified SVD approximation algorithm to the problem of text retrieval. We show that, in the case of latent semantic indexing, we can achieve near optimal approximations of the exact SVD using considerably less computation by using an appropriate distribution to sample the documents we include in our SVD analysis.
Information Retrieval on the Web: Selected Topics
 IBM research, Tokyo Research Laboratory, IBM
, 1999
"... In this paper we review studies on the growth of the Internet and technologies which are useful for information search and retrieval on the Web. In the rst section, we present data on the Internet from several dierent sources, e.g., current as well as projected number of users, hosts and Web sites. ..."
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Cited by 5 (0 self)
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In this paper we review studies on the growth of the Internet and technologies which are useful for information search and retrieval on the Web. In the rst section, we present data on the Internet from several dierent sources, e.g., current as well as projected number of users, hosts and Web sites. Although the numerical gures vary, the overall trends cited by the sources are consistent and point to exponential growth during the coming decade. And Internet users are increasingly using search engines and search services to nd speci c information of interest. However, users are not satis ed with the performance of the current generation of search engines; the slow speed of retrieval, communication delays, and poor quality of retrieved results (e.g., noise and broken links) are commonly cited problems. The main body of our paper focuses on linear algebraic models and techniques for solving these problems. keywords: clustering, indexing, information retrieval, Internet, late...