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87
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Computing the Singular Value Decomposition with High Relative Accuracy
 Linear Algebra Appl
, 1997
"... We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the a ..."
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Cited by 55 (12 self)
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We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...
FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
"... Abstract. Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful t ..."
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Cited by 40 (0 self)
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Abstract. Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing lowrank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired lowrank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition
RELATIVEERROR CUR MATRIX DECOMPOSITIONS
 SIAM J. MATRIX ANAL. APPL
, 2008
"... Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the ..."
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Cited by 39 (9 self)
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Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an m × n matrix A and a rank parameter k. In our first algorithm, C is chosen, and we let A ′ = CC + A, where C + is the Moore–Penrose generalized inverse of C. In our second algorithm C, U, R are chosen, and we let A ′ = CUR. (C and R are matrices that consist of actual columns and rows, respectively, of A, and U is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least 1 − δ, ‖A − A ′ ‖F ≤ (1 + ɛ) ‖A − Ak‖F, where Ak is the “best ” rankk approximation provided by truncating the SVD of A, and where ‖X‖F is the Frobenius norm of the matrix X. The number of columns of C and rows of R is a lowdegree polynomial in k, 1/ɛ, and log(1/δ). Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants
A fast randomized algorithm for the approximation of matrices
, 2007
"... We introduce a randomized procedure that, given an m×n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l × m random matrix R to each column of A, where l is an integer near to, but greater than, k. The structure of R allows u ..."
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Cited by 28 (6 self)
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We introduce a randomized procedure that, given an m×n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l × m random matrix R to each column of A, where l is an integer near to, but greater than, k. The structure of R allows us to apply it to an arbitrary m × 1 vector at a cost proportional to m log(l); the resulting procedure can construct a rankk approximation Z from the entries of A at a cost proportional to mn log(k)+l 2 (m+n). We prove several bounds on the accuracy of the algorithm; one such bound guarantees that the spectral norm ‖A − Z ‖ of the discrepancy between A and Z is of the same order as √ max{m, n} times the (k + 1) st greatest singular value σk+1 of A, with small probability of large deviations. In contrast, the classical pivoted “Q R ” decomposition algorithms (such as GramSchmidt or Householder) require at least kmn floatingpoint operations in order to compute a similarly accurate rankk approximation. In practice, the algorithm of this paper is faster than the classical algorithms, as long as k is neither very small nor very large. Furthermore, the algorithm operates reliably independently of the structure of the matrix A, can access each column of A independently and at most twice, and parallelizes naturally. The results are illustrated via several numerical examples.
An Improved Approximation Algorithm for the Column Subset Selection Problem
"... We consider the problem of selecting the “best ” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel twostage algorithm that runs in O(min{mn 2, m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stag ..."
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Cited by 28 (2 self)
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We consider the problem of selecting the “best ” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel twostage algorithm that runs in O(min{mn 2, m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciouslychosen probability distribution that depends on information in the topk right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic columnselection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let PC denote the projection matrix onto the span of those columns, and let Ak denote the “best ” rankk approximation to the matrix A as computed with the singular value decomposition. Then, we prove that ‖A − PCA‖2 ≤ O k 3 4 log 1
Fast linear algebra is stable
 In preparation
, 2006
"... In [23] we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of nbyn matrices can be done by any algorithm in O(n ω+η) operations for any η> 0, then it can be done stably in O(n ω+η) operations for any η> ..."
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Cited by 25 (15 self)
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In [23] we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of nbyn matrices can be done by any algorithm in O(n ω+η) operations for any η> 0, then it can be done stably in O(n ω+η) operations for any η> 0. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(n ω+η) operations. 1
FINDING STRUCTURE WITH RANDOMNESS: STOCHASTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
, 2009
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing l ..."
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Cited by 25 (2 self)
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Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing lowrank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. In particular, these techniques offer a route toward principal component analysis (PCA) for petascale data. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired lowrank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider
A randomized algorithm for principal component analysis
 SIAM Journal on Matrix Analysis and Applications
"... Principal component analysis (PCA) requires the computation of a lowrank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rankdeficient approximation is at most a few digits (measured in the spectral norm, relative to the ..."
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Cited by 21 (0 self)
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Principal component analysis (PCA) requires the computation of a lowrank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rankdeficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, existing efficient algorithms have not guaranteed good accuracy for the approximations they produce, unless one or both dimensions of the matrix being approximated are small. We describe an efficient algorithm for the lowrank approximation of matrices that produces accuracy very close to the best possible, for matrices of arbitrary sizes. We illustrate our theoretical results via several numerical examples. 1
Expressions And Bounds For The GMRES Residual
 BIT
, 1999
"... . Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal mat ..."
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Cited by 20 (0 self)
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. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the twonorm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...