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53
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 86 (17 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 63 (7 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.
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 56 (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
Fast algorithms for spherical harmonic expansions
 SIAM J. Sci. Comput
, 2004
"... An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the twodimensional sphere S 2 in � 3 of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmon ..."
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Cited by 37 (4 self)
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An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the twodimensional sphere S 2 in � 3 of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmonic expansions of functions specified by their values at appropriately chosen points on S 2 (known as the forward spherical harmonic transform). The procedure is numerically stable and requires an amount of CPU time proportional to N(log N) log(1/ε), where N is the number of nodes in the discretization of S 2, and ε is the precision of computations. The performance of the algorithm is illustrated via several numerical examples. 1
TENSORCUR DECOMPOSITIONS FOR TENSORBASED DATA
 SIAM J. MATRIX ANAL. APPL.
, 2008
"... Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensorbased extension of the matrix CUR decomposition. The tensorCUR decomposition is most relevant as a data analysis tool when the data consist of one mode that i ..."
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Cited by 36 (10 self)
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Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensorbased extension of the matrix CUR decomposition. The tensorCUR decomposition is most relevant as a data analysis tool when the data consist of one mode that is qualitatively different from the others. In this case, the tensorCUR decomposition approximately expresses the original data tensor in terms of a basis consisting of underlying subtensors that are actual data elements and thus that have a natural interpretation in terms of the processes generating the data. Assume the data may be modeled as a (2+1)tensor, i.e., an m×n×p tensor A in which the first two modes are similar and the third is qualitatively different. We refer to each of the p different m × n matrices as “slabs ” and each of the mn different pvectors as “fibers.” In this case, the tensorCUR algorithm computes an approximation to the data tensor A that is of the form CUR, where C is an m×n×c tensor consisting of a small number c of the slabs, R is an r × p matrix consisting of a small number r of the fibers, and U is an appropriately defined and easily computed c × r encoding matrix. Both C and R may be chosen by randomly sampling either slabs or fibers according to a judiciously chosen and datadependent probability distribution, and both c and r depend on a rank parameter k, an error parameter ɛ, and a failure probability δ. Under
A randomized algorithm for the approximation of matrices
 In review. Yale CS research report YALEU/DCS/RR1361
, 2006
"... Abstract. Given an m×n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is effici ..."
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Cited by 23 (4 self)
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Abstract. Given an m×n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A T can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as √ lm times the (k + 1) st greatest singular value σk+1 of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l − k is a fixed small nonnegative integer. For example, according to one of our estimates for l − k = 20, the probability that the spectral norm �A − Z � is greater than 10 p (k + 20) m σk+1 is less than 10 −17. The paper contains a number of estimates for �A − Z�, including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A T can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.
How to find a good submatrix
 Research Report 0810, ICM HKBU, Kowloon Tong, Hong Kong
, 2008
"... Pseudoskeleton approximation and some other problems require the knowledge of sufficiently wellconditioned submatrix in a largescale matrix. The quality of a submatrix can be measured by modulus of its determinant, also known as volume. In this paper we discuss a search algorithm for the maximumv ..."
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Cited by 22 (8 self)
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Pseudoskeleton approximation and some other problems require the knowledge of sufficiently wellconditioned submatrix in a largescale matrix. The quality of a submatrix can be measured by modulus of its determinant, also known as volume. In this paper we discuss a search algorithm for the maximumvolume submatrix which already proved to be useful in several matrix and tensor approximation algorithms. We investigate the behavior of this algorithm on random matrices and present some its applications, including maximization of a bivariate functional. 1
Hybrid Cross Approximation of Integral Operators
, 2005
"... The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the Hmatrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdom ..."
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Cited by 21 (6 self)
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The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the Hmatrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by lowrank matrices. The lowrank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA).
Approximate Iterations for Structured Matrices
, 2005
"... Important matrixvalued functions f(A) are, e.g., the inverse A −1, the square root √ A, the sign function and the exponent. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f(A) (often f(A) possess ..."
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Cited by 20 (11 self)
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Important matrixvalued functions f(A) are, e.g., the inverse A −1, the square root √ A, the sign function and the exponent. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f(A) (often f(A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixedpoint like process for the evaluation of f(A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and √ A.