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FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
"... 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 ..."
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Cited by 253 (6 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 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 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.
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 62 (4 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
The fast JohnsonLindenstrauss transform and approximate nearest neighbors
 SIAM J. COMPUT
, 2009
"... We introduce a new lowdistortion embedding of ℓd n) 2 into ℓO(log p (p =1, 2) called the fast Johnson–Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a random ..."
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Cited by 61 (0 self)
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We introduce a new lowdistortion embedding of ℓd n) 2 into ℓO(log p (p =1, 2) called the fast Johnson–Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for lowdistortion embeddings. We overcome this handicap by exploiting the “Heisenberg principle ” of the Fourier transform, i.e., its localglobal duality. The FJLT can be used to speed up search algorithms based on lowdistortion embeddings in ℓ1 and ℓ2. We consider the case of approximate nearest neighbors in ℓd 2. We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.
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 Approximation of Matrix Coherence and Statistical Leverage
"... The statistical leverage scores of a matrix A are the squared rownorms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recentlypopular problems such as matrix completion and Nyströmbased lowrank matrix ..."
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Cited by 53 (11 self)
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The statistical leverage scores of a matrix A are the squared rownorms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recentlypopular problems such as matrix completion and Nyströmbased lowrank matrix approximation as well as in largescale statistical data analysis applications more generally; moreover, they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms. Our main result is a randomized algorithm that takes as input an arbitrary n×d matrix A, with n ≫ d, and that returns as output relativeerror approximations to all n of the statistical leverage scores. The proposed algorithm runs (under assumptions on the precise values of n and d) in O(nd logn) time, as opposed to the O(nd 2) time required by the naïve algorithm that involves computing an orthogonal basis for the range of A. Our analysis may be viewed in terms of computing a relativeerror approximation to an underconstrained leastsquares approximation problem, or, relatedly, it may be viewed as an application of JohnsonLindenstrauss type ideas. Several practicallyimportant extensions of our basic result are also described, including the approximation of socalled crossleverage scores, the extension of these ideas to matrices with n≈d, and the extension to streaming environments.
Faster least squares approximation
 Numerische Mathematik
"... Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. Methods dating back to Gauss and Legendre find a solution in O(nd 2) time, where n is the number of constraints and d is the number of variables. We present two rand ..."
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Cited by 53 (13 self)
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Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. Methods dating back to Gauss and Legendre find a solution in O(nd 2) time, where n is the number of constraints and d is the number of variables. We present two randomized algorithms that provide very accurate relativeerror approximations to the solution of a least squares approximation problem more rapidly than existing exact algorithms. Both of our algorithms preprocess the data with a randomized Hadamard transform. One then uniformly randomly samples constraints and solves the smaller problem on those constraints, and the other performs a sparse random projection and solves the smaller problem on those projected coordinates. In both cases, the solution to the smaller problem provides a relativeerror approximation to the exact solution and can be computed in o(nd 2) time. 1
Doulion: Counting Triangles in Massive Graphs with a Coin
 PROCEEDINGS OF ACM KDD,
, 2009
"... Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a ..."
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Cited by 53 (16 self)
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Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a triangle counting algorithm. Furthermore, several interesting graph mining applications rely on computing the number of triangles in the graph of interest. In this paper, we focus on the problem of counting triangles in a graph. We propose a practical method, out of which all triangle counting algorithms can potentially benefit. Using a straightforward triangle counting algorithm as a black box, we performed 166 experiments on realworld networks and on synthetic datasets as well, where we show that our method works with high accuracy, typically more than 99 % and gives significant speedups, resulting in even ≈ 130 times faster performance.