Improved Approximation Algorithms for Large Matrices via Random Projections

Sarlós, Tamás

doi:10.1109/focs.2006.37

Cited by 562 publications

(756 citation statements)

References 44 publications

(96 reference statements)

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“…Other constructions of random projection matrices have been discovered since [2,3,4,5,6]. Their properties make random projections a key player in rank-k approximation algorithms [7,8,9,10,11,12,13,14], other algorithms in numerical linear algebra [15,16,17], compressed sensing [18,19,20], and various other applications, e.g, [21,22].…”

Section: Introductionmentioning

confidence: 99%

Dense Fast Random Projections and Lean Walsh Transforms

Liberty

Ailon

Singer

2010

Discrete Comput Geom

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Random projection methods give distributions over k × d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors xi ∈ R d the resulting vectors Ψxi ∈ R k approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random ±1 diagonal matrix) is a good random projector for a subset of vectors in R d . Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms or running time, the best known current result (due to Matousek) under comparable assumptions.

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Section: Introductionmentioning

confidence: 99%

Dense Fast Random Projections and Lean Walsh Transforms

Liberty

Ailon

Singer

2010

Discrete Comput Geom

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“…Other suggested methods as [1,7,38,39] seem to have the same complexity O(kmn), since they project each row of A on some k-dimensional subspace of R n . Clearly, a best CUR approximation is chosen by the least squares principle:…”

Section: Low Rank Approximations Using Sampling Of Rowsmentioning

confidence: 99%

Low-Rank Approximation of Tensors

Friedland

Tammali

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CURdecomposition are most suitable.For d-mode tensors T ∈ ⊗ d i=1 R n i , with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1, . . . , r d )-approximation, which maximizes the 2 norm of the projection of T on ⊗ d i=1 Ui, where Ui is an ri-dimensional subspace R n i . One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1, . . . , d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors.The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r1, . . . , r d )-approximation. We compare numerically different approximation methods.2000 Mathematics Subject Classification. 14M15, 15A18, 15A69, 65H10, 65K10.

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“…In the case of "power-law" networks it was shown in [32] that the spectral counting of triangles can be efficient due to their special spectral properties and [33] extended this idea using the randomized algorithm by [12] by proposing a simple biased node sampling. This algorithm can be viewed as a special case of a streaming algorithm, since there exist algorithms, e.g., [29], that perform a constant number of passes over the non-zero elements of the matrix to produce a good low rank matrix approximation. In [5] the semi-streaming model for counting triangles is introduced, which allows log n passes over the edges.…”

Section: Existing Workmentioning

confidence: 99%

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Kolountzakis

Peng

Tsourakakis

2010

Algorithms and Models for the Web-Graph

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Abstract. The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [15]. The key idea of our algorithm is to combine the sampling algorithm of [34,35] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [2], treating each set appropriately. We obtain a running time O m + m 3/2 ∆ log n tǫ 2 and an ǫ approximation (multiplicative error), where n is the number of vertices, m the number of edges and ∆ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage O m 1/2 log n + m 3/2 ∆ log n tǫ 2 and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

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Improved Approximation Algorithms for Large Matrices via Random Projections

Cited by 562 publications

References 44 publications

Dense Fast Random Projections and Lean Walsh Transforms

Dense Fast Random Projections and Lean Walsh Transforms

Low-Rank Approximation of Tensors

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

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