Near Optimal Column-Based Matrix Reconstruction

Boutsidis, Christos; Drineas, Petros; Magdon‐Ismail, Malik

doi:10.1109/focs.2011.21

Cited by 72 publications

(196 citation statements)

References 26 publications

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“…Before discussing the details of some of the available CUR algorithms in [17,20,23,36,27,49], we briefly mention a similar problem which constructs factorizations of the form A = CX + E, where C contains columns of A and X has rank at most k. Unlike CUR, there are optimal algorithms for this problem [6,31], in both the spectral and the Frobenius norm. Indeed, to obtain a relative-error optimal CUR in this paper we use a sampling method from [6], which allows to select O(k) columns and rows.…”

Section: Deterministic Curmentioning

confidence: 99%

“…Lemma 3.3 (Lemma 3.4 in [6]). Given A ∈ R m×n of rank ρ, a target rank 2 ≤ k < ρ, and 0 < ǫ ≤ 1, there exists a randomized algorithm that computes Z ∈ R n×k with Z T Z = I k and…”

Section: Randomized Linear-time Approximate Svdmentioning

confidence: 99%

“…First, we design an input-sparsity-time version of the BSS sampling step (see Lemma 4.3). To do that, we combine the method from [6] with ideas from the sparse subspace embedding literature [11]. Second, we develop inputsparsity-time versions of the adaptive sampling algorithms of [15,49] (see Lemma 4.4 and Lemma 4.5).…”

Section: Cur Proto-algorithmmentioning

confidence: 99%

“…To address primitive (1), we combine known ideas for column subset selection including leverage-scores sampling [23], BSS sampling [6] (i.e., deterministic sampling similar to the method of Batson, Spielman, and Srivastava [3]) and adaptive sampling [15] (see Section 3.4). To find Z 1 , we use techniques for approximating the SVD (see Section 3.3).…”

Section: Cur Proto-algorithmmentioning

confidence: 99%

“…Section 2 summarizes results from prior literature and puts our CUR algorithms in context (see Table I). To design our CUR algorithms in Sections 5,6, and 7 we need several "subset selection tools" from prior literature, which we summarize in Section 3, as well as new tools, which we present in Section 4. Finally, we give a lower bound for a CUR algorithm in Section 8.…”

Section: Introduction Given As Inputs a Matrix A ∈ Rmentioning

confidence: 99%

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Optimal CUR matrix decompositions

Boutsidis

Woodruff

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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Abstract. The CUR decomposition of an m × n matrix A finds an m × c matrix C with a subset of c < n columns of A, together with an r × n matrix R with a subset of r < m rows of A, as well as a c × r low-rank matrix U such that the matrix CUR approximates the matrix A, that is, A − CUR 2 F ≤ (1 + ε) A − A k 2 F , where . F denotes the Frobenius norm and A k is the best m × n matrix of rank k constructed via the SVD. We present input-sparsity-time and deterministic algorithms for constructing such a CUR decomposition where c = O(k/ε) and r = O(k/ε) and rank(U) = k. Up to constant factors, our algorithms are simultaneously optimal in c, r, and rank(U).

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Section: Deterministic Curmentioning

confidence: 99%

Section: Randomized Linear-time Approximate Svdmentioning

confidence: 99%

Section: Cur Proto-algorithmmentioning

confidence: 99%

Section: Cur Proto-algorithmmentioning

confidence: 99%

Section: Introduction Given As Inputs a Matrix A ∈ Rmentioning

confidence: 99%

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Optimal CUR matrix decompositions

Boutsidis

Woodruff

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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Block CUR: Decomposing Matrices Using Groups of Columns

Oswal

Jain

et al. 2019

Machine Learning and Knowledge Discovery in Databases

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A common problem in large-scale data analysis is to approximate a matrix using a combination of specifically sampled rows and columns, known as CUR decomposition. Unfortunately, in many real-world environments, the ability to sample specific individual rows or columns of the matrix is limited by either system constraints or cost. In this paper, we consider matrix approximation by sampling predefined blocks of columns (or rows) from the matrix. We present an algorithm for sampling useful column blocks and provide novel guarantees for the quality of the approximation. This algorithm has application in problems as diverse as biometric data analysis to distributed computing. We demonstrate the effectiveness of the proposed algorithms for computing the Block CUR decomposition of large matrices in a distributed setting with multiple nodes in a compute cluster, where such blocks correspond to columns (or rows) of the matrix stored on the same node, which can be retrieved with much less overhead than retrieving individual columns stored across different nodes. In the biometric setting, the rows correspond to different users and columns correspond to users' biometric reaction to external stimuli, e.g., watching video content, at a particular time instant. There is significant cost in acquiring each user's reaction to lengthy content so we sample a few important scenes to approximate the biometric response. An individual time sample in this use case cannot be queried in isolation due to the lack of context that caused that biometric reaction. Instead, collections of time segments (i.e., blocks) must be presented to the user. The practical application of these algorithms is shown via experimental results using real-world user biometric data from a content testing environment.

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Accuracy-Preserving and Scalable Column-Based Low-Rank Matrix Approximation

Jian-gang

Liao

2015

Knowledge Science, Engineering and Management

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Near Optimal Column-Based Matrix Reconstruction

Cited by 72 publications

References 26 publications

Optimal CUR matrix decompositions

Optimal CUR matrix decompositions

Block CUR: Decomposing Matrices Using Groups of Columns

Accuracy-Preserving and Scalable Column-Based Low-Rank Matrix Approximation

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