Robust CUR Decomposition: Theory and Imaging Applications

Cai, HanQin; Hamm, Keaton; Huang, Longxiu; Needell, Deanna

doi:10.1137/20m1388322

Cited by 25 publications

(10 citation statements)

References 37 publications

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“…This implies that all singular values of SU X are strictly positive, and thus that rank(SU X )= rank(U X ) = rank(X), which establishes the claim (7).…”

Section: Appendix: Proofmentioning

confidence: 61%

“…In this context, CUR decompositions are particularly interesting, as they directly sample actual rows or columns of matrices to form the random sample and preserve the interpretability of the original data [1]. Therefore, the CUR matrix decompositions have been extensively discussed in the theoretical computer science, the machine learning, and the numerical linear algebra community [2]- [7]. The more applications of CUR decomposition include modeling largescale traffic networks, large-scale retrieval, and compression sensing, and so on.…”

Section: Introductionmentioning

confidence: 99%

“…The more applications of CUR decomposition include modeling largescale traffic networks, large-scale retrieval, and compression sensing, and so on. Moreover, the related studies have been extended to tensor CUR which can be thought of as multidimensional generalizations of CUR decompositions [8]- [10]. The computational complexity of sampling probabilities and approximate accuracy are two fundamental problems.…”

Section: Introductionmentioning

confidence: 99%

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Energy-Based Adaptive CUR Matrix Decomposition

Zhao

Zhang

2023

IEEE Access

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CUR decompositions are interpretable data analysis tools that express a data matrix in terms of a small number of actual columns and/or actual rows of the data matrix. One bottleneck of existing relative-error CUR algorithms lies on high computational complexity for computing important sampling probabilities. In this paper, we provide a simple yet effective framework that considers energy-based sampling algorithm. On one hand, we provide an intuitive and fast relative-error sampling algorithm for column selection problem. On the other hand, by combining the relative-error sampling algorithm with adaptive sampling algorithm we provide a novel CUR matrix approximation algorithms which is referred to as energy-based adaptive sampling algorithm. The sampling algorithm is the first adaptive relative-error CUR decomposition in the coherent sense. Specially, in each stage of our algorithm, we sample columns or rows from data matrix using sampling probabilities that are directly proportional to Euclidean norms of the columns or rows of the original data and residual matrix, respectively. Our empirical results exactly indicate that the new adaptive sampling algorithm typically achieves a good balance between computational complexity and approximate accuracy.

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“…This implies that all singular values of SU X are strictly positive, and thus that rank(SU X )= rank(U X ) = rank(X), which establishes the claim (7).…”

Section: Appendix: Proofmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Energy-Based Adaptive CUR Matrix Decomposition

Zhao

Zhang

2023

IEEE Access

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“…Follow the same argument for Assumption 1, the observed outliers are also presented in the vector form. Note that Assumption 3 holds with high probability provided s ♮ is (α/2)-sparse [7,Proposition 4.5]. This assumption also implies that HΠ Ω s ♮ has no more than αpn non-zero entries in each of it rows and columns.…”

Section: Assumptions and Notationmentioning

confidence: 99%

Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

Cai¹,

Cai²,

You³

2022

Preprint

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We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we explore the convenient Hankel structure and propose a novel non-convex algorithm, coined Hankel Structured Gradient Descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing-and sample-efficient compared to the state-ofthe-arts. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.

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“…There is extensive work on CUR-type decompositions in both numerical linear algebra and theoretical computer science; see [7,8,20,44]. Recently, in [17], Gidisu and Hochstenbach developed a generalized CUR decomposition (GCUR) for matrix pair A and B with the same number of columns: A is m × n, B is d × n and both are of full column rank, which can be viewed as a CUR decomposition of A relative to B.…”

Section: Introductionmentioning

confidence: 99%

Randomized GCUR decompositions

Cao¹,

Wei²,

Xie³

2023

Preprint

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By exploiting the random sampling techniques, this paper derives an efficient randomized algorithm for computing a generalized CUR decomposition, which provides low-rank approximations of both matrices simultaneously in terms of some of their rows and columns. For large-scale data sets that are expensive to store and manipulate, a new variant of the discrete empirical interpolation method known as L-DEIM, which needs much lower cost and provides a significant acceleration in practice, is also combined with the random sampling approach to further improve the efficiency of our algorithm. Moreover, adopting the randomized algorithm to implement the truncation process of restricted singular value decomposition (RSVD), combined with the L-DEIM procedure, we propose a fast algorithm for computing an RSVD based CUR decomposition, which provides a coordinated low-rank approximation of the three matrices in a CUR-type format simultaneously and provides advantages over the standard CUR approximation for some applications. We establish detailed probabilistic error analysis for the algorithms and provide numerical results that show the promise of our approaches.

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Robust CUR Decomposition: Theory and Imaging Applications

Cited by 25 publications

References 37 publications

Energy-Based Adaptive CUR Matrix Decomposition

Energy-Based Adaptive CUR Matrix Decomposition

Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

Randomized GCUR decompositions

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