Perspectives on CUR decompositions

Hamm, Keaton; Huang, Longxiu

doi:10.1016/j.acha.2019.08.006

Cited by 39 publications

(21 citation statements)

References 39 publications

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“…It also illustrates its relationship with the CUR decomposition studied e.g. by Mahoney and Drineas in [13], Wang and Zhang [28], Sorensen and Embree [21], Hamm and Huang [8], and predicted by Penrose in [19].…”

Section: Related Workmentioning

confidence: 69%

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A theory of meta-factorization

Karpowicz¹

2021

Preprint

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This paper introduces meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with the example of SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nyström method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

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Section: Related Workmentioning

confidence: 69%

“…As demonstrated by Nakatsukasa, the preferred way to compute the mixing matrix, G = (Ω * r AΩ c ) + , is to perform a QR factorization Ω * r AΩ c = QR and then take Wang and Zhang [28], Martinsson and Tropp [14], Hamm and Huang [8], and others.…”

Section: Generalized Nyström's Methods and The Cur Decompositionmentioning

confidence: 99%

A theory of meta-factorization

Karpowicz¹

2021

Preprint

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“…For the reader's convenience, we recall the following characterization of CUR decompositions of low-rank matrices given in [20].…”

Section: Characterization Of Cur Decompositionsmentioning

confidence: 99%

“…Note that if C and R are column and row submatrices of A which has low-rank, and U is the matrix formed from entries where C and R overlap -i.e., if C = A(:, J) and R = A(I, :) then U := A(I, J)then the classical statement of the CUR decomposition is that A = CU † R if and only if rank (U ) = rank (A). This exact decomposition goes back at least as far as the 1950s [24] in the case that U is square and invertible (this case also follows from rank additivity for Schur decompositions [17]); for a history, the reader is invited to consult [20], but the main theorem therein which characterizes this exact decomposition is restated in Section 3. Our initial sampling result is obtained from some established results of Rudelson and Vershynin [27].…”

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confidence: 99%

Stability of sampling for CUR decompositions

Hamm¹,

Huang²

2020

Foundations of Data Science

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This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix of a rank k matrix also has rank k. For random column sampling schemes, there is typically a tradeoff between the number of columns needed to be chosen and the complexity of determining the sampling probabilities. We discuss several sampling methods and their complexities as well as stability of the method under perturbations of both the probabilities and the underlying matrix. As an application, we give a high probability guarantee of the exact solution of the Subspace Clustering Problem via CUR decompositions when columns are sampled according to their Euclidean lengths.

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“…This enforced structure makes the bases, a.k.a. singular vectors, hard to interpret [9]. On the other hand, it is shown that borrowing bases from the actual samples of a dataset provides a robust representation, which can be employed in interesting applications where sampling is their key factor [20].…”

Section: Introductionmentioning

confidence: 99%

Two-Way Spectrum Pursuit for CUR Decomposition and its Application in Joint Column/Row Subset Selection

Esmaeili

Joneidi

Salimitari

et al. 2021

2021 IEEE 31st International Workshop on Machine Learning for Signal Processing (MLSP)

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The problem of simultaneous column and row subset selection is addressed in this paper. The column space and row space of a matrix are spanned by its left and right singular vectors, respectively. However, the singular vectors are not within actual columns/rows of the matrix. In this paper, an iterative approach is proposed to capture the most structural information of columns/rows via selecting a subset of actual columns/rows. This algorithm is referred to as two-way spectrum pursuit (TWSP) which provides us with an accurate solution for the CUR matrix decomposition. TWSP is applicable in a wide range of applications since it enjoys a linear complexity w.r.t. number of original columns/rows. We demonstrated the application of TWSP for joint channel and sensor selection in cognitive radio networks, informative users and contents detection, and efficient supervised data reduction.

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Perspectives on CUR decompositions

Cited by 39 publications

References 39 publications

A theory of meta-factorization

A theory of meta-factorization

Stability of sampling for CUR decompositions

Two-Way Spectrum Pursuit for CUR Decomposition and its Application in Joint Column/Row Subset Selection

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