Efficient algorithms for cur and interpolative matrix decompositions

Voronin, Sergey; Martinsson, Per-Gunnar

doi:10.1007/s10444-016-9494-8

Cited by 56 publications

(69 citation statements)

References 20 publications

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“…See also [46]. December 13, 2018 DRAFT Subsequently, others have considered algorithms for computing CUR decompositions which still provide approximately optimal error bounds in the sense described above; see, for example, [2], [38], [47], [48], [49]. For applications of the CUR decomposition in various aspects of data analysis across scientific disciplines, consult [50], [51], [52], [53].…”

Section: Cur Decompositionmentioning

confidence: 99%

CUR Decompositions, Similarity Matrices, and Subspace Clustering

Aldroubi

Hamm

Koku

et al. 2019

Front. Appl. Math. Stat.

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A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces U = M i=1 S i . The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method.Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.

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Section: Cur Decompositionmentioning

confidence: 99%

CUR Decompositions, Similarity Matrices, and Subspace Clustering

Aldroubi

Hamm

Koku

et al. 2019

Front. Appl. Math. Stat.

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“…While structured random matrices can achieve interesting speed-ups in the offline stage, the complexity reduction is only possible without power iterations. 44 Omitting power iterations can lead to a loss of accuracy, which may not be attractive for some MOR problems (see Section 4.5).…”

Section: Randomized Methods For Computing Low-rank Approximationsmentioning

confidence: 99%

“…() Error bounds similar to the ones for Gaussian random variables have also been derived for the SRFT and the SRHT,() as well as other theoretical results on the effects of introducing structured random matrices for dimension reduction (see also the work of Ailon and Chazelle). While structured random matrices can achieve interesting speed‐ups in the offline stage, the complexity reduction is only possible without power iterations . Omitting power iterations can lead to a loss of accuracy, which may not be attractive for some MOR problems (see Section 4.5).…”

Section: State Of the Artmentioning

confidence: 99%

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

Bach

Duddeck

Song

2019

Numerical Meth Engineering

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Summary Many model order reduction (MOR) methods employ a reduced basis boldV∈Rm×k to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low‐rank approximation of the snapshot matrix boldA∈Rm×n can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of scriptOfalse(minfalse(mn2,m2nfalse)false), which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed‐rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed‐precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.

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“…Most of these works focus on randomly sampling columns and rows to form the approximation; however, these works consider many different choices for the middle matrix X in the CUR approximation. Nonetheless, there are deterministic methods of selecting columns given in [28,36], the latter of which first uses a fast QR factorization of A and subsequently implicitly forms the CUR approximation.…”

Section: Proposition 21 ([31]mentioning

confidence: 99%

Perspectives on CUR decompositions

Hamm

Huang

2020

Applied and Computational Harmonic Analysis

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The CUR decomposition is a factorization of a low-rank matrix obtained by selecting certain column and row submatrices of it. We perform a thorough investigation of what happens to such decompositions in the presence of noise. Since CUR decompositions are non-uniquely formed, we investigate several variants and give perturbation estimates for each in terms of the magnitude of the noise matrix in a broad class of norms which includes all Schatten p-norms. The estimates given here are qualitative and illustrate how the choice of columns and rows affects the quality of the approximation, and additionally we obtain new state-of-the-art bounds for some variants of CUR approximations.2010 Mathematics Subject Classification. 15A23,65F30,68P99,68W20.

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Efficient algorithms for cur and interpolative matrix decompositions

Cited by 56 publications

References 20 publications

CUR Decompositions, Similarity Matrices, and Subspace Clustering

CUR Decompositions, Similarity Matrices, and Subspace Clustering

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

Perspectives on CUR decompositions

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