A DEIM Induced CUR Factorization

Sorensen, Danny C.; Embree, Mark

doi:10.1137/140978430

Cited by 85 publications

(128 citation statements)

References 23 publications

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“…Corresponding rank r Actual error Rank-r STHOSVD error* 0.25 (3, 10, 1) 0.1995 0.1995 0.2 (10, 23, 1) 0.1799 0.1796 0.15 (22,51,5) 0.1421 0.1403 0.1 (32,114,8) 0.0965 0.0946 0.05 (38,237,10) 0.0400 0.0381 0.01 (40,381,10) 0.0057 0.0055 Table 5 A comparison of the adaptive R-STHOSVD algorithm Algorithm 4.2 to the STHOSVD. We first obtained the rank of the core tensor with the requested relative error tolerance from the adaptive algorithm.…”

Section: Error Tolerancementioning

confidence: 99%

Randomized Algorithms for Low-Rank Tensor Decompositions in the Tucker Format

Minster¹,

Saibaba²,

Kilmer³

2020

SIAM Journal on Mathematics of Data Science

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Many applications in data science and scientific computing involve large-scale datasets that are expensive to store and compute with, but can be efficiently compressed and stored in an appropriate tensor format. In recent years, randomized matrix methods have been used to efficiently and accurately compute low-rank matrix decompositions. Motivated by this success, we focus on developing randomized algorithms for tensor decompositions in the Tucker representation. Specifically, we present randomized versions of two well-known compression algorithms, namely, HOSVD and STHOSVD. We present a detailed probabilistic analysis of the error of the randomized tensor algorithms. We also develop variants of these algorithms that tackle specific challenges posed by large-scale datasets. The first variant adaptively finds a low-rank representation satisfying a given tolerance and it is beneficial when the target-rank is not known in advance. The second variant preserves the structure of the original tensor, and is beneficial for large sparse tensors that are difficult to load in memory. We consider several different datasets for our numerical experiments: synthetic test tensors and realistic applications such as the compression of facial image samples in the Olivetti database and word counts in the Enron email dataset.

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Section: Error Tolerancementioning

confidence: 99%

Randomized Algorithms for Low-Rank Tensor Decompositions in the Tucker Format

Minster¹,

Saibaba²,

Kilmer³

2020

SIAM Journal on Mathematics of Data Science

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“…The unitary matrices U, V are obtained by drawing a random Gaussian matrix, and taking its QR factorization. We distinguish between the following cases The first example is adapted from [22], whereas the second and third examples are drawn from [24]. In all the examples, the random matrices were fixed by setting the random seed and we the set the parameter r = 15.…”

Section: Test Matricesmentioning

confidence: 99%

Randomized Subspace Iteration: Analysis of Canonical Angles and Unitarily Invariant Norms

Saibaba¹

2019

SIAM J. Matrix Anal. Appl.

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This paper analyzes the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the exact and the approximate singular subspaces. Second, we derive bounds for the low-rank approximation in any unitarily invariant norm (including the Schatten-p norm). This generalizes the bounds for Spectral and Frobenius norms found in the literature. Third, we present bounds for the accuracy of the singular values. The bounds are structural in that they are applicable to any starting guess, be it random or deterministic, that satisfies some minimal assumptions. Specialized bounds are provided when a Gaussian random matrix is used as the starting guess. Numerical experiments demonstrate the effectiveness of the proposed bounds.Many advances have been made in the analysis of randomized algorithms for low-rank approximations. The analysis typically has two stages: a structural, or deterministic stage, in which minimal assumption about the distribution of the random matrix is made, and a probabilistic stage, in which the distribution of the random matrix is taken into account to derive bounds for expected and tail bounds of the error distribution. As mentioned earlier, existing literature only targets the error in the low-rank representation [11,12]. When the low-rank representation is in the SVD format, it is desirable to understand the quality of the approximate subspaces and the individual singular triplets. This paper aims to fill in some of the missing gaps in the literature by a rigorous analysis of the accuracy of approximate singular values, vectors and subspaces obtained using randomized subspace iteration. This analysis will be beneficial in applications where an analysis beyond the low-rank approximation is desired.We survey the contents and the main contributions of this paper.Canonical angles. We have developed bounds for all the canonical angles between the spaces spanned by the exact and the approximate singular vectors. Several different flavors of bounds are provided:1. The bounds in Section 3.1 relate the canonical angles between the exact and the approximate singular subspaces. Analysis is also provided for unitarily invariant norms of the canonical angles.2. In applications where lower dimensional subspaces are extracted from the approximate singular subspaces, the bounds in Section 3.2 quantifies the accuracy in the extraction process.3. Section 3.2 also presents bounds for the angles between the individual exact and approximate singular vectors, extracted from the appropriate subspaces.

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“…Enrichment may be performed using iterative greedy or Tabu algorithms. Some of them make a link between the DEIM and the CUR factorization [36]. The coefficients in are then computed such that the reduced order model minimizes a given error between the approximation of the matrix and the highfidelity matrix.…”

Section: Cur Low Rank Approximationmentioning

confidence: 99%