A randomized tensor singular value decomposition based on the t‐product

Zhang, Jiani; Saibaba, Arvind K.; Kilmer, Misha E.; Aeron, Shuchin

doi:10.1002/nla.2179

Cited by 63 publications

(67 citation statements)

References 46 publications

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“…An error bound for Algorithm 2.1 in the Frobenius norm is presented below, and we will use this result frequently in our analysis. This theorem and its proof can be found in [46,Theorem 3].…”

Section: Randomized Svdmentioning

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: Randomized Svdmentioning

confidence: 99%

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

Minster¹,

Saibaba²,

Kilmer³

2020

SIAM Journal on Mathematics of Data Science

Self Cite

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“…Let us denote A the tensor obtained by applying the DCT on all the tubes of the tensor A. This operation and its inverse are implemented in the Matlab by the commands dct and idct as A = dct(A, [ ], 3), and idct( A, [ ], 3) = A,…”

Section: Definitions and Properties Of The Cosine Productmentioning

confidence: 99%

“…In the matrix case, it involves the computation of eigenvalues or singular decompositions. In the tensor case, even though various factorization techniques have been developed over the last decades (high-order SVD (HOSVD), Candecomp-Parafac (CP) and Tucker decomposition), the recent tensor SVDs (t-SVD and c-SVD), based on the use of the tensor t-product or c-products offer a matrix-like framework for third-order tensors, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] for more details on recent work related to tensors and applications. In the present work, we consider third order tensors that could be defined as three dimensional arrays of data.…”

Section: Introductionmentioning

confidence: 99%

A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition

Hached

Jbilou²,

Koukouvinos

et al. 2021

Mathematics

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Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness.

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“…Zhang et al [41] use the tensor SVD to store video sequences efficiently and also to fill in missing entries in video sequences. Zhang et al [39] use a randomized version of the tensor SVD to produce low-rank approximations to matrices. Ren et al [28] define a tensor version of principal component analysis and use it to extract features from hyperspectral images.…”

Section: Related Workmentioning

confidence: 99%

Generalized Visual Information Analysis Via Tensorial Algebra

Liao

Maybank

2020

J Math Imaging Vis

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High-order data are modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are referred to as t-matrices. The t-matrices can be scaled, added and multiplied in the usual way. There are t-matrix generalizations of positive matrices, orthogonal matrices and Hermitian symmetric matrices. With the t-matrix model, it is possible to generalize many well-known matrix algorithms. In particular, the t-matrices are used to generalize the singular value decomposition (SVD), high-order SVD (HOSVD), principal component analysis (PCA), two-dimensional PCA (2DPCA) and Grassmannian component analysis (GCA). The generalized t-matrix algorithms, namely TSVD, THOSVD, TPCA, T2DPCA and TGCA, are applied to low-rank approximation, reconstruction and supervised classification of images. Experiments show that the t-matrix algorithms compare favorably with standard matrix algorithms.

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A randomized tensor singular value decomposition based on the t‐product

Cited by 63 publications

References 46 publications

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

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

A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition

Generalized Visual Information Analysis Via Tensorial Algebra

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