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Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data

Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data
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Year:    2017

Author:    Yangyang Xu

Journal of Computational Mathematics, Vol. 35 (2017), Iss. 4 : pp. 397–422

Abstract

Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition.
In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update (BCU). Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI.
In addition, we compare the proposed method to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1608-m2016-0641

Journal of Computational Mathematics, Vol. 35 (2017), Iss. 4 : pp. 397–422

Published online:    2017-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    26

Keywords:    multilinear data analysis higher-order singular value decomposition (HOSVD) low-rank tensor completion non-convex optimization higher-order orthogonality iteration (HOOI) global convergence.

Author Details

Yangyang Xu

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