Volume 35, Issue 4
Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data

Yangyang Xu

J. Comp. Math., 35 (2017), pp. 397-422.

Published online: 2017-08

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65F99, 9008, 90C06, 90C26.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yangyang.xu@ua.edu (Yangyang Xu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-397, author = {Xu , Yangyang}, title = {Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {4}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1608-m2016-0641}, url = {http://global-sci.org/intro/article_detail/jcm/10023.html} }
TY - JOUR T1 - Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data AU - Xu , Yangyang JO - Journal of Computational Mathematics VL - 4 SP - 397 EP - 422 PY - 2017 DA - 2017/08 SN - 35 DO - http://doi.org/10.4208/jcm.1608-m2016-0641 UR - https://global-sci.org/intro/article_detail/jcm/10023.html KW - multilinear data analysis, higher-order singular value decomposition (HOSVD), low-rank tensor completion, non-convex optimization, higher-order orthogonality iteration (HOOI), global convergence. AB -

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.

Yangyang Xu. (2020). Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data. Journal of Computational Mathematics. 35 (4). 397-422. doi:10.4208/jcm.1608-m2016-0641
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