Low-M-Rank Tensor Completion and Robust Tensor PCA

Bo Jiang; Shiqian Ma; Shuzhong Zhang

doi:10.1109/JSTSP.2018.2873144

Low-M-Rank Tensor Completion and Robust Tensor PCA

Bo Jiang, Shiqian Ma, Shuzhong Zhang

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12 Scopus citations

Abstract

In this paper, we propose a new approach to solve low-rank tensor completion and robust tensor PCA. Our approach is based on some novel notion of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the connections between these new tensor ranks and the CP-rank and the symmetric CP-rank of an even-order tensor. We show that the M-rank provides a reliable and easy-computable approximation to the CP-rank. As a result, we propose to replace the CP-rank by the M-rank in the low-CP-rank tensor completion and robust tensor PCA. Numerical results suggest that our new approach based on the M-rank outperforms existing methods that are based on low-n-rank, t-SVD, and KBR approaches for solving low-rank tensor completion and robust tensor PCA when the underlying tensor has low CP-rank.

Original language	English (US)
Article number	8477031
Pages (from-to)	1390-1404
Number of pages	15
Journal	IEEE Journal on Selected Topics in Signal Processing
Volume	12
Issue number	6
DOIs	https://doi.org/10.1109/JSTSP.2018.2873144
State	Published - Dec 2018
Externally published	Yes

Bibliographical note

Funding Information:
The work of B. Jiang was supported in part by the National Natural Science Foundation of China under Grants 11771269 and 11831002 and in part by the Program for Innovative Research Team of Shanghai University of Finance and Economics. The work of S. Ma was supported by a startup package in the Department of Mathematics at UC Davis. The work of S. Zhang was supported in part by the National Science Foundation under Grant CMMI-1462408 and in part by the Shenzhen Fundamental Research Fund under Grant KQTD2015033114415450. The guest editor coordinating the review of this paper and approving it for publication was Prof. Thierry Bouwmans.

Funding Information:
Manuscript received April 13, 2018; revised August 3, 2018; accepted September 26, 2018. Date of publication October 1, 2018; date of current version December 17, 2018. The work of B. Jiang was supported in part by the National Natural Science Foundation of China under Grants 11771269 and 11831002 and in part by the Program for Innovative Research Team of Shanghai University of Finance and Economics. The work of S. Ma was supported by a startup package in the Department of Mathematics at UC Davis. The work of S. Zhang was supported in part by the National Science Foundation under Grant CMMI-1462408 and in part by the Shenzhen Fundamental Research Fund under Grant KQTD2015033114415450. The guest editor coordinating the review of this paper and approving it for publication was Prof. Thierry Bouwmans. (Corresponding author: Shiqian Ma.) B. Jiang is with the Research Institute for Interdisciplinary Sciences, School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China (e-mail:,isyebojiang@gmail.com).

Publisher Copyright:
© 2018 IEEE.

Keywords

CP-rank
Low-rank tensor completion
M-rank
Tucker-rank
matrix unfolding
robust tensor PCA

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title = "Low-M-Rank Tensor Completion and Robust Tensor PCA",

abstract = "In this paper, we propose a new approach to solve low-rank tensor completion and robust tensor PCA. Our approach is based on some novel notion of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the connections between these new tensor ranks and the CP-rank and the symmetric CP-rank of an even-order tensor. We show that the M-rank provides a reliable and easy-computable approximation to the CP-rank. As a result, we propose to replace the CP-rank by the M-rank in the low-CP-rank tensor completion and robust tensor PCA. Numerical results suggest that our new approach based on the M-rank outperforms existing methods that are based on low-n-rank, t-SVD, and KBR approaches for solving low-rank tensor completion and robust tensor PCA when the underlying tensor has low CP-rank.",

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note = "Funding Information: The work of B. Jiang was supported in part by the National Natural Science Foundation of China under Grants 11771269 and 11831002 and in part by the Program for Innovative Research Team of Shanghai University of Finance and Economics. The work of S. Ma was supported by a startup package in the Department of Mathematics at UC Davis. The work of S. Zhang was supported in part by the National Science Foundation under Grant CMMI-1462408 and in part by the Shenzhen Fundamental Research Fund under Grant KQTD2015033114415450. The guest editor coordinating the review of this paper and approving it for publication was Prof. Thierry Bouwmans. Funding Information: Manuscript received April 13, 2018; revised August 3, 2018; accepted September 26, 2018. Date of publication October 1, 2018; date of current version December 17, 2018. The work of B. Jiang was supported in part by the National Natural Science Foundation of China under Grants 11771269 and 11831002 and in part by the Program for Innovative Research Team of Shanghai University of Finance and Economics. The work of S. Ma was supported by a startup package in the Department of Mathematics at UC Davis. The work of S. Zhang was supported in part by the National Science Foundation under Grant CMMI-1462408 and in part by the Shenzhen Fundamental Research Fund under Grant KQTD2015033114415450. The guest editor coordinating the review of this paper and approving it for publication was Prof. Thierry Bouwmans. (Corresponding author: Shiqian Ma.) B. Jiang is with the Research Institute for Interdisciplinary Sciences, School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China (e-mail:,isyebojiang@gmail.com). Publisher Copyright: {\textcopyright} 2018 IEEE.",

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Low-M-Rank Tensor Completion and Robust Tensor PCA

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