### A TRUST-REGION-BASED ALTERNATING LEAST-SQUARES ALGORITHM FOR TENSOR DECOMPOSITIONS

Fan Jiang1, Deren Han1, Xiaofei Zhang2

1. 1. Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China;
2. Information Technology Department, Chinascope, Nanjing 210023, China
• Received:2017-01-05 Revised:2017-03-20 Online:2018-05-15 Published:2018-05-15
• Supported by:

This research is supported by a project funded by PAPD of Jiangsu Higher Education Institutions and the NSFC grants 11625105, 11371197, 11431002, 11571178.

Fan Jiang, Deren Han, Xiaofei Zhang. A TRUST-REGION-BASED ALTERNATING LEAST-SQUARES ALGORITHM FOR TENSOR DECOMPOSITIONS[J]. Journal of Computational Mathematics, 2018, 36(3): 351-373.

Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposes a tensor as a sum of rank-one tensors, which finds numerous applications in signal processing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is one of the most popular numerical algorithms for solving it. While there have been lots of efforts for enhancing its efficiency, in general its convergence can not been guaranteed.
In this paper, we cooperate the ALS and the trust-region technique from optimization field to generate a trust-region-based alternating least-squares (TRALS) method for CP. Under mild assumptions, we prove that the whole iterative sequence generated by TRALS converges to a stationary point of CP. This thus provides a reasonable way to alleviate the swamps, the notorious phenomena of ALS that slow down the speed of the algorithm. Moreover, the trust region itself, in contrast to the regularization alternating least-squares (RALS) method, provides a self-adaptive way in choosing the parameter, which is essential for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS in[26], which only proved the cluster point of the iterative sequence generated by RALS is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation scheme. We apply our algorithm to the amino acid fluorescence data decomposition from chemometrics, BCM decomposition and rank-(Lr, Lr, 1) decomposition arising from signal processing, and compare it with ALS and RALS. The numerical results show that TRALS is superior to ALS and RALS, both from the number of iterations and CPU time perspectives.

CLC Number:

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