The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence

Yuning Yang

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arXiv · 1911.10921

The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence

Yuning Yang

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Abstract

The epsilon alternating least squares ($ε$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, \emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $ε$-ALS exhibits a promising performance in terms of efficiency and effectiveness.

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Yuning Yang. 2019-12-05. The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence. https://arxiv.org/abs/1911.10921

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