Practical Alternating Least Squares for Tensor Ring Decomposition

TL;DR

This paper introduces practical ALS-based algorithms for tensor ring decomposition that address intermediate data explosion and stability issues, demonstrating improved efficiency and robustness through theoretical strategies and extensive experiments.

Contribution

It proposes two novel strategies to improve ALS for tensor ring decomposition, enhancing computational efficiency and numerical stability.

Findings

Algorithms outperform standard ALS in speed.

Enhanced stability with QR-based ALS methods.

Validated effectiveness on synthetic and real data.

Abstract

Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low-rank patterns in multidimensional data. A well-known method for TR decomposition is the alternating least squares (ALS). However, it often suffers from the notorious intermediate data explosion issue, especially for large-scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS-based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR-cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms.