Low-Rank Tensor Completion: A Pseudo-Bayesian Learning Approach

Abstract

Low rank tensor completion, which solves a linear inverse problem with the principle of parsimony, is a powerful technique used in many application domains in computer vision and pattern recognition. As a surrogate function of the matrix rank that is non-convex and discontinuous, the nuclear norm is often used instead to derive efficient algorithms for recovering missing information in matrices and higher order tensors. However, the nuclear norm is a loose approximation of the matrix rank, and what is more, the tensor nuclear norm is not guaranteed to be the tightest convex envelope of a multilinear rank. Alternative algorithms either require specifying/tuning several parameters (e.g., the tensor rank), and/or have a performance far from reaching the theoretical limit where the number of observed elements equals the degree of freedom in the unknown low-rank tensor. In this paper, we propose a pseudo-Bayesian approach, where a Bayesian-inspired cost function is adjusted using appropriate approximations that lead to desirable attributes including concavity and symmetry. Although deviating from the original Bayesian model, the resulting non-convex cost function is proved to have the ability to recover the true tensor with a low multilinear rank. A computational efficient algorithm is derived to solve the resulting non-convex optimization problem. We demonstrate the superior performance of the proposed algorithm in comparison with state-of-the-art alternatives by conducting extensive experiments on both synthetic data and several visual data recovery tasks.