A General Multi-Factor Norm Based Low-Rank Tensor Completion Framework

Low-rank tensor completion aims to recover the missing entries of the tensor from its partially observed data by using the low-rank property of the tensor. Since rank minimization is an NP-hard problem, the convex surrogate nuclear norm is usually used to replace the rank norm and has obtained promising results. However, the nuclear norm is not a tight envelope of the rank norm and usually over-penalizes large singular values. In this paper, inspired by the effectiveness of the matrix Schatten-q norm, which is a tighter approximation of rank norm when 0 < q <  1, we generalize the matrix Schatten-q norm to tensor case and propose a Unitary Transformed Tensor Schatten-q Norm (UTT-Sq) with an arbitrary unitary transform matrix. More importantly, the factor tensor norm surrogate theorem is derived. We prove large-scale UTT-Sq norm (which is nonconvex and not tractable when 0 < q < 1) is equivalent to minimizing the weighted sum formulation of multiple small-scale UTT- $S_{q_{i}}$ (with different qi and qi ≥ 1). Based on this equivalence, we propose a low-rank tensor completion framework using Unitary Transformed Tensor Multi-Factor Norm (UTTMFN) penalty. The optimization problem is solved using the Alternating Direction Method of Multipliers (ADMM) with the proof of convergence. Experimental results on synthetic data, images and videos show that the proposed UTTMFN can achieve competitive results with the state-of-the-art methods for tensor completion.