Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence

TL;DR

This paper introduces a nonconvex tensor ring rank minimization method using logdet relaxation, which improves tensor completion accuracy and convergence guarantees over existing convex approaches.

Contribution

It proposes a novel nonconvex relaxation for tensor ring rank and develops an efficient algorithm with proven convergence for tensor completion tasks.

Findings

Outperforms state-of-the-art methods in image and video tensor completion

Provides convergence guarantees for the proposed algorithm

Achieves better visual and quantitative results

Abstract

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. In this paper, we propose to use the logdet-based function as a nonconvex smooth relaxation of the TR rank for tensor completion, which can more accurately approximate the TR rank and better promote the low-rankness of the solution. To solve the proposed nonconvex model efficiently, we develop an alternating direction method of multipliers algorithm and theoretically prove that, under some mild assumptions, our algorithm converges to a stationary point. Extensive experiments on color images, multispectral images, and color videos demonstrate that the proposed method outperforms several state-of-the-art competitors in both visual and quantitative comparison. Key words: nonconvex optimization, tensor ring rank, logdet function, tensor completion, alternating direction method of multipliers.