Enhanced Nonconvex Low-Rank Approximation of Tensor Multi-Modes for Tensor Completion

Higher-order low-rank tensor arises in many data processing applications and has attracted great interests. Inspired by low-rank approximation theory, researchers have proposed a series of effective tensor completion methods. However, most of these methods directly consider the global low-rankness of underlying tensors, which is not sufficient for a low sampling rate; in addition, the single nuclear norm or its relaxation is usually adopted to approximate the rank function, which would lead to suboptimal solution deviated from the original one. To alleviate the above problems, in this paper, we propose a novel low-rank approximation of tensor multi-modes (LRATM), in which a double nonconvex L-gamma norm is designed to represent the underlying joint-manifold drawn from the factorization factors of each mode in the underlying tensor. A block successive upper-bound minimization method-based algorithm is designed to efficiently solve the proposed model, and it can be demonstrated that our numerical scheme converges to the coordinatewise minimizers. Numerical results on three types of public multi-dimensional datasets have tested and shown that our algorithm can recover a variety of low-rank tensors with significantly fewer samples than the compared methods.

Automated summary

This paper proposes a novel low-rank approximation model for tensor multi-modes, addressing the inadequacy of traditional methods in terms of global low rankness of underlying tensors and suboptimal solutions. By utilizing a block successive upper-bound minimization algorithm, the numerical scheme is proven to converge to coordinatewise minimizers, and significant recovery results are achieved on public multi-dimensional datasets.