Nonconvex third-order Tensor Recovery Based on Logarithmic Minimax Function

Recent researches have shown that low-rank tensor recovery based non-convex relaxation has gained extensive attention. In this context, we propose a new Logarithmic Minimax (LM) function. The comparative analysis between the LM function and the Logarithmic, Minimax concave penalty (MCP), and Minimax Logarithmic concave penalty (MLCP) functions reveals that the proposed function can protect large singular values while imposing stronger penalization on small singular values. Based on this, we define a weighted tensor LM norm as a non-convex relaxation for tensor tubal rank. Subsequently, we propose the TLM-based low-rank tensor completion (LRTC) model and the TLM-based tensor robust principal component analysis (TRPCA) model respectively. Furthermore, we provide theoretical convergence guarantees for the proposed methods. Comprehensive experiments were conducted on various real datasets, and a comparison analysis was made with the similar EMLCP method. The results demonstrate that the proposed method outperforms the state-of-the-art methods

A dual framework for low-rank tensor completion

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201