Nonconvex Low Tubal Rank Tensor Minimization

In the sparse vector recovery problem, the L
₀
-norm can be approximated by a convex function or a nonconvex function to achieve sparse solutions. In the low-rank matrix recovery problem, the nonconvex matrix rank can be replaced by a convex function or a nonconvex function on the singular value of matrix to achieve low-rank solutions. Although the convex relaxation can easily lead to the optimal solution, the nonconvex approximation tends to yield more sparse or lower rank local solutions. As a natural extension of vector and matrix to high order structure, tensor can better represent the essential structure of data for modeling the high-dimensional data. In this paper, we study the low tubal rank tensor recovery problem by nonconvex optimization. Instead of using convex tensor nuclear norm, we use nonconvex surrogate functions to approximate the tensor tubal rank, and propose a tensor based iteratively reweighted nuclear norm solver. We further provide the convergence analysis of our new solver. Sufficient experiments on synthetic data and real images verify the effectiveness of our new method.