Abstract
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.
Highlights
In recent years, low-rank matix minimization has been widely studied and applied in machine learning tasks, such as image and video recovery, image segmentation, motion segmentation, background modeling, etc. [1]–[3]
The matrix completion problem [4] is a known example of matrix rank minimization problem, which aims at recovering low-rank matrix X from incomplete observed matrix M
We study a family of general nonconvex functions which satisfy certain conditions
Summary
Low-rank matix minimization has been widely studied and applied in machine learning tasks, such as image and video recovery, image segmentation, motion segmentation, background modeling, etc. [1]–[3]. The Iteratively Reweighted Nuclear Norm (IRNN) algorithm [9] proposed to solve the general nonconvex nonsmooth low-rank minimization problem and provided the convergence analysis. The work [12] studies a low tubal rank tensor completion problem by convex tensor nuclear norm minimization min X ∗, s.t. P (X ) = P (M). [22] proposed a t-SVD based nonconvex approximation of tensor tubal-rank named partial sum of the tensor nuclear norm (PSTNN). The work [23] proposes a t-SVD based tensor Schatten-p norm, which approximates the tensor tubal rank better than the convex tensor nuclear norm. Their supergradients (see Section II-B) are nonnegative and monotonically decreasing In this work, these nonconvex surrogate functions are extended to tensor model to better approximate the tensor tubal rank.
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