Convex–Concave Tensor Robust Principal Component Analysis

Tensor robust principal component analysis (TRPCA) aims at recovering the underlying low-rank clean tensor and residual sparse component from the observed tensor. The recovery quality heavily depends on the definition of tensor rank which has diverse construction schemes. Recently, tensor average rank has been proposed and the tensor nuclear norm has been proven to be its best convex surrogate. Many improved works based on the tensor nuclear norm have emerged rapidly. Nevertheless, there exist three common drawbacks: (1) the neglect of consideration on relativity between the distribution of large singular values and low-rank constraint; (2) the prior assumption of equal treatment for frontal slices hidden in tensor nuclear norm; (3) the missing convergence of whole iteration sequences in optimization. To address these problems together, in this paper, we propose a convex–concave TRPCA method in which the notion of convex–convex singular value separation (CCSVS) plays a dominant role in the objective. It can adjust the distribution of the first several largest singular values with low-rank controlling in a relative way and emphasize the importance of frontal slices collaboratively. Remarkably, we provide the rigorous convergence analysis of whole iteration sequences in optimization. Besides, a low-rank tensor recovery guarantee is established for the proposed CCSVS model. Extensive experiments demonstrate that the proposed CCSVS significantly outperforms state-of-the-art methods over toy data and real-world datasets, and running time per image is also the fastest.