Tensor Recovery Based on Tensor Equivalent Minimax-Concave Penalty

Tensor recovery is an important problem in computer vision and machine learning. It usually uses the convex relaxation of tensor rank and $l_{0}$ norm, i.e., the nuclear norm and $l_{1}$ norm respectively, to solve such problem. Convex approximations are known to produce biased estimators. To overcome this problem, a corresponding non-convex regularizer is adopted and designed. Inspired by the recently developed matrix equivalent Minimax-Concave Penalty (EMCP) theorem, a theorem of tensor equivalent Minimax-Concave Penalty (TEMCP) is established in this paper. Tensor equivalent MCP (TEMCP) as the non-convex regularizer part and equivalent weighted tensor $\gamma$ norm (EWTGN) as the low-rank part are constructed, both of which can achieve weight adaptive. Meanwhile, we propose two corresponding adaptive models for two classical tensor recovery problems, namely, low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA), in which the optimization algorithm is based on alternating direction multiplier (ADMM). This novel iterative adaptive algorithm is devised, which can produce more accurate tensor recovery effect. For the tensor completion model, multispectral image (MSI), magnetic resonance imaging (MRI) and color video (CV) data are considered, while for the tensor robust principal component analysis model, hyperspectral image (HSI) denoising under gaussian noise plus salt and pepper noise is considered. The proposed algorithm is superior to the state-of-arts method, and the reduction and convergence of which are guaranteed through experiments.