Tensor Robust Principal Component Analysis Based on a Two-Layer Tucker Rank Minimization Model

Tensor robust principal component analysis (TRPCA), which aims to remove sparse noise or outliers of high-dimensional data with intrinsic low rank properties, has attracted extensive research and been widely used in various areas. In this paper, we focus on TRPCA based on the Tucker rank. Considering that computing singular value decompositions (SVDs) of all unfolding matrices in the convex relaxation model of TRPCA based on the Tucker rank is highly time consuming, we propose a two-layer TRPCA model (TTRPCA) based on the convex relaxation model. In TTRPCA, we select a mode according to the nuclear norm of all unfolding matrices, and only need to compute SVD of the matrix unfolded along this mode, which can capture more information of the original data with low-rankness compared with other unfolding matrices. Moreover, we establish a generalized nonconvex two-layer TRPCA model (NTRPCA). Unlike existing methods which usually use a specific nonconvex function, NTRPCA uses a class of nonconvex functions to approximate the rank function and the [Formula: see text] norm to more accurately capture the low rank structure and the sparsity. After that, we establish an error bound of the proposed NTRPCA model, which still holds for the TTRPCA model, and give some comparisons of cases using specific nonconvex functions. An alternating direction method of multipliers algorithm with convergence guarantee is then developed to solve the NTRPCA (as well as the TTRPCA) model. Finally, extensive numerical experiments on various datasets demonstrate the superior performance of proposed models in comparison with several state-of-the-art TRPCA methods.