Robust principal component analysis based on tensor train rank and Schatten p-norm

For a given data, robust principal component analysis (RPCA) aims to exactly recover the low-rank and sparse components from it. To date, as the convex relaxations of tensor rank, a number of tensor nuclear norms have been defined and applied to approximate the tensor rank because of their convexity. However, those relaxations may make the solution seriously deviate from the original solution for real-world data recovery. In this paper, we define the tensor Schatten p-norm based on tensor train rank and propose a new model for tensor robust principal component analysis (named TTSp). We solve the proposed model iteratively by using the ADMM algorithm. In addition, a tensor augmentation tool called ket augmentation is introduced to convert lower-order tensors to higher-order tensors to exploit the low-TT-rank structure. We report higher PSNR and SSIM values in numerical experiments to image recovery problems which demonstrate the superiority of our method. Further experiments on real data also illustrate the effectiveness of the proposed method.