Transforms Based Tensor Robust PCA: Corrupted Low-Rank Tensors Recovery via Convex Optimization

Abstract

This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank and the corresponding tensor nuclear norm. Then we solve the TRPCA problem by convex optimization whose objective is a weighted combination of the new tensor nuclear norm and l_1-norm. In theory, we prove that under some incoherence conditions, the convex program exactly recovers the underlying low-rank and sparse components with high probability. Our new TRPCA is much more general since it allows to use any invertible linear transforms. Thus, we have more choices in practice for different tasks and different type of data. Numerical experiments verify our results and the application on image recovery demonstrates the superiority of our method.