Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X in R^n*n*n_3 such that X=L_0+S_0, where L_0 has low tubal rank and S_0 is sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the l1-norm, i.e., min L,E s.t. ||L||_*+lambda||E||_1 s.t. X=L+E. where lambda=1/sqrtmax(n_1,n_2)n_3. Interestingly, TRPCA involves RPCA as a special case when n_3=1 and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method.