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 [4] to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) [14] and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X is an element of R-n1xn2xn3 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 l(1)-norm, i.e., min(L,E) parallel to L parallel to(*) + lambda parallel to E parallel to(1), s.t. X = L + E, where lambda = 1/root max(n1, n2)n3. 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.