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

Abstract

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA (Candes et al. 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin 2011) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor XR^n_1 n_2 n_3 such that X=L_0+E_0, where L_0 has low tubal rank and E_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 _1-norm, i.e., _L,\ E \ \|L\|_*+\|E\|_1, \ s.t. \ X=L+E, where = 1/(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.

Tasks

Reproductions