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

08/14/2017 ∙ by Canyi Lu, et al. ∙ 0

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA RPCA to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) kilmer2011factorization and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X∈R^n_1× n_2× 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 ℓ_1-norm, i.e., _L,EL_*+λE_1, s.t.X=L+E, where λ= 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.

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