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 ε R^n1×n2×n3 such that X = L₀ + S₀, where L₀ has low tubal rank and _{S0 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 ||L||* + λ||ε||1, s.t. X = L + ε, where λ = 1/√max(n₁, n₂)n₃. Interestingly, TRPCA involves RPCA as a special case when n₃ = 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.