Title: Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm,
Abstract: In this paper, we consider the Tensor Robust Principal Component Analysis
(TRPCA) problem, which aims to exactly recover the low-rank and sparse
components from their sum. Our model is based on the recently proposed
tensor-tensor product (or t-product) [13]. Induced by the t-product, we first
rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor
average rank, and show that the tensor nuclear norm is the convex envelope of
the tensor average rank within the unit ball of the tensor spectral norm. These
definitions, their relationships and properties are consistent with matrix
cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA
problem by solving a convex program and provide the theoretical guarantee for
the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA
as a special case. Numerical experiments verify our results, and the
applications to image recovery and background modeling problems demonstrate the
effectiveness of our method.