Robust Low-Rank Tensor Minimization via a New Tensor Spectral k - Support Norm

Recently, based on a new tensor algebraic framework for third-order tensors, the tensor singular value decomposition (t-SVD) and its associated tubal rank definition have shed new light on low-rank tensor modeling. Its applications to robust image/video recovery and background modeling show promising performance due to its superior capability in modeling cross-channel/frame information. Under the t-SVD framework, we propose a new tensor norm called tensor spectral k-support norm (TSP- $k$ ) by an alternative convex relaxation. As an interpolation between the existing tensor nuclear norm (TNN) and tensor Frobenius norm (TFN), it is able to simultaneously drive minor singular values to zero to induce low-rankness, and to capture more global information for better preserving intrinsic structure. We provide the proximal operator and the polar operator for the TSP- $k$ norm as key optimization blocks, along with two showcase optimization algorithms for medium- and large-size tensors. Experiments on synthetic, image and video datasets in medium and large sizes, all verify the superiority of the TSP- $k$ norm and the effectiveness of both optimization methods in comparison with the existing counterparts.