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In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) isproposed based on tensor singular value decomposition (t-SVD). In particular,it can be proved that p-TNN is a better approximation of the tensor averagerank than the tensor nuclear norm when p < 1. Therefore, by employing thep-shrinkage nuclear norm, a novel low-rank tensor completion (LRTC) model isproposed to estimate a tensor from its partial observations. Statistically, theupper bound of recovery error is provided for the LRTC model. Furthermore, anefficient algorithm, accelerated by the adaptive momentum scheme, is developedto solve the resulting nonconvex optimization problem. It can be furtherguaranteed that the algorithm enjoys a global convergence rate under thesmoothness assumption. Numerical experiments conducted on both synthetic andreal-world data sets verify our results and demonstrate the superiority of ourp-TNN in LRTC problems over several state-of-the-art methods.

Tensor p-shrinkage nuclear norm for low-rank tensor completion