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