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

## TL;DR

This paper introduces a tensor p-shrinkage nuclear norm (p-TNN) based on tensor SVD, which better approximates tensor rank for low-rank tensor completion, along with an efficient algorithm and theoretical guarantees.

## Contribution

It proposes a novel tensor p-shrinkage nuclear norm and a low-rank tensor completion model with convergence guarantees, outperforming existing methods.

## Key findings

- p-TNN provides a better approximation of tensor rank than traditional nuclear norm.
- The proposed algorithm converges globally under certain conditions.
- Numerical results show superior performance on synthetic and real data.

## Abstract

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.