Robust low tubal rank tensor recovery via [formula omitted]E criterion

Abstract

Tensor analysis has received enormous attention as the increasing prevalence of high-dimensional data in science research and engineering application. Tensor recovery is an important and meaningful problem for tensor analysis. It aims to complete a tensor from an observed subset of its entries disturbed by noise. However, classical methods either develop on the second-order statistics or Lasso-type penalty, leading to them not effectively dealing with gross or dense noise effectively. To address such issues, we propose a robust tensor recovery model for simultaneously completing a low tubal rank tensor with complex noise and missing data. Based on tensor–tensor product (t-product), we first develop a tensor factor Frobenius norm to exploit the low tubal rank property which is closely related to tensor nuclear norm and tubal rank. By utilizing the robust L2 criterion, we derive the nonconvex objective function to accommodate the low tubal rank tensor. An implementable alternating minimization algorithm has been developed to estimate the low tubal rank tensor. It is worth noting that our method is able to jointly estimate the precision parameter to capture the hidden complex noise pattern. Furthermore, some convergence properties of the proposed algorithm are presented. A series of numerical experiments are conducted on both synthetic and real-world data to demonstrate the effectiveness and robustness of the proposed approach in comparison with the state-of-the-art methods.