Robust low-rank tensor regression via clipping and Huber loss

Abstract

In this paper, we construct a parameter estimation framework for robust low-rank tensor regression based on a truncation method and Huber loss, specifically focusing on models with random noise having only finite second-order moments. Through a robust gradient descent method, our proposed Huber-type estimator is theoretically optimal in two aspects: (1) its statistical error rate matches the optimal upper bound established for the traditional least squares method under sub-Gaussian error; and (2) the sample complexity for recovering the tensor parameter is also optimal. Extensive numerical experiments demonstrate the robustness of our estimator, indicating that the utilization of truncation and Huber loss significantly enhances stability and statistical effectiveness, outperforming the traditional least squares method. Additionally, the phenomenon of phase transition in the convergence rate of the proposed estimator is confirmed through simulation. Furthermore, applications to image recovery and the Beijing air-quality dataset demonstrate the practical effectiveness of our method.