Robust low-rank tensor regression via clipping and Huber loss

TL;DR

This paper introduces a robust low-rank tensor regression method using truncation and Huber loss, achieving optimal statistical error and sample complexity under finite second-moment noise, with demonstrated practical advantages.

Contribution

It develops a theoretically optimal robust tensor regression framework combining truncation and Huber loss, outperforming traditional least squares in noisy settings.

Findings

Estimator achieves optimal error rate under finite second-moment noise.

Sample complexity for tensor recovery is optimal.

Numerical experiments confirm robustness and phase transition phenomena.

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.