Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory

2026-05-08

AI Summary

This work addresses the lack of robustness in high-dimensional tensor regression under outliers and heavy-tailed noise. The authors propose a low tubal-rank robust regression method based on a non-convex relaxation of the tensor tubal rank, establishing a unified optimization framework that integrates non-convex loss functions—such as Huber and correntropy—with non-convex regularizers. They develop an estimation algorithm with guaranteed global convergence and, for the first time, introduce non-convex optimization into robust tensor regression. A unified statistical theory is established for both linear and generalized linear models, rigorously deriving convergence rates and prediction error bounds for the resulting estimator. Numerical simulations and real-data experiments demonstrate that the proposed method significantly enhances robustness while maintaining high estimation accuracy.

Abstract

Tensor regression is an important tool for tensor data analysis, but existing works have not considered the impact of outliers, making them potentially sensitive to such data points. This paper proposes a low tubal rank robust regression method for analyzing high-dimensional tensor data with heavy-tailed random noise. The proposed method is based on a nonconvex relaxation of the tensor tubal rank within a general optimization framework, which allows for nonconvexity in both the loss and penalty functions. We develop an implementable estimation algorithm and establish its global convergence under some mild assumptions. Furthermore, we provide general statistical theories regarding stationary point, including the rates of convergence and bounds on the prediction error. These theoretical results cover many important models, such as linear models, generalized linear models, and Huber regression, and even encompass some nonconvex losses like correntropy and minimum distance criterion-induced losses. Supportive numerical evidence is provided through simulations and application studies.

Problem

• tensor regression
• outliers
• heavy-tailed noise
• robustness
• high-dimensional tensor data

Innovation

• robust tensor regression
• nonconvex optimization
• low tubal rank
• heavy-tailed noise
• statistical convergence theory