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.

翻译：张量回归是张量数据分析的重要工具，但现有工作未考虑异常值的影响，使其对此类数据点可能较为敏感。本文提出一种低管秩稳健回归方法，用于分析具有重尾随机噪声的高维张量数据。该方法基于张量管秩的非凸松弛，在通用优化框架下同时允许损失函数和惩罚函数的非凸性。我们开发了一种可实现的估计算法，并在温和假设下证明了其全局收敛性。此外，我们提供了关于驻点的一般统计理论，包括收敛速率和预测误差界。这些理论结果涵盖多个重要模型，如线性模型、广义线性模型和Huber回归，甚至包含某些非凸损失（如相关熵和最小距离准则诱导的损失）。通过数值模拟和应用研究，我们提供了支持性的数值证据。