Low-rank tensor models are widely used in statistics and machine learning. However, most existing methods rely heavily on the assumption that data follows a sub-Gaussian distribution. To address the challenges associated with heavy-tailed distributions encountered in real-world applications, we propose a novel robust estimation procedure based on truncated gradient descent for general low-rank tensor models. We establish the computational convergence of the proposed method and derive optimal statistical rates under heavy-tailed distributional settings of both covariates and noise for various low-rank models. Notably, the statistical error rates are governed by a local moment condition, which captures the distributional properties of tensor variables projected onto certain low-dimensional local regions. Furthermore, we present numerical results to demonstrate the effectiveness of our method.

翻译：低秩张量模型在统计学与机器学习中应用广泛。然而，现有方法大多严重依赖于数据服从亚高斯分布的假设。为应对实际应用中常遇到的厚尾分布带来的挑战，本文针对一般低秩张量模型提出了一种基于截断梯度下降的新型鲁棒估计方法。我们建立了所提方法的计算收敛性，并在协变量与噪声均服从厚尾分布的场景下，推导了多种低秩模型的最优统计速率。值得注意的是，统计误差率受局部矩条件控制，该条件刻画了张量变量在特定低维局部区域上投影的分布特性。此外，我们通过数值实验验证了该方法的有效性。