Robust Gradient Descent Estimation for Tensor Models under Heavy-Tailed Distributions

2024/12/06 by Xiaoyu Zhang, Zhang, Xiaoyu, Di Wang +5 · 1 citation
Engineering · Mathematics · Computer Science · #Elasticity and Material Modeling #Tensor decomposition and applications #Medical Image Segmentation Techniques

paper · pdf · doi:10.48550/arxiv.2412.04773

Abstract

Low-rank tensor models are widely used in statistics. 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.

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• High-dimensional low-rank matrix regression with unknown latent structures

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