# Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory

arXiv: https://arxiv.org/abs/2605.07448 · HTML: https://mlsanity.com/paper/2605.07448/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 60 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 15 of 15)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods](https://mlsanity.com/paper/1602.02915/) (2016)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [Tucker Tensor Regression and Neuroimaging Analysis](https://mlsanity.com/paper/1304.5637/) (2013)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm](https://mlsanity.com/paper/1510.06895/) (2015)
- [Generalized High-Dimensional Trace Regression via Nuclear Norm Regularization](https://mlsanity.com/paper/1710.08083/) (2017)
- [Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance](https://mlsanity.com/paper/2012.10249/) (2020)
- [Convergence rate analysis of a sequential convex programming method with line search for a class of constrained difference-of-convex optimization problems](https://mlsanity.com/paper/2001.06998/) (2020)
- [Towards Faster Rates and Oracle Property for Low-Rank Matrix Estimation](https://mlsanity.com/paper/1505.04780/) (2015)
- [Calculus rules of the generalized concave Kurdyka-L ojasiewicz property](https://mlsanity.com/paper/2110.03795/) (2021)

## Abstract (arXiv)

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.