In recent years, promising statistical modeling approaches to tensor data analysis have been rapidly developed. Traditional multivariate analysis tools, such as multivariate regression and discriminant analysis, are generalized from modeling random vectors and matrices to higher-order random tensors. One of the biggest challenges to statistical tensor models is the non-Gaussian nature of many real-world data. Unfortunately, existing approaches are either restricted to normality or implicitly using least squares type objective functions that are computationally efficient but sensitive to data contamination. Motivated by this, we adopt a simple tensor t-distribution that is, unlike the commonly used matrix t-distributions, compatible with tensor operators and reshaping of the data. We study the tensor response regression with tensor t-error, and develop penalized likelihood-based estimation and a novel one-step estimation. We study the asymptotic relative efficiency of various estimators and establish the one-step estimator's oracle properties and near-optimal asymptotic efficiency. We further propose a high-dimensional modification to the one-step estimation procedure and show that it attains the minimax optimal rate in estimation. Numerical studies show the excellent performance of the one-step estimator.

翻译：近年来，针对张量数据处理的统计建模方法迅速发展。传统多变量分析工具（如多变量回归和判别分析）已从随机向量和矩阵的建模推广至高阶随机张量。统计张量模型面临的最大挑战之一是许多真实数据的非高斯性。遗憾的是，现有方法要么局限于正态分布假设，要么隐式使用最小二乘类目标函数——这类函数计算高效但对数据污染敏感。受此启发，我们采用一种简单的张量t分布，与常用的矩阵t分布不同，该分布与张量算子及数据重塑操作兼容。我们研究了带有张量t误差的张量响应回归，并开发了基于惩罚似然的估计方法及新的一步估计法。我们分析了各估计量的渐近相对效率，建立了一步估计量的Oracle性质及近最优渐近效率。进一步提出高维改进的一步估计过程，证明其达到极小化最优估计率。数值研究表明一步估计量具有优越性能。