  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.
