  While matrix variate regression models have been studied in many existing
works, classical statistical and computational methods for the analysis of the
regression coefficient estimation are highly affected by high dimensional and
noisy matrix-valued predictors. To address these issues, this paper proposes a
framework of matrix variate regression models based on a rank constraint,
vector regularization (e.g., sparsity), and a general loss function with three
special cases considered: ordinary matrix regression, robust matrix regression,
and matrix logistic regression. We also propose an alternating projected
gradient descent algorithm. Based on analyzing our objective functions on
manifolds with bounded curvature, we show that the algorithm is guaranteed to
converge, all accumulation points of the iterates have estimation errors in the
order of $O(1/\sqrt{n})$ asymptotically and substantially attaining the minimax
rate. Our theoretical analysis can be applied to general optimization problems
on manifolds with bounded curvature and can be considered an important
technical contribution to this work. We validate the proposed method through
simulation studies and real image data examples.
