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While matrix-covariate 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 matrix-valued covariates. To address these issues, this paper proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization term for structured signals, with considerations of models of both continuous and binary responses. We propose an efficient Riemannian-steepest-descent algorithm for regression coefficient estimation. We prove that the consistency of the proposed estimator is in the order of O(sqrt{r(q+m)+p}/sqrt{n}), where r is the rank, p x m is the dimension of the coefficient matrix and p is the dimension of the coefficient vector. When the rank r is small, this rate improves over O(sqrt{qm+p}/sqrt{n}), the consistency of the existing work (Li et al. in Electron J Stat 15:1909-1950, 2021) that does not apply a rank constraint. In addition, we prove that all accumulation points of the iterates have similar estimation errors asymptotically and substantially attaining the minimax rate. We validate the proposed method through a simulated dataset on two-dimensional shape images and two real datasets of brain signals and microscopic leucorrhea images.