Modern technology often generates data with complex structures in which both response and explanatory variables are matrix-valued. Existing methods in the literature are able to tackle matrix-valued predictors but are rather limited for matrix-valued responses. In this article, we study matrix-variate regressions for such data, where the response Y on each experimental unit is a random matrix and the predictor X can be either a scalar, a vector, or a matrix, treated as non-stochastic in terms of the conditional distribution Y|X. We propose models for matrix-variate regressions and then develop envelope extensions of these models. Under the envelope framework, redundant variation can be eliminated in estimation and the number of parameters can be notably reduced when the matrix-variate dimension is large, possibly resulting in significant gains in efficiency. The proposed methods are applicable to high dimensional settings.
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