Abstract: This study examines the problem of estimating a large coefficient matrix in a multiple response linear regression model when the coefficient matrix could be both of low rank and sparse, in the sense that most nonzero entries are concentrated in a few rows and columns. We are especially interested in high-dimensional settings in which the numbers of predictors and/or response variables can be much larger than the number of observations. We propose a new estimation scheme, and show that it achieves both competitive numerical performance and fast computation. Moreover, we show that (a slight variant of) the proposed estimator simultaneously achieves near optimal nonasymptotic minimax rates of estimation under a collection of squared Schatten norm losses by providing both the error bounds for the estimator and the minimax lower bounds. The effectiveness of the proposed algorithm is also demonstrated using an in vivo calcium imaging data set.

Key words and phrases: Adaptive estimation, dimension reduction, group sparsity, high dimensionality, low rank matrices, minimax rates, neuroimaging, ariable selection.