Majorize-minimize algorithm for multiresponse sparse regression

Multiresponse sparse regression is the problem of estimating many response variables using a common subset of input variables. Our model is linear, so row sparsity of the coefficient matrix implies subset selection. This is formulated as the problem of minimizing the residual sum of squares, where the row norms of the coefficient matrix are penalized. The proposed approach differs from existing ones in that any penalty function that is increasing, differentiable, and concave can be used. A convergent majorize-minimize algorithm is adopted for minimization. We also propose an active set strategy for tracking the nonzero rows of the coefficient matrix when the minimization is performed for a sequence of descending values of the penalty parameter. Numerical experiments are given to illustrate the active set strategy and analyze penalization with different degrees of concavity.