  We consider the problem of multivariate regression in a setting where the
relevant predictors could be shared among different responses. We propose an
algorithm which decomposes the coefficient matrix into the product of a long
matrix and a wide matrix, with an elastic net penalty on the former and an
$\ell_1$ penalty on the latter. The first matrix linearly transforms the
predictors to a set of latent factors, and the second one regresses the
responses on these factors. Our algorithm simultaneously performs dimension
reduction and coefficient estimation and automatically estimates the number of
latent factors from the data. Our formulation results in a non-convex
optimization problem, which despite its flexibility to impose effective
low-dimensional structure, is difficult, or even impossible, to solve exactly
in a reasonable time. We specify an optimization algorithm based on alternating
minimization with three different sets of updates to solve this non-convex
problem and provide theoretical results on its convergence and optimality.
Finally, we demonstrate the effectiveness of our algorithm via experiments on
simulated and real data.
