Fast algorithms for sparse reduced-rank regression

We consider a reformulation of Reduced-Rank Regression (RRR) and Sparse Reduced-Rank Regression (SRRR) as a non-convex non-differentiable function of a single of the two matrices usually introduced to parametrize low-rank matrix learning problems. We study the behavior of proximal gradient algorithms for the minimization of the objective. In particular, based on an analysis of the geometry of the problem, we establish that a proximal Polyak-Łojasiewicz inequality is satisfied in a neighborhood of the set of optima under a condition on the regularization parameter. We consequently derive linear convergence rates for the proximal gradient descent with line search and for related algorithms in a neighborhood of the optima. Our experiments show that our formulation leads to much faster learning algorithms for RRR and especially for SRRR.