Sparse multivariate factor regression

Milad Kharratzadeh, Mark Coates

Abstract: We introduce a sparse multivariate regression algorithm which simultaneously performs dimensionality reduction and parameter estimation. We decompose the coefficient matrix into two sparse matrices: a long matrix mapping the predictors to a set of factors and a wide matrix estimating the responses from the factors. We impose an elastic net penalty on the former and an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> penalty on the latter. 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 a greedy optimization algorithm based on alternating minimization 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.