Sparse Multivariate Factor Regression

TL;DR

This paper introduces a novel multivariate regression method that decomposes the coefficient matrix into latent factors with regularization, enabling simultaneous dimension reduction and coefficient estimation, and automatically determining the number of factors.

Contribution

It proposes a new algorithm for multivariate regression that decomposes the coefficient matrix with elastic net and l1 penalties, including an efficient optimization scheme and theoretical convergence guarantees.

Findings

Effective in simulated data

Demonstrates advantages on real datasets

Automatically estimates number of latent factors

Abstract

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.