Sparse Multivariate Factor Regression
This addresses regression problems where predictors are shared across responses, but it appears incremental as it builds on existing factor-based methods with specific penalties.
The paper tackles multivariate regression with shared predictors by proposing an algorithm that decomposes the coefficient matrix into latent factors, using penalties for sparsity and automatically estimating the number of factors. It demonstrates effectiveness through experiments on simulated and real data, though no concrete numbers are provided in the 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.