High-Dimensional Support Union Recovery in Multivariate Regression

Abstract

We study the behavior of block (cid:96)1/(cid:96)2 regularization for multivariate regression, where a K-dimensional response vector is regressed upon a fixed set of p co- variates. The problem of support union recovery is to recover the subset of covariates that are active in at least one of the regression problems. Study- ing this problem under high-dimensional scaling (where the problem parame- ters as well as sample size n tend to infinity simultaneously), our main result is to show that exact recovery is possible once the order parameter given by θ(cid:96)1/(cid:96)2(n, p, s) : = n/[2ψ(B∗) log(p − s)] exceeds a critical threshold. Here n is the sample size, p is the ambient dimension of the regression model, s is the size of the union of supports, and ψ(B∗) is a sparsity-overlap function that measures a combination of the sparsities and overlaps of the K-regression coefficient vectors that constitute the model. This sparsity-overlap function reveals that block (cid:96)1/(cid:96)2 regularization for multivariate regression never harms performance relative to a naive (cid:96)1-approach, and can yield substantial improvements in sample complexity (up to a factor of K) when the regression vectors are suitably orthogonal rela- tive to the design. We complement our theoretical results with simulations that demonstrate the sharpness of the result, even for relatively small problems.