High-dimensional support union recovery in multivariate regression

We study the behavior of block 1/ 2 regularization for multivariate regression, where a K-dimensional response vector is regressed upon a fixed set of p covariates. The problem of support union recovery is to recover the subset of covariates that are active in at least one of the regression problems. Studying this problem under high-dimensional scaling (where the problem parameters 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 1/ 2 (n, p, s) : = n/[2(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 (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 1/ 2 regularization for multivariate regression never har...

• Type: Conference
• Year: 2008
• Where: NIPS
• Authors: Guillaume Obozinski, Martin J. Wainwright, Michael I. Jordan
• Keywords: Information Technology, Multivariate Regression, NIPS 2008, Sample Size, Sparsity-overlap Function