SUPPORT UNION RECOVERY IN HIGH-DIMENSIONAL MULTIVARIATE REGRESSION

• 成果类型: Article
• 署名作者: Obozinski, Guillaume; Wainwright, Martin J.; Jordan, Michael I.
• 署名单位: University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS776
• 发表日期: 2011
• 页码: 1-47
• 关键词: VARIABLE SELECTION; sparsity recovery; model selection; group lasso; Consistency

摘要

In multivariate regression, a K-dimensional response vector is regressed upon a common set of p covariates, with a matrix B* is an element of R-pxK of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the l(1)/l(2) norm is used for support union recovery, or recovery of the set of s rows for which B* is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter theta(n, p, s): = n/[2 psi(B*) log(p - s)]. Here n is the sample size, and psi(B*) is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the K-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences (n, p, s) such that theta(n, p, s) exceeds a critical level theta(u), and fails for sequences such that theta(n, p, s) lies below a critical level theta(l). For the special case of the standard Gaussian ensemble, we show that theta(l) = theta(u) so that the characterization is sharp. The sparsity-overlap function psi(B*) reveals that, if the design is uncorrelated on the active rows, l(1)/l(2) regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of K) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.