Union Support Recovery in High-Dimensional Multivariate Regression
CALIFORNIA UNIV BERKELEY DEPT OF STATISTICS
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In the problem of multivariate regression, a K-dimensional response vector is regressed upon a common set of p covariates, with a matrix of regression coefficients. We study the behavior of the group Lasso using l1l2 regularization for the union support problem, meaning that the set of s rows for which B is non-zero is recovered exactly. Studying this problem under high-dimensional scaling, we show that group Lasso recovers the exact row pattern with high probability over the random design and noise for scalings of such that the sample complexity parameter exceeds a critical threshold. Here n is the sample size, p is the ambient dimension of the regression model, s is the number of non-zero rows, 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.
- Statistics and Probability