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Sharp Threshold for Multivariate Multi-Response Linear Regression via Block Regularized Lasso

In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using $l_1/l_2$-regularized Lasso. We characterize sufficient and necessary conditions on sample complexity \emph{as a sharp threshold} to guarantee successful recovery of the support union. Namely, if the sample size is above the threshold, then $l_1/l_2$-regularized Lasso correctly recovers the support union; and if the sample size is below the threshold, $l_1/l_2$-regularized Lasso fails to recover the support union. In particular, the threshold precisely captures the impact of the sparsity of regression vectors and the statistical properties of the design matrices on sample complexity. Therefore, the threshold function also captures the advantages of joint support union recovery using multi-task Lasso over individual support recovery using single-task Lasso.