  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.
