The multivariate multi-response (MVMR) linear regression problem is
investigated, in which design matrices can be distributed differently across K
linear regressions. The support union of K p-dimensional regression vectors is
recovered via block regularized Lasso which uses the $l_1/l_2$ norm for
regression vectors across K tasks. Sufficient and necessary conditions to
guarantee successful recovery of the support union are characterized. In
particular, the advantage in sample size for joint support union recovery using
multi-task Lasso over performing each task individually is characterized
analytically and demonstrated numerically.