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

for K linear regressions. The support union of K p-dimensional regression vectors (collected as columns of matrix B∗) is recovered using l1/l2-regularized Lasso. Sufficient and necessary conditions on sample complexity are characterized as a sharp threshold to guarantee successful recovery of the support union. This model has been previously studied via l1/l∞regularized Lasso by Negahban & Wainwright (2011) and via l1/l1 + l1/l∞-regularized Lasso by Jalali et al. (2010), in which sharp threshold on sample complexity is characterized only for K = 2 and under special conditions. In this work, using l1/l2-regularized Lasso, sharp threshold on sample complexity is characterized under only standard regularization conditions. Namely, if n > cp1ψ(B,Σ) log(p − s) where cp1 is a constant, and s is the size of the support set, then l1/l2-regularized Lasso correctly recovers the support union; and if n < cp2ψ(B ,Σ) log(p − s) where cp2 is a constant, then l1/l2-regularized Lasso fails to recover the support union. In particular, the function ψ(B,Σ) captures the impact of the sparsity of K regression vectors and the statistical properties of the design matrices on the threshold on sample complexity. Therefore, such threshold function also demonstrates the advantages of joint support union recovery using multi-task Lasso over individual support recovery using single-task Lasso.