Collective Support Recovery for Multi-Design Multi-Response Linear Regression

The multi-design multi-response linear regression problem is investigated, in which design matrices are Gaussian with covariance matrices Σ^(1:K) = Σ⁽¹⁾, . . .,Σ^(K) for K linear regression tasks. Design matrices across tasks are assumed to be independent. 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 lassoand via l₁/l₁ + l₁/l_∞-regularized lasso, in which sharp threshold on sample complexity is characterized only for K = 2 and under special conditions. In this paper, using l₁/l₂-regularized lasso, sharp threshold on sample complexity is characterized under standard regularization conditions. Namely, if n > c_p1Ψ(B∗,Σ(1:K)) log( p - s) where c_p1 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 > n > c_p2Ψ(B∗,Σ(1:K)) log( p - s) where c_p2 is a constant, then l₁/l₂-regularized lasso fails to recover the support union. In particular, the function Ψ(B∗,Σ^(1:K)) 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 multitask lasso over individual support recovery using single-task lasso.