# Block Regularized Lasso for Multivariate Multi-Response Linear Regression

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Weiguang Wang,
Yingbin Liang,
Eric Xing
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Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics, PMLR 31:608-617, 2013.

#### Abstract

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 are 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. More specifically, it is shown that under certain conditions on the distributions of design matrices, if n > c_p1 ψ(B^*,Σ^(1:K))\log(p-s) where c_p1 is a constant and s is the size of the support set, then the l_1/l_2 regularized Lasso correctly recovers the support union; and if n < c_p2 ψ(B^*,Σ^(1:K))\log(p-s) where c_p2 is a constant, then the l_1/l_2 regularized Lasso fails to recover the support union. In particular, ψ(B^*,Σ^(1:K)) captures the sparsity of K regression vectors and the statistical properties of the design matrices. Numerical results are provided to demonstrate the advantages of joint support union recovery using multi-task Lasso problem over studying each problem individually.

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