# SEARCH

# ISTC-CC NEWSLETTER

# RESEARCH HIGHLIGHTS

Ling Liu's SC13 paper "Large Graph Processing Without the Overhead" featured by HPCwire.

ISTC-CC provides a listing of useful benchmarks for cloud computing.

Another list highlighting Open Source Software Releases.

Second GraphLab workshop should be even bigger than the first! GraphLab is a new programming framework for graph-style data analytics.

# ISTC-CC Abstract

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

*IEEE Transactions on Information Theory, in press, 2014.*

**Weiguang Wang, Yingbin Liang, Eric P. Xing***

Syracuse University, Syracuse, NY

* Carnegie Mellon University

The multivariate multi-response (MVMR) linear regression problem is investigated, in which design matrices are Gaussian with covariance matrices ∑^{(1:K)} =(∑^{(1)} ,...,∑^{(K)}) for *K* linear regressions. The support union of *K* *p*-dimensional regression vectors (collected as columns of matrix *B**) is recovered using *l*_{1}/*l*_{2}-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 *l*_{1}/*l*_{∞}-regularized Lasso in [1] and via* l*_{1}/*l*_{1} + *l*_{1}/*l*_{∞}-regularized Lasso in [2], in which sharp threshold on sample complexity is characterized only for *K *= 2 and under special conditions. In this work, using *l*_{1}/*l*_{2}-regularized Lasso, sharp threshold on sample complexity is characterized under standard regularization conditions. Namely, if *n* > *c*_{p1}𝛙 (B*,∑^{(1:K)}) log(*p* - *s*) where

*c*_{p1} is a constant, and *s* is the size of the support set, then *l*_{1}/*l*_{2}-regularized Lasso correctly recovers the support union; and if n < *c*_{p2}𝛙 (B*,∑^{(1:K)}) log(*p - s*) where *c*_{p2} is a constant, then *l*_{1}/*l*_{2}-regularized Lasso fails to recover the support union. In particular, the function 𝛙 (B*,∑^{(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 multi-task Lasso over individual support recovery using single-task Lasso.

**FULL PAPER: pdf**