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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 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 in  and via l1/l1 + l1/l∞-regularized Lasso in , 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 standard regularization conditions. Namely, if n > cp1𝛙 (B*,∑(1:K)) 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*,∑(1:K)) 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*,∑(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