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We consider the problem of learning a high-dimensional multi-task regressionmodel, under sparsity constraints induced by presence of grouping structures onthe input covariates and on the output predictors. This problem is primarilymotivated by expression quantitative trait locus (eQTL) mapping, of which thegoal is to discover genetic variations in the genome (inputs) that influencethe expression levels of multiple co-expressed genes (outputs), eitherepistatically, or pleiotropically, or both. A structured input-output lasso(SIOL) model based on an intricate l1/l2-norm penalty over the regressioncoefficient matrix is employed to enable discovery of complex sparseinput/output relationships; and a highly efficient new optimization algorithmcalled hierarchical group thresholding (HiGT) is developed to solve theresultant non-differentiable, non-separable, and ultra high-dimensionaloptimization problem. We show on both simulation and on a yeast eQTL datasetthat our model leads to significantly better recovery of the structured sparserelationships between the inputs and the outputs, and our algorithmsignificantly outperforms other optimization techniques under the same model.Additionally, we propose a novel approach for efficiently and effectivelydetecting input interactions by exploiting the prior knowledge available frombiological experiments.

Structured Input-Output Lasso, with Application to eQTL Mapping, and a Thresholding Algorithm for Fast Estimation