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

We consider the problem of learning a high-dimensional multi-task regression model, under sparsity constraints induced by presence of grouping structures on the input covariates and on the output predictors. This problem is primarily motivated by expression quantitative trait locus (eQTL) mapping, of which the goal is to discover genetic variations in the genome (inputs) that influence the expression levels of multiple co-expressed genes (outputs), either epistatically, or pleiotropically, or both. A structured input-output lasso (SIOL) model based on an intricate l1/l2-norm penalty over the regression coefficient matrix is employed to enable discovery of complex sparse input/output relationships; and a highly efficient new optimization algorithm called hierarchical group thresholding (HiGT) is developed to solve the resultant non-differentiable, non-separable, and ultra high-dimensional optimization problem. We show on both simulation and on a yeast eQTL dataset that our model leads to significantly better recovery of the structured sparse relationships between the inputs and the outputs, and our algorithm significantly outperforms other optimization techniques under the same model. Additionally, we propose a novel approach for efficiently and effectively detecting input interactions by exploiting the prior knowledge available from biological experiments.