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We consider the problem of estimating a sparse multi-response regressionfunction, with an application to expression quantitative trait locus (eQTL)mapping, where the goal is to discover genetic variations that influencegene-expression levels. In particular, we investigate a shrinkage techniquecapable of capturing a given hierarchical structure over the responses, such asa hierarchical clustering tree with leaf nodes for responses and internal nodesfor clusters of related responses at multiple granularity, and we seek toleverage this structure to recover covariates relevant to eachhierarchically-defined cluster of responses. We propose a tree-guided grouplasso, or tree lasso, for estimating such structured sparsity undermulti-response regression by employing a novel penalty function constructedfrom the tree. We describe a systematic weighting scheme for the overlappinggroups in the tree-penalty such that each regression coefficient is penalizedin a balanced manner despite the inhomogeneous multiplicity of groupmemberships of the regression coefficients due to overlaps among groups. Forefficient optimization, we employ a smoothing proximal gradient method that wasoriginally developed for a general class of structured-sparsity-inducingpenalties. Using simulated and yeast data sets, we demonstrate that our methodshows a superior performance in terms of both prediction errors and recovery oftrue sparsity patterns, compared to other methods for learning amultivariate-response regression.

Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping