We consider the problem of estimating a sparse multi-response
regression function, with an application to expression
quantitative trait locus (eQTL) mapping, where the goal is to
discover genetic variations that influence gene-expression
levels. In particular, we investigate a shrinkage technique
capable of capturing a given hierarchical structure over the
responses, such as a hierarchical clustering tree with leaf
nodes for responses and internal nodes for clusters of related
responses at multiple granularity, and we seek to leverage this
structure to recover covariates relevant to each
hierarchically-defined cluster of responses. We propose a
tree-guided group lasso, or tree lasso, for estimating
such structured sparsity under multi-response regression by
employing a novel penalty function constructed from the tree. We
describe a systematic weighting scheme for the overlapping
groups in the tree-penalty such that each regression coefficient
is penalized in a balanced manner despite the inhomogeneous
multiplicity of group memberships of the regression coefficients
due to overlaps among groups. For efficient optimization, we
employ a smoothing proximal gradient method that was originally
developed for a general class of structured-sparsity-inducing
penalties. Using simulated and yeast data sets, we demonstrate
that our method shows a superior performance in terms of both
prediction errors and recovery of true sparsity patterns,
compared to other methods for learning a multivariate-response
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