Abstract
Feature selection is important for many biological studies, especially when the number of available samples is limited (in order of hundreds) while the number of input features is large (in order of millions), such as eQTL (expression quantitative trait loci) mapping, GWAS (genome wide association study) and environmental microbial community study. We study the problem of multiple output regression which leverages the underlying common relationship shared by multiple output features and propose an efficient and accurate approach for feature selection. Our approach considers both intra- and inter-group sparsities. The intergroup sparsity assumes that only small set of input features are related to the output features. The intragroup sparsity assumes that each input features may relate to multiple output features which should have different kinds of sparsity. Most existing methods do not model the intragroup sparsity well by either assuming uniform regularization on each group, i.e. each input feature relates to similar number of output features, or requiring prior knowledge of the relationship of input and output features. By modelling the regression coefficients as a mixture distributions of Laplacian and Gaussian, we can shrink group regression coefficients to be small adaptively and learn the intergroup, intragroup sparsity and shrinkage estimation patterns. Empirical studies on the synthetic and real environmental microbial community datasets show that our model has better predictions on test dataset than existing methods such as Lasso, Elastic Net, dirty model and rMTFL (robust multi-task feature learning). Moreover, by using least angle regression or coordinate descent and projected gradient descent techniques for optimization, we can obtain the optimal regression efficiently. Availability: Software is available from the authors upon request.