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Abstract:
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Lasso is of fundamental importance in high-dimensional statistics and has been routinely used to regress a response on a high-dimensional set of predictors. In many scientific applications, there exists external information that encodes the predictive power and sparsity structure of the predictors. We develop a new method, called the Structure Adaptive Lasso (SA-Lasso), to incorporate these potentially useful side information into a penalized regression. The basic idea is to translate the external information into different penalization strengths for the regression coefficients. We study the risk properties of the resulting estimator. In particular, we generalize the state evolution framework recently introduced for the analysis of the approximate message-passing algorithm to the SA-Lasso setting. We show that the finite sample risk of the SA-Lasso estimator is consistent with the theoretical risk predicted by the state evolution equation. Our theory suggests that the SA-Lasso with an informative group or covariate structure can significantly outperform the Lasso, the Adaptive Lasso, and the Sparse Group Lasso. This is further confirmed through synthetic and real data analyses.
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