Consistent group selection in high-dimensional linear regression

TL;DR

This paper analyzes the group Lasso method for high-dimensional linear regression with grouped covariates, providing conditions for model selection consistency and proposing an adaptive version to improve selection accuracy.

Contribution

It establishes theoretical properties of the group Lasso in high-dimensional settings and introduces an adaptive group Lasso for better variable selection.

Findings

Group Lasso can be estimation consistent but not always selection consistent.

Sufficient conditions are provided for high-probability model selection.

Adaptive group Lasso improves selection accuracy under certain conditions.

Abstract

In regression problems where covariates can be naturally grouped, the group Lasso is an attractive method for variable selection since it respects the grouping structure in the data. We study the selection and estimation properties of the group Lasso in high-dimensional settings when the number of groups exceeds the sample size. We provide sufficient conditions under which the group Lasso selects a model whose dimension is comparable with the underlying model with high probability and is estimation consistent. However, the group Lasso is, in general, not selection consistent and also tends to select groups that are not important in the model. To improve the selection results, we propose an adaptive group Lasso method which is a generalization of the adaptive Lasso and requires an initial estimator. We show that the adaptive group Lasso is consistent in group selection under certain conditions if the group Lasso is used as the initial estimator.