Structured Variable Selection with Sparsity-Inducing Norms

TL;DR

This paper investigates structured sparsity-inducing norms for linear models, introducing algorithms to connect group structures with nonzero patterns, and analyzing variable selection consistency in various dimensions.

Contribution

It develops methods to design structured sparsity norms based on prior knowledge and provides algorithms to identify nonzero patterns efficiently.

Findings

Algorithms linking group structures to nonzero patterns.

Efficient active set algorithm for structured sparsity.

Consistency analysis of variable selection in different dimensions.

Abstract

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual $\ell_1$-norm and the group $\ell_1$-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.