Structured sparsity through convex optimization

TL;DR

This paper introduces a convex optimization framework for structured sparsity, extending traditional sparse estimation methods to incorporate prior structural knowledge for improved data and model representation.

Contribution

It develops structured norms based on groupings of variables, enabling flexible sparsity patterns in both supervised and unsupervised learning tasks.

Findings

Effective structured sparsity norms for overlapping groups

Applications to sparse PCA and hierarchical dictionary learning

Enhanced variable selection with structural priors

Abstract

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the $\ell_1$-norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the $\ell_1$-norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of non-linear variable selection.