Collaborative Hierarchical Sparse Modeling

TL;DR

This paper introduces hierarchical and collaborative sparse modeling frameworks that combine group and feature sparsity, with efficient optimization algorithms and applications like source separation.

Contribution

It proposes the hierarchical Lasso and its collaborative extension, integrating group and feature sparsity with guaranteed convergence optimization methods.

Findings

Hierarchical Lasso captures structured sparsity effectively.

Collaborative model shares group sparsity across signals.

Optimization guarantees convergence to the global optimum.

Abstract

Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is done by solving an l_1-regularized linear regression problem, usually called Lasso. In this work we first combine the sparsity-inducing property of the Lasso model, at the individual feature level, with the block-sparsity property of the group Lasso model, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the hierarchical Lasso, which shows important practical modeling advantages. We then extend this approach to the collaborative case, where a set of simultaneously coded signals share the same sparsity pattern at the higher (group) level but not necessarily at the lower one. Signals then share the same active groups, or classes, but not necessarily the same active set. This is very well suited for applications such as source separation. An efficient optimization procedure, which guarantees convergence to the global optimum, is developed for these new models. The underlying presentation of the new framework and optimization approach is complemented with experimental examples and preliminary theoretical results.