Sparse learning with concave regularization: relaxation of the irrepresentable condition

TL;DR

This paper investigates a concave regularization method for sparse linear regression, demonstrating it relaxes the irrepresentable condition required by Lasso, thus potentially reducing the number of measurements needed for accurate feature selection.

Contribution

It introduces a non-convex concave regularization approach that relaxes the irrepresentable condition, improving sparse model selection over traditional Lasso.

Findings

Concave regularization relaxes the irrepresentable condition.

Reduced number of measurements needed compared to Lasso.

Numerical experiments confirm improved feature selection.

Abstract

Learning sparse models from data is an important task in all those frameworks where relevant information should be identified within a large dataset. This can be achieved by formulating and solving suitable sparsity promoting optimization problems. As to linear regression models, Lasso is the most popular convex approach, based on an $\ell_1$-norm regularization. In contrast, in this paper, we analyse a concave regularized approach, and we prove that it relaxes the irrepresentable condition, which is sufficient and essentially necessary for Lasso to select the right significant parameters. In practice, this has the benefit of reducing the number of necessary measurements with respect to Lasso. Since the proposed problem is non-convex, we also discuss different algorithms to solve it, and we illustrate the obtained enhancement via numerical experiments.