LASSO reloaded: a variational analysis perspective with applications to compressed sensing

TL;DR

This paper offers a variational analysis of the LASSO problem, revealing smoothness and sensitivity properties of solutions, with applications to compressed sensing and validation through numerical experiments.

Contribution

It introduces a novel variational analysis framework for LASSO, providing insights into solution smoothness and parameter sensitivity in compressed sensing.

Findings

Optimal value and solutions are Lipschitz continuous with respect to measurements and parameters.

The analysis enhances understanding of LASSO's stability and sensitivity in inverse problems.

Numerical experiments confirm theoretical predictions.

Abstract

This paper provides a variational analysis of the unconstrained formulation of the LASSO problem, ubiquitous in statistical learning, signal processing, and inverse problems. In particular, we establish smoothness results for the optimal value as well as Lipschitz properties of the optimal solution as functions of the right-hand side (or measurement vector) and the regularization parameter. Moreover, we show how to apply the proposed variational analysis to study the sensitivity of the optimal solution to the tuning parameter in the context of compressed sensing with subgaussian measurements. Our theoretical findings are validated by numerical experiments.