# Re-Weighted $\ell_1$ Algorithms within the Lagrange Duality Framework: Bringing Interpretability to Weights

## TL;DR

This paper introduces a novel approach to updating weights in Re-Weighted $ ext{l}_1$ algorithms by interpreting them as Lagrange multipliers, enhancing interpretability while maintaining competitive performance.

## Contribution

The paper proposes a new methodology for weight updates in Re-Weighted $ ext{l}_1$ algorithms, linking weights to Lagrange multipliers for better interpretability.

## Key findings

- Comparable performance to traditional methods
- Effective in noisy linear systems as Re-Weighted LASSO
- Provides interpretability of weights as Lagrange multipliers

## Abstract

We consider an important problem in signal processing, which consists in finding the sparsest solution of a linear system $\Phi x=b$. This problem has applications in several areas, but is NP-hard in general. Usually an alternative convex problem is considered, based on minimizing the (weighted) $\ell_{1}$ norm. For this alternative to be useful, weights should be chosen as to obtain a solution of the original NP-hard problem. A well known algorithm for this is the Re-Weighted $\ell_{1}$, proposed by Cand\`es, Wakin and Boyd. In this article we introduce a new methodology for updating the weights of a Re-Weighted $\ell_{1}$ algorithm, based on identifying these weights as Lagrange multipliers. This is then translated into an algorithm with performance comparable to the usual methodology, but allowing an interpretation of the weights as Lagrange multipliers. The methodology may also be used for a noisy linear system, obtaining in this case a Re-Weighted LASSO algorithm, with a promising performance according to the experimental results.