# A Laplacian Approach to $\ell_1$-Norm Minimization

## TL;DR

This paper introduces a differentiable Laplacian-inspired reformulation of the basis pursuit problem, enabling new gradient-based methods and analyzing their iteration complexities for solving $ ext{l}_1$ minimization tasks.

## Contribution

It presents a novel Laplacian-based reformulation of $ ext{l}_1$ minimization that facilitates gradient methods and provides complexity bounds for iterative algorithms.

## Key findings

- Derived bounds on iteration complexity for multiplicative weights update scheme.
- Analyzed accelerated gradient scheme complexity for the reformulated problem.
- Connected the approach to IRLS methods, offering new theoretical insights.

## Abstract

We propose a novel differentiable reformulation of the linearly-constrained $\ell_1$ minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based methods for the solution of $\ell_1$ minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The results can be seen as bounds on the complexity of iteratively reweighted least squares (IRLS) type methods of basis pursuit.