# Recovery and convergence rate of the Frank-Wolfe Algorithm for the m-EXACT-SPARSE Problem

## TL;DR

This paper analyzes the Frank-Wolfe algorithm's ability to efficiently recover sparse signals using a dictionary, proving support-recovery and exponential convergence under certain coherence conditions.

## Contribution

It provides theoretical guarantees for support recovery and exponential convergence of the Frank-Wolfe algorithm in sparse reconstruction problems.

## Key findings

- Atoms recruited are limited to the signal's support.
- Exponential convergence occurs after a certain iteration.
- Support recovery is guaranteed under coherence conditions.

## Abstract

We study the properties of the Frank-Wolfe algorithm to solve the m-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as a sparse linear combination of a predefined set of atoms, called dictionary. We prove that when the signal is sparse enough with respect to the coherence of the dictionary, then the iterative process implemented by the Frank-Wolfe algorithm only recruits atoms from the support of the signal, that is the smallest set of atoms from the dictionary that allows for a perfect reconstruction of y. We also prove that under this same condition, there exists an iteration beyond which the algorithm converges exponentially.