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

arXiv: https://arxiv.org/abs/1905.10443 · HTML: https://mlsanity.com/paper/1905.10443/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2019.

## Built on by (most cited first, 1 of 1)

- [Sketch and shift: a robust decoder for compressive clustering](https://mlsanity.com/paper/2312.09940/) (2023)

## Abstract (arXiv)

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.