# Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry

## TL;DR

This paper investigates the convergence of the forward-backward algorithm using geometric conditions, extending classical notions to more general sets and infinite-dimensional spaces, with applications in inverse problems and signal processing.

## Contribution

It extends geometric convergence analysis of the forward-backward algorithm to arbitrary sets and infinite dimensions, introducing new inequalities and connections to inverse problem conditions.

## Key findings

- First Lojasiewicz inequality for a quadratic function with a compact operator
- New linear convergence rates for inverse problems with low-complexity priors
- Unified framework connecting geometry and inverse problem conditions

## Abstract

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or {\L}ojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first {\L}ojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.