From Contraction Theory to Fixed Point Algorithms on Riemannian and Non-Euclidean Spaces

TL;DR

This paper reviews recent advances in contraction theory and fixed point algorithms on Euclidean and non-Euclidean spaces, emphasizing Riemannian manifolds and non-Euclidean norms, with applications in optimization and control.

Contribution

It introduces new insights into contraction conditions and fixed point schemes on Riemannian and non-Euclidean spaces, bridging gaps in existing theory.

Findings

Enhanced understanding of Demidovich and Lipschitz conditions

Development of fixed point algorithms for non-Euclidean spaces

Application of weak pairings to $\, ext{l}_1$ and $ ext{l}_$ norms

Abstract

The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the relationship between Demidovich conditions, one-sided Lipschitz conditions, and contractivity theorems. We review the standard contraction theory on Euclidean spaces as well as little-known results for Riemannian manifolds. Special emphasis is placed on the setting of non-Euclidean norms and the recently introduced weak pairings for the $\ell_1$ and $\ell_\infty$ norms. We highlight recent results on explicit and implicit fixed point schemes for non-Euclidean contracting systems.