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

arXiv: https://arxiv.org/abs/2110.03623 · HTML: https://mlsanity.com/paper/2110.03623/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2021.

## Built on by (most cited first, 2 of 2)

- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [Small-Gain Nash: Certified Contraction to Nash Equilibria in Differentiable Games](https://mlsanity.com/paper/2512.06791/) (2025)

## Abstract (arXiv)

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 ℓ1 and ℓ_∞ norms. We highlight recent results on explicit and implicit fixed point schemes for non-Euclidean contracting systems.