# Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces

arXiv: https://arxiv.org/abs/2304.05239 · HTML: https://mlsanity.com/paper/2304.05239/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2023.

## Built on by (most cited first, 4 of 4)

- [Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport](https://mlsanity.com/paper/2403.12887/) (2024)
- [Training Infinitely Deep and Wide Transformers](https://mlsanity.com/paper/2605.17660/) (2026)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)
- [Man, Machine, and Mathematics](https://mlsanity.com/paper/2604.27052/) (2026)

## Abstract (arXiv)

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka--Lojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on ℝn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on ℝn.