# Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds

arXiv: https://arxiv.org/abs/2307.02378 · HTML: https://mlsanity.com/paper/2307.02378/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 5 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows](https://mlsanity.com/paper/1607.08654/) (2016)
- [Balancing Geometry and Density: Path Distances on High-Dimensional Data](https://mlsanity.com/paper/2012.09385/) (2020)
- [Ollivier-Ricci curvature convergence in random geometric graphs](https://mlsanity.com/paper/2008.01209/) (2020)
- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Forman-Ricci flow for change detection in large dynamic data sets](https://mlsanity.com/paper/1604.06634/) (2016)
- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)
- [Approximating geodesics via random points](https://mlsanity.com/paper/1711.06952/) (2017)
- [Continuum Limit of Lipschitz Learning on Graphs](https://mlsanity.com/paper/2012.03772/) (2020)
- [Efficient Weingarten Map and Curvature Estimation on Manifolds](https://mlsanity.com/paper/1905.10725/) (2019)
- [Gromov-Hausdorff limit of Wasserstein spaces on point clouds](https://mlsanity.com/paper/1702.03464/) (2017)

## Built on by (most cited first, 6 of 6)

- [Hardness of Learning Neural Networks under the Manifold Hypothesis](https://mlsanity.com/paper/2406.01461/) (2024)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Neural Feature Geometry Evolves as Discrete Ricci Flow](https://mlsanity.com/paper/2509.22362/) (2025)
- [Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures](https://mlsanity.com/paper/2510.04936/) (2025)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [From Direction to Magnitude: How Multimodal Instruction-Tuning Reorganizes the Geometric Encoding of Identity-Specifying Prompts in Transformer Hidden States](https://mlsanity.com/paper/2607.09842/) (2026)

## Abstract (arXiv)

Let M denote a low-dimensional manifold embedded in Euclidean space and let X=  x1, dots, xn  be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from X and the curvature of the manifold M via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if M has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We discuss applications of the global discrete curvature bounds to contraction properties of heat kernels on graphs, as well as implications for manifold learning from data clouds. In particular, we show that our consistency results allow for estimating the intrinsic curvature of a manifold by first estimating concrete extrinsic quantities.