# Ricci Curvature and the Manifold Learning Problem

arXiv: https://arxiv.org/abs/1410.3351 · HTML: https://mlsanity.com/paper/1410.3351/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2014.

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

- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Model-centric Data Manifold: the Data Through the Eyes of the Model](https://mlsanity.com/paper/2104.13289/) (2021)
- [Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix](https://mlsanity.com/paper/2409.07412/) (2024)
- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)

## Abstract (arXiv)

Consider a sample of n points taken i.i.d from a submanifold Sigma of Euclidean space. We show that there is a way to estimate the Ricci curvature of Sigma with respect to the induced metric from the sample. Our method is grounded in the notions of Carr'e du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis.