# An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds

arXiv: https://arxiv.org/abs/2308.02615 · HTML: https://mlsanity.com/paper/2308.02615/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 8 of 8)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [PCPNET: Learning Local Shape Properties from Raw Point Clouds](https://mlsanity.com/paper/1710.04954/) (2017)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [Geodesic Distance Estimation with Spherelets](https://mlsanity.com/paper/1907.00296/) (2019)
- [Diffusion Curvature for Estimating Local Curvature in High Dimensional Data](https://mlsanity.com/paper/2206.03977/) (2022)

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

- [Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach](https://mlsanity.com/paper/2411.02058/) (2024)
- [Principal Curvatures Estimation with Applications to Single Cell Data](https://mlsanity.com/paper/2502.03750/) (2025)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model](https://mlsanity.com/paper/2601.19567/) (2026)

## Abstract (arXiv)

We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space. Our estimator depends only on the metric structure of the data and not on an embedding in ℝn. We show that the estimator is consistent in the sense that for points sampled from a probability measure on a compact Riemannian manifold, the estimator converges to the scalar curvature as the number of points increases. To justify its use in applications, we show that the estimator is stable with respect to perturbations of the metric structure, e.g., noise in the sample or error estimating the intrinsic metric. We validate our estimator experimentally on synthetic data that is sampled from manifolds with specified curvature.