# Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

arXiv: https://arxiv.org/abs/2206.03977 · HTML: https://mlsanity.com/paper/2206.03977/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 38 references that are in the map.
- First posted on arXiv 8 Jun 2022.

## Builds on (most cited first, 6 of 6)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Negative eigenvalues of the Hessian in deep neural networks](https://mlsanity.com/paper/1902.02366/) (2019)
- [Visualizing the PHATE of Neural Networks](https://mlsanity.com/paper/1908.02831/) (2019)
- [Exploring the Geometry and Topology of Neural Network Loss Landscapes](https://mlsanity.com/paper/2102.00485/) (2021)
- [Variational Quantum Classifiers Through the Lens of the Hessian](https://mlsanity.com/paper/2105.10162/) (2021)

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

- [Topological Singularity Detection at Multiple Scales](https://mlsanity.com/paper/2210.00069/) (2022)
- [An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds](https://mlsanity.com/paper/2308.02615/) (2023)
- [Neural FIM for learning Fisher Information Metrics from point cloud data](https://mlsanity.com/paper/2306.06062/) (2023)
- [HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell Data](https://mlsanity.com/paper/2502.07746/) (2025)
- [Sequential Flow Straightening for Generative Modeling](https://mlsanity.com/paper/2402.06461/) (2024)
- [Principal Curvatures Estimation with Applications to Single Cell Data](https://mlsanity.com/paper/2502.03750/) (2025)

## Abstract (arXiv)

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.