Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

Abstract

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.

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Text

Bhaskar et al. "Diffusion Curvature for Estimating Local Curvature in High Dimensional Data." Neural Information Processing Systems, 2022.

Markdown

[Bhaskar et al. "Diffusion Curvature for Estimating Local Curvature in High Dimensional Data." Neural Information Processing Systems, 2022.](https://mlanthology.org/neurips/2022/bhaskar2022neurips-diffusion/)

BibTeX

@inproceedings{bhaskar2022neurips-diffusion,
  title     = {{Diffusion Curvature for Estimating Local Curvature in High Dimensional Data}},
  author    = {Bhaskar, Dhananjay and MacDonald, Kincaid and Fasina, Oluwadamilola and Thomas, Dawson and Rieck, Bastian and Adelstein, Ian and Krishnaswamy, Smita},
  booktitle = {Neural Information Processing Systems},
  year      = {2022},
  url       = {https://mlanthology.org/neurips/2022/bhaskar2022neurips-diffusion/}
}