Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

TL;DR

This paper introduces diffusion curvature, a new intrinsic measure of local curvature for high-dimensional point-cloud data, leveraging diffusion maps and neural networks to analyze geometric properties and neural network loss landscapes.

Contribution

It proposes a novel curvature measure based on diffusion maps and extends it to a quadratic form using neural networks, applicable to various data types and neural network analysis.

Findings

Diffusion curvature correlates with volume comparison in Riemannian geometry.

Neural network estimations of the quadratic form effectively analyze local Hessian matrices.

Applications demonstrate usefulness on toy data, single-cell data, and neural network loss landscapes.

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.