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We introduce a new intrinsic measure of local curvature on point-cloud datacalled diffusion curvature. Our measure uses the framework of diffusion maps,including the data diffusion operator, to structure point cloud data and definelocal curvature based on the laziness of a random walk starting at a point orregion of the data. We show that this laziness directly relates to volumecomparison results from Riemannian geometry. We then extend this scalarcurvature notion to an entire quadratic form using neural network estimationsbased on the diffusion map of point-cloud data. We show applications of bothestimations on toy data, single-cell data, and on estimating local Hessianmatrices of neural network loss landscapes.

Diffusion Curvature for Estimating Local Curvature in High Dimensional Data