  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.
