Diffusion Curvature for Estimating Local Curvature in High Dimensional DataDownload PDF

Published: 31 Oct 2022, 18:00, Last Modified: 12 Oct 2022, 16:19NeurIPS 2022 AcceptReaders: Everyone
Keywords: local curvature estimation, neural network loss landscape, diffusion
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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