We introduce a new intrinsic measure of local curvature on pointcloud 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
pointcloud data. We show applications of both estimations on toy data, singlecell
data and on estimating local Hessian matrices of neural network loss landscapes.
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Neural FIM for learning Fisher information metrics from point cloud data.
Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data  allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM’s utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two singlecell datasets of IPSC reprogramming and PBMCs (immune cells).
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 Award ID(s):
 2047856
 NSFPAR ID:
 10434433
 Date Published:
 Journal Name:
 Proceedings of the 40th International Conference on Machine Learning
 Volume:
 202
 Page Range / eLocation ID:
 98149826
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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