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Diffusion Curvature for Estimating Local Curvature in High Dimensional Data
Dhananjay Bhaskar · Kincaid MacDonald · Oluwadamilola Fasina · Dawson Thomas · Bastian Rieck · Ian Adelstein · Smita Krishnaswamy

Thu Dec 01 02:00 PM -- 04:00 PM (PST) @ Hall J #514

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.

Author Information

Dhananjay Bhaskar (Yale University)
Kincaid MacDonald (Yale University)
Oluwadamilola Fasina (Yale University)
Dawson Thomas (Yale University)

I am an undergraduate at Yale University studying mathematics and physics. My background includes experimental particle physics and data geometry and topology. Broadly, I am interested in using novel machine learning methods to take advantage of symmetries for experimental physics tasks.

Bastian Rieck (AIDOS Lab, Institute of AI for Health, Helmholtz Munich)
Ian Adelstein (Yale University)
Smita Krishnaswamy (Yale University)

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