Diffusion Geometry

The diffusion geometry preprint is available on arxiv. The code is on GitHub. The Diffusion Geometry notebooks show the computation in practice.

Diffusion geometry is a framework for geometric and topological data analysis. It uses the Bakry-Emery gamma calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide range of probability spaces.

This lets us extend the classical tools of Riemannian geometry beyond manifolds and apply them to data. We obtain all the vector fields, differential forms, and other tensor spaces, and the differential operators between them, like the gradient.

Function and gradient vector field

We also obtain the second-order objects, like the Hessian.

Hessian and critical points

Diffusion geometry can be used for topological data analysis by computing the spectrum of the Hodge Laplacian on data. This lets us find harmonic forms that represent the ‘holes’ in the data (its cohomology).

Harmonic forms and circular coordinates

I gave these talks on diffusion geometry to AATRN in February 2025 and the Erlangen AI Hub in March 2026.