Diffusion geometry

The diffusion geometry preprint is available on arxiv. The code is on GitHub.

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.

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).

Diffusion geometry can be used for geometric representation learning and is very robust to noise. The following ABM simulations are by Josh Bull, and model the infiltration of immune cells into a tumour at different speeds (columns) and with different amounts of noise (rows).

We can use persistent homology or diffusion geometry as a biomarker for this process. Colour shows the rate of infiltration: red is fastest (right column above), and blue is slowest (left column above). The persistent homology representations (top two rows below) are not robust to noise and break down as the noise level increases. The diffusion geometry representation (bottom row below) is clearer and robust to very large amounts of noise .

I gave this talk on diffusion geometry to AATRN in February 2025: