Papers › Diffusion Geometry
Diffusion Geometry
Iolo Jones
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery Γ-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide range of probability spaces. We construct statistical estimators for these objects from a sample of data, and so introduce a whole family of new methods for geometric data analysis and computational geometry. This includes vector fields and differential forms on the data, and many of the important operators in exterior calculus. Unlike existing methods like persistent homology and local principal component analysis, diffusion geometry is explicitly related to Riemannian geometry, and is significantly more robust to noise, significantly faster to compute, provides a richer topological description (like the cup product on cohomology), and is naturally vectorised for statistics and machine learning. We find that diffusion geometry outperforms multiparameter persistent homology as a biomarker for real and simulated tumour histology data and can robustly measure the manifold hypothesis by detecting singularities in manifold-like data.
In Syntology Open this paper in Syntology's Atlas, the map of the papers in Syntology's graph and their citations.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections