# Computing Diffusion Geometry

arXiv: https://arxiv.org/abs/2602.06006 · HTML: https://mlsanity.com/paper/2602.06006/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2026.

## Built on by (most cited first, 7 of 7)

- [Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior](https://mlsanity.com/paper/2605.05115/) (2026)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [Diffusion Maps is not Dimensionality Reduction](https://mlsanity.com/paper/2603.28037/) (2026)
- [Neural Point-Forms](https://mlsanity.com/paper/2605.15524/) (2026)
- [Disentanglement Beyond Generative Models with Riemannian ICA](https://mlsanity.com/paper/2605.22531/) (2026)

## Abstract (arXiv)

Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates classical calculus and geometry in terms of a diffusion process, allowing these theories to generalise beyond manifolds and be computed from data. This work introduces a new computational framework for diffusion geometry that substantially broadens its practical scope and improves its precision, robustness to noise, and computational complexity. We present a range of new computational methods, including all the standard objects from vector calculus and Riemannian geometry, and apply them to solve spatial PDEs and vector field flows, find geodesic (intrinsic) distances, curvature, and several new topological tools like de Rham cohomology, circular coordinates, and Morse theory. These methods are data-driven, scalable, and can exploit highly optimised numerical tools for linear algebra.