Computing Diffusion Geometry

TL;DR

This paper introduces a computational framework for diffusion geometry that extends classical calculus and geometry to data, enabling practical analysis of complex spatial phenomena with improved robustness and scalability.

Contribution

It presents new data-driven computational methods for diffusion geometry, broadening its application scope and enhancing precision, robustness, and efficiency.

Findings

Developed methods for vector calculus and Riemannian geometry from data

Applied to solve spatial PDEs and analyze vector fields

Enabled computation of geodesic distances, curvature, and topological features

Abstract

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.