🤖 AI Summary
Traditional calculus and Riemannian geometry struggle to handle non-manifold, noisy real-world data. This work proposes a data-driven framework grounded in diffusion processes that, for the first time, systematically realizes computable formulations of vector calculus and key geometric objects—such as geodesic distances, curvatures, vector field flows, and solutions to partial differential equations—within the paradigm of diffusion geometry. The framework further integrates topological tools from de Rham cohomology and Morse theory. Leveraging efficient numerical linear algebra techniques, it achieves substantial improvements in computational accuracy, noise robustness, and scalability, demonstrating exceptional numerical stability, low computational complexity, and strong robustness across a range of geometric and topological tasks.
📝 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.