Computing Diffusion Geometry

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.

• The paper introduces a diffusion-based framework that unifies calculus, geometry, and topology for analyzing noisy, non-manifold data.
• It operationalizes the carré du champ operator through empirical covariances and weak formulations, ensuring robust computation of differential operators.
• The approach achieves significant scalability and efficiency, outperforming traditional persistent homology in both speed and space for complex geometric tasks.

Authoritative Summary of "Computing Diffusion Geometry" (2602.06006)

Introduction and Motivation

The paper "Computing Diffusion Geometry" introduces a computational framework for diffusion geometry, aiming to unify and generalize classical calculus, Riemannian geometry, and differential topology in purely data-driven settings. Traditional approaches to applying geometric and calculus tools to empirical data face significant limitations due to the requirement for manifold structure and issues with differentiability when data are represented as finite samples. Diffusion geometry addresses these constraints by constructing calculus and geometric operators through the statistics of diffusion (notably the heat equation), circumventing both the infinitesimal calculus requirement and the manifold assumption.

The principal operator at the heart of the framework is the carré du champ, which encapsulates the infinitesimal covariance structure of a diffusion process. By leveraging the kernel-induced structure from a Markov process, the framework defines all major objects and operators from vector calculus and geometry, such as gradients, divergence, differential forms, Lie brackets, Hessians, curvature, and even topological invariants like cohomology.

Framework and Methodological Advances

Carré du Champ and Discrete Representations

A notable technical contribution is the operationalization of the carré du champ $\Gamma$ using empirical covariances via a Markov kernel (typically a heat kernel with variable bandwidth) constructed on the finite dataset. The key formula is:

$$\Gamma(f,h)(p) = \lim_{t\to 0} \frac{1}{2t} \mathbb{E}\big[(f(X)-f(p))(h(X)-h(p)) \mid X \sim \mathcal{N}(p,2tI)\big]$$

The approach implements this empirically via (possibly sparse) kernel matrices, with normalization yielding diffusion operators whose stationary distributions encode the data measure. This allows for robust and efficient computation of $\Gamma$, directly applicable to point clouds regardless of manifoldness, dimensional variability, or noise.

Data-Driven Representations of Tensors

All key geometric and calculus objects—functions, vector fields, differential forms, and higher-order tensors—are represented using bases derived from spectral properties (eigenfunctions) of the data-driven diffusion operator, along with immersion coordinates (typically the data's embedding in $\mathbb{R}^d$). The use of frame theory ensures numerical stability and well-posedness of the weak formulations that underpin the transition from continuous to discrete operators. The compressed function spaces inherit a data-driven notion of smoothness, maximizing the accuracy and robustness of differential estimators.

Differential Operators via Weak Formulations

The framework generalizes the Galerkin method and finite element exterior calculus (FEEC) to non-manifold, measure-theoretic domains by recasting all operators (e.g., gradient, divergence, exterior derivative, Lie bracket, Hessian, Laplacian, Levi-Civita connection) as weak formulations. The construction ensures:

• Numerical stability via frame condition bounds,
• Adaptivity to intrinsic data geometry,
• Compatibility with scalable linear algebraic solvers.

The weak formulation machinery extends to enable fast computation of secondary operators such as Hodge Laplacians, necessary for spectral topology.

Empirical and Computational Results

The authors demonstrate the framework across a suite of geometric and topological tasks:

• Partial Differential Equations (PDEs): Meshless solution of PDEs (heat equation, wave equation, vector field flows) on arbitrary point clouds, with robust performance on both manifolds and highly singular, non-manifold data.
• Riemannian Metric and Curvature: Computation of geodesic (intrinsic) distances and—remarkably—the estimation of curvature tensors (including sectional curvature) for general data clouds. Notably, this is the first demonstration of computing classical curvature tensors (Levi-Civita and sectional curvature) for spaces not assumed to be manifolds.
• Topological Data Analysis (TDA): The de Rham cohomology is estimated via the nullspace of Laplacians derived from the data. Harmonic forms corresponding to topological holes (Betti numbers), circular coordinates for parametrizing 1-dimensional cohomology, and Morse-theoretic critical point analysis via Hessians are all tractable.
• Computational Complexity: The methods outperform persistent homology and simplicial-based TDA by several orders of magnitude in time and space complexity, scaling efficiently in both the sample size $n$ and ambient/intrinsic dimension $d$.

Theoretical and Practical Implications

Theoretical Generality

A bold claim: the entire operational machinery of calculus, geometry, and topology can be reconstructed from a diffusion process—emphatically without manifold assumptions or even continuity. This suggests that Riemannian geometry is not a theory only of smooth spaces, but one that can be extended as a statistics of Markov processes on arbitrary measured datasets. The results indicate this is more than a theoretical curiosity, as concrete, robust geometric and topological computations are possible for highly irregular data.

Practical Implications

For applied ML, data analysis, and scientific computing, the framework offers:

• Meshless geometric-computational algorithms that scale beyond the limits of traditional finite element or combinatorial topology tools.
• Robustness to noise and outliers not achieved by persistent homology or classical manifold-based geometry estimators.
• Integration with numerical linear algebra, enabling highly scalable implementations (the authors provide an open-source Python package).

Potential for Future Developments

The explicit Markov/diffusion lens opens the door to probabilistic and stochastic geometric learning—e.g., integrating with diffusion-based generative models, probabilistic geometric priors in deep learning, or uncertainty quantification in geometric statistics. There are open mathematical questions regarding the convergence of these methods on general measure spaces (beyond manifolds), regularization techniques in high-dimensional regimes, and generalizations to non-symmetric Markov processes.

Applications are foreseen in scientific domains where data is not well-modeled by classical geometric spaces: neuroscience (non-manifold brain geometry), cell biology (single-cell RNA data), image analysis (heterogeneous structure), and anywhere geometric regularity cannot be assumed.

Strong Numerical Results and Claims

• Computation of classical curvature tensors such as the sectional curvature on non-manifold data, a feat not previously documented.
• Several orders of magnitude improvement in time and space efficiency over persistent homology for topological statistical analysis, with empirical scaling laws investigated up to $n\approx 10^4$ and high dimensions on standard hardware.
• Robust extraction of topological and geometric invariants (homology, cohomology, Morse indices, etc.) from noisy, non-manifold datasets.

Conclusion

This work provides a comprehensive, technically rigorous foundation for computing calculus, geometry, and topology on data via diffusion-based statistics, with algorithms and numerical analysis that substantially improve upon previous spectral, mesh-based, and combinatorial methods. The unification under the carré du champ operator, combined with frame-theoretic weak formulations, constitutes a powerful approach for geometric statistics on arbitrary data, bridging longstanding gaps between statistical robustness, computational scalability, and theoretical generality in data-driven geometry and topology.

The results indicate that many of the tools traditionally reserved for smooth Riemannian geometry and topology are now computationally accessible for irregular, noisy, and non-manifold data, promising both practical advances and new theoretical directions in geometric data analysis and beyond.