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Computing Diffusion Geometry

Iolo Jones, David Lanners

arXiv:2602.06006v1math.DGcs.CGmath.AT

TL;DR

The paper addresses the difficulty of applying calculus and Riemannian geometry to finite data and non-manifold spaces. It develops a heat-kernel and carré-du-champ computational framework using weak formulations, then applies it across calculus, geometry, differential equations, and topology. The resulting tools provide scalable data analysis methods, while their scope includes author-identified limitations of combinatorial and spectral approaches.

  • Problem

    Finite data prevent infinitesimal derivative measurements, and classical Riemannian geometry applies only to manifolds, whereas many real datasets have variable dimensions or branching structures.

  • Method

    The paper computes a heat-kernel-based carré du champ on point clouds and uses weak formulations to estimate differential operators and represent geometric and topological objects.

  • Results

    The framework translates vector calculus, Riemannian geometry, geometric analysis, and differential topology into scalable tools for point-cloud data, including equations, intrinsic distances, curvature, and topological methods.

  • Takeaways & Limitations

    Diffusion geometry provides data-driven computational access to calculus, geometry, and topology on spaces that may be non-manifold, noisy, variable-dimensional, or singular.

  • Takeaways & Limitations

    Combinatorial point-cloud models depend sensitively on complex construction and become computationally expensive as the number of data points grows, while the paper notes no formal guarantee that its Hodge-theoretic cohomology connection extends beyond manifolds.

Abstract

from arXiv · show

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.

1 Introduction

Classical calculus and Riemannian geometry are difficult to estimate from finite data and often do not apply to non-manifold spaces. The paper develops a diffusion-geometry framework that translates these theories into scalable data-analysis tools.

  • The problem of data-driven calculus: Finite samples prevent infinitesimal derivative measurements, while most real data violate the manifold assumptions required by classical Riemannian geometry.The paper identifies these as the two major obstacles to data-driven calculus.
  • Diffusion geometry: Diffusion geometry addresses both obstacles by reformulating calculus and geometry through heat diffusion, whose heat kernel can be evaluated on finite data and remains robust to noise.The diffusion process can also be defined on spaces more general than manifolds.
  • Applications: The methods cover spatial PDEs, vector-field flows, geodesic distances, sectional curvature, de Rham cohomology, circular coordinates, and Morse theory.These applications extend computational calculus, geometry, and topology to point-cloud data.
  • Evaluation and dissemination: The framework is presented as scalable and data-driven, with topological-data-analysis computations outperforming comparable persistent-homology methods by several orders of magnitude.The paper releases a Python software package implementing the framework.
  • Computational framework: The framework represents vector fields, differential forms, and tensors using embedding or immersion coordinates and computes them from Markov processes through heat-kernel-based statistics.This representation is intended to improve statistical and computational performance.
  • Computational framework: Weak formulations represent major differential operators and support a meshless system for differential equations and geometric analysis on arbitrary data geometries.The paper also solves geodesic distance equations to recover intrinsic metrics from point clouds.

2 Overview and main concepts

Diffusion geometry constructs calculus objects from a diffusion process and its carré du champ, allowing gradients and vector fields to be estimated directly from point clouds. The same procedure extends beyond manifolds to noisy, variable-dimensional, and singular data.

  • Vector fields: A vector field assigns a direction to each point, and its coordinate representation records the directional derivatives of the coordinate functions.On manifolds, ambient vector fields are projected onto tangent spaces through pullbacks.
  • Diffusion geometry: Diffusion geometry generalises calculus and geometry beyond manifolds by reformulating them in terms of a diffusion process.This reformulation is designed both for general spaces and for computation from data.
  • Carré du champ: The carré du champ encodes pointwise inner products of gradient fields and can be defined from general diffusions, thereby extending the associated geometry.For Euclidean spaces and manifolds, the relevant diffusion is heat flow.
  • Data-driven computation: The data-driven carré du champ is computed by evaluating and row-normalising a heat kernel on the data to obtain a discrete Markov process.Functions on the data are represented as vectors, after which the covariance formula is applied.
  • Gradient vector fields: Gradients are obtained from the carré du champ and measure the speed and direction of fastest increase relative to the underlying spatial geometry.The resulting vector field can be visualised through directional derivatives of coordinate functions.
  • General recipe: The general recipe defines an object using Γ, computes Γ through covariance, and substitutes those computations into the object’s formula.The gradient example verifies agreement with ground truth on sparse manifold samples and produces meaningful results on non-manifold data with noise, variable density, variable dimension, and singularities.

3 Functions, vector fields, forms, and tensors

The framework computes functions, vector fields, differential forms, and tensors from a Markov-chain diffusion using the measure and carrée du champ. It compresses representations with smooth eigenfunctions, supports variable-density data and scalable sparse computations, and improves robustness to noise and precision.

  • 3.1 Carrée du champ and measure from a Markov chain: A Markov chain constructed from an affinity kernel supplies the measure and carrée du champ used to compute functions, vector fields, forms, and tensors.The kernel is row-normalised into transition probabilities, while its row sums estimate the underlying sampling density.
  • 3.1.1 Constructing a Markov chain from a kernel: The kernel bandwidth must balance locality and smoothing: excessively small values produce an identity chain, whereas excessively large values produce nearly uniform transitions.The bandwidth therefore determines whether the Markov chain captures relationships and subtle structure between data points.
  • 3.1.1 Constructing a Markov chain from a kernel: Variable-bandwidth kernels adapt their length scale to local density, improving performance on heterogeneous data; nearest-neighbour heuristics provide a practical bandwidth estimate.Bandwidth increases in low-density regions and decreases in high-density regions, with ρ(pi) commonly based on the kth-nearest-neighbour distance.
  • 3.1.1 Constructing a Markov chain from a kernel: Sparse nearest-neighbour kernels reduce construction from dense O(n^2d) time and O(n^2) space to O(n log n + d) time and O(kn) space.Only k nearest neighbours are retained, yielding O(kn) nonzero entries.
  • 3.1.3 Carrée du champ: The carrée du champ is computed from covariance of functions under the Markov transition distribution, and the covariance formulation yields scalable, robust results on high-dimensional data.For variable bandwidths, the covariance is scaled by the local bandwidth; with constant bandwidth it reduces to the simpler formulation.
  • 3.1.3 Carrée du champ: Mean-centred covariance is slightly more robust to noise and more precise on manifold data than the graph-Laplacian-equivalent carrée du champ.The alternative uses the generator L = diag(ρ(x_i)^−2)(P − I), but the covariance form is preferred for these empirical properties.
  • 3.2 Functions: Rather than retaining all n discrete functions, the framework restricts computations to a smooth subspace because smoothness controls carrée du champ numerical error and reduces downstream complexity.The full function space is R^n, while the compressed space uses a chosen lower dimension n0.
  • 3.2.1 Compressed function space: The compressed function basis consists of the first n0 eigenvectors of the Markov chain, ordered from smoothest to increasingly oscillatory and stored in an n × n0 matrix U.These functions are orthonormal under μ, form a bandlimited representation, and can be computed from a sparse P in O(nn0) time and space.

4 Frame theory and weak formulations

The frame-theoretic formulation makes weak problems stable and convergent under boundedness, density, and matrix-convergence conditions, while exposing important limits for noisy or general spectral truncations.

  • 4.1.2 Weak formulations in Bessel sequences: For Bessel sequences, boundedness of G+ is equivalent to closed range, resolving the two principal obstacles to using G+ in weak formulations.
  • 4.1.3 Frames: A frame ensures every x ∈ H has coefficient representations, enabling numerically stable weak-formulation solutions through a bounded pseudoinverse G+.The stability is quantified by the condition number B/A.
  • 4.3.1 Upper bound: Empirical estimates suggest stable upper bounds for Gram-matrix spectra and weak-formulation vectors as n increases with fixed bandwidth, although self-tuned bandwidth weakens formal guarantees.Using the implementation’s standard method, the observed upper bound remains between 1 and 2.
  • 4.3.2 Lower bound: The lower-bound choice has little practical effect: the condition number can range from 5 to 10^12 because excluded or retained low-norm vectors contribute little.
  • 4.5 Convergence to continuous differential operators: Finite-data spanning sets converge toward continuous operators as sample size grows, bandwidth decreases, and truncation size n1 increases, provided relevant matrix and vector entries converge.
  • 4.5.1 Convergence of weak formulations in truncated frames: Frame truncations are numerically stable and converge to the true solution because their subspaces become dense, yielding ∥x−xk∥→0 as k→∞.The finite approximation xk is the orthogonal projection of x onto Hk.
  • 4.5.2 The problem of stable convergence in noisy truncated frames: Tikhonov regularisation supports truncation convergence because regularisation commutes with frame projection, whereas spectral cutoff can improve conditioning and remain unbiased under its stated threshold condition.Tikhonov has condition number κ(Gk,ϵ) ≤ (B + ϵ)/ϵ, while spectral cutoff has κ(Gk,ϵ) ≤ B/ϵ and aϵ = a for ϵ < A.
  • 4.5.5 Truncation convergence of spectral cutoff: General spectral regularisations lack guaranteed truncation convergence because finite-section spectra may fail to converge to the spectrum of G and can exhibit spectral pollution.

5 Differential operators

The framework computes differential operators on point-cloud data using weak formulations and diffusion-geometry representations. It covers gradients, divergence, exterior derivatives, vector-field actions, and the Hodge Laplacian, enabling differential-topological analysis.

  • Operator construction: Differential operators generalise vector calculus to general spaces and are discretised from diffusion geometry using weak formulations.The framework represents operators as finite matrices acting on coefficient vectors.
  • Vector calculus: The gradient maps functions to vector fields, measuring the direction and rate of steepest increase.Its strong matrix representation is recovered from the weak gradient and the vector-field metric.
  • Vector calculus: Divergence is the negative adjoint of the gradient and measures local expansion or contraction under a vector-field flow.With an orthonormal function basis, it is represented by the negative transpose of the weak gradient.
  • Vector-field actions: Vector fields act on functions as directional-derivative operators represented by square matrices, while the Hessian supports critical-point classification in Morse theory.The same operator framework connects differential actions with topological analysis.
  • Hodge theory: The Hodge Laplacian combines up and down differential energies, and its spectrum supplies harmonic forms for topological data analysis.The discretised operator is self-adjoint and can be handled through a generalised eigenproblem.

6 Differential equations

Diffusion geometry provides mesh-free methods for solving spatial differential equations and computing vector-field flows directly on point clouds. The framework handles heat, wave, damped-wave, and flow equations through matrix operator representations.

  • Motivation: Mesh-based spatial PDE methods are difficult to apply to point clouds, whereas diffusion geometry offers a mesh-free alternative.The method uses diffusion-geometry representations of differential operators rather than a pre-existing mesh.
  • Geometric operators: The Levi-Civita connection recovers expected covariant derivatives for rotational fields, flat-space coordinate fields, and an orthonormal frame on a torus.The torus example correctly recovers the ground-truth covariant derivative.
  • Heat and wave equations: The heat equation is solved by setting the linear operator to the negative Hodge Laplacian and exponentiating its matrix representation.The resulting evolution is visualised on point-cloud and histology data.
  • Heat and wave equations: The wave equation is handled as a second-order system with zero initial speed, while damping adds a friction coefficient γ > 0.The second-order equation is converted into a first-order system on an augmented state vector.
  • Vector-field flows: Vector-field flows are computed by evolving coordinate functions, producing integral curves and a one-parameter family of point transformations.Solving globally for all points avoids the instability of representing individual points as Dirac delta functions.

7 Geometric data analysis

The framework extends diffusion geometry to intrinsic distances and curvature on data, including settings without the manifold assumption. Geodesic distances are estimated through Lipschitz constraints, while curvature is computed intrinsically from the connection.

  • Geodesic distances: Geodesic distance functions are characterised as 1-Lipschitz functions that vanish at a chosen point and maximise their mean value.The discretisation becomes a convex optimisation problem with pointwise quadratic constraints from the carré du champ.
  • Geodesic distances: Because geodesic distance is nonsmooth at its source, the method models it as ambient Euclidean distance plus a non-negative smooth correction.This addresses approximation error caused by the low-frequency compressed function space.
  • Curvature: The framework computes sectional curvature from the Riemann curvature tensor and the Levi-Civita connection.The resulting curvature is illustrated for a sphere, torus, and hyperboloid.
  • Curvature: The presented sectional-curvature method is, to the authors’ knowledge, the first attempt to compute classical curvature tensors without assuming the data form a manifold.Earlier approaches described in the passage either used manifold assumptions or targeted different curvature quantities.

8 Topological data analysis

Diffusion geometry supplies topological tools based on harmonic forms, including de Rham cohomology, circular coordinates, and cup products. These methods aim to distinguish and analyse topology while improving scalability and robustness relative to persistent homology.

  • de Rham cohomology: The Hodge theorem identifies cohomology classes with harmonic forms found from the kernel of the Hodge Laplacian.The authors use this correspondence to motivate cohomology statistics beyond manifolds, while noting that a formal guarantee is unavailable there.
  • Circular coordinates: The first eigenforms of the Hodge Laplacian approximate harmonic forms and provide circular coordinates that parameterise holes in data.The circular-coordinate colour intensity is proportional to the corresponding harmonic form’s pointwise norm.
  • Topological structure: The framework also introduces a cup-product-based comparison that captures topological interactions not encoded by Betti numbers alone.This extends the topological toolkit beyond counting holes.
  • Cup products: The cup product projects wedge products of harmonic representatives onto the harmonic subspace, distinguishing spaces with identical Betti numbers but different topologies.The construction is illustrated by comparing a torus with a sphere carrying two attached circles.
  • Scalability and robustness: Diffusion geometry uses less computational time and memory than persistent homology by several orders of magnitude.Persistent homology is rarely applied beyond datasets of approximately 10^3 points or degrees greater than one.
  • Scalability and robustness: Heat-kernel aggregation makes the resulting harmonic forms robust to large perturbations and outliers, whereas Vietoris-Rips filtrations are more easily distorted.The paper contrasts this behaviour directly in its robustness analysis.

9 Computational complexity and scaling

The framework exposes tunable complexity controls and scales favorably for cohomology computations as dataset size and dimension increase. Compared with persistent homology, diffusion geometry avoids cumulative degree computation and shows degree-independent scaling in n.

  • Complexity controls: Complexity is controlled by nearest-neighbour sparsity k, compressed function-space dimension n0, and coefficient-function count n1.These parameters can be tuned to available hardware and desired numerical precision.
  • Complexity controls: The framework’s main operations exploit sparsity, including the kernel matrix at O(n log(n) + d) and the carré du champ at O(kn).The kernel uses only k-nearest neighbours, while the carré du champ benefits directly from sparsity through k.
  • Comparison with persistent homology: Diffusion geometry can isolate degree k, whereas persistent homology requires computing all degrees 0 through k−1.The comparison concerns de Rham cohomology via the Hodge Laplacian versus Vietoris–Rips persistent homology.
  • Scaling with dataset size: Diffusion geometry’s scaling in dataset size does not depend on degree, while Vietoris–Rips persistent homology has complexity exponential in k.For k = 0, 1, and 2, the diffusion-geometry curves have the same slope, whereas persistent-homology slopes increase with k.
  • Scaling with dataset size: Diffusion geometry computes homology in all degrees in ≈1 second for n = 12,000 on standard hardware.The benchmark uses a torus distribution and consumer hardware, with n1 = 50 for the diffusion-geometry Hodge Laplacian.
  • Scaling with dimension: In dimension-scaling tests, diffusion geometry remains manageable at n = 5,000 through degree 9, with computation taking ≈10 seconds.Its complexity is minimal at extremal degrees because of the combinatorial symmetry of the k-form space, enabling fast inference near k = 0 or the ambient dimension.

10 Background and context

Computational geometry models for point clouds trade off topology, geometric structure, robustness, and scalability. Diffusion geometry addresses these constraints with a diffusion-based carre9 du champ operator, weak formulations, and flexible overcomplete representations.

  • Combinatorial models: Combinatorial models represent functions, vector fields, and forms on discrete structures, but their construction from point clouds is statistically sensitive and computationally expensive.They are primarily topological and cannot support a Riemannian metric because function-vector multiplication is not associative and commutative in these models.
  • Diffusion methods: Diffusion methods use kernels to approximate heat diffusion and the Laplacian, providing statistically robust and computationally efficient analysis of point clouds, including non-manifold data.Their theoretical justification has mainly covered functions on manifolds, despite successful applications to non-manifold data.
  • Spectral exterior calculus: The earlier spectral approach can become numerically unstable when truncated and prohibitively expensive at full resolution, where computing c_ijk costs O(n^4).Truncation compounds multiplication errors and can produce Gram matrices with negative eigenvalues comparable to positive ones.
  • Spectral exterior calculus: Spectral exterior calculus also depends on potentially many eigenfunction gradients, whose ordering can delay representation of small geometric features and increase complexity.A circle with a short attached interval may require many eigenfunctions before variation along the interval is represented.
  • Diffusion geometry: Unlike the spectral approach, the framework treats the carre9 du champ as a general bilinear operator that can be composed for second derivatives and evaluated without relying directly on Laplacian spectral information.This operator also permits alternative choices of immersion for generating the spanning set.
  • Diffusion geometry: The proposed model combines associative and commutative multiplication, weak formulations, an overcomplete spanning set, and a diffusion-based pointwise positive-definite metric.Using data coordinates as an immersion can encode inductive biases, reduce complexity, and improve robustness to noise.

11 Conclusions

The paper applies diffusion geometry computationally by estimating the carre9 du champ from heat kernels on point clouds and using weak formulations for differential operators. This translates vector calculus, Riemannian geometry, geometric analysis, and differential topology into data-driven statistics.

  • The framework computes a carre9 du champ operator on point clouds from a heat kernel and estimates differential operators through weak formulations.
  • Diffusion geometry translates vector calculus, Riemannian geometry, geometric analysis, and differential topology into computational statistics.

Carr´e du champ as infinitesimal covariance

For heat diffusion, the carre9 du champ is obtained as the small-time limit of a scaled covariance of function values under the transition kernel. Taylor expansion and Gaussian moments establish the result for differentiable functions.

  • The same covariance formula extends from heat diffusion to a general Markov semigroup with carre9 du champ operator and transition kernel.The general case is cited to an external proof.
  • Proposition 2.1 characterizes the carre9 du champ for differentiable functions under heat diffusion on R^d.
  • Taylor expansion expresses local function increments through gradients plus higher-order terms, enabling the infinitesimal covariance calculation.
  • The heat-kernel integral of increment products equals an expectation over X drawn from N(p, 2tI).
  • Scaling the expectation by 1/(2t) yields the carre9 du champ limit, and replacing the expectation with covariance preserves that limit.

Symmetric kernels are self-adjoint Markov chains

For a finite Markov chain, row-normalizing a symmetric kernel, reversibility with respect to a measure, and self-adjointness in the corresponding L2 space are equivalent. These properties also make the measure stationary and motivate symmetric kernels computationally.

  • A Markov chain is characterized equivalently by symmetric-kernel row normalization, reversibility, or self-adjointness in L2(A, bc).
  • When these conditions hold, bc is a stationary distribution for the Markov chain.
  • A reversible chain yields a symmetric kernel by defining K_ij = bc_iP_ij, while symmetric-kernel normalization recovers the original transition probabilities.
  • Self-adjointness implies reversibility through the detailed-balance relation P_klbc_l = P_lkbc_k.
  • Symmetric kernels are natural because self-adjointness is theoretically expected and accelerates eigenvector computation.

Operator form of the Hessian

The Hessian is formulated as an operator that maps functions to bilinear operators on vector fields, then discretized into tensor representations for computation.

  • Operator form of the Hessian: The operator-form Hessian maps each function to a bilinear operator on pairs of vector fields.This provides the abstract input-output structure for the Hessian.
  • Operator form of the Hessian: Its discretization is an n × n1d × n1d × n0 4-tensor acting on functions in Rn0.The tensor represents bilinear maps from Rn1d × Rn1d to Rn.
  • Operator form of the Hessian: The implementation first evaluates the Hessian on basis-derived gradient vector fields at a point, producing a higher-order tensor.The intermediate tensor has dimensions n × n1 × d × n1 × d × n0.
  • Operator form of the Hessian: Reshaping the tensor lets functions and vector fields be supplied to compute the Hessian of a function in directions X and Y.The resulting operation returns the directional Hessian evaluation.
  • Operator form of the Hessian: For selected gradient directions, π′j1j2 acts as a linear map from Rn0 to the d × d Hessian matrix at each point.This gives a reduced computation when only H(f)(∇xi, ∇xj) is required.

(1, 1)-tensor form of the Levi-Civita connection.

The Levi-Civita connection is represented as a (1, 1)-tensor that maps a vector field to a linear operator on vector fields, with a tensor discretization providing its strong form.

  • (1, 1)-tensor form of the Levi-Civita connection.: The Levi-Civita connection is interpreted as a linear operator ∇X on vector fields when X is fixed.This expresses the connection as an operator acting on vector fields.
  • (1, 1)-tensor form of the Levi-Civita connection.: In tensor form, the connection maps a vector field X to a linear map from vector fields to vector fields.The discretization uses an n1d × n1d × n1d 3-tensor.
  • (1, 1)-tensor form of the Levi-Civita connection.: The weak connection representation is reshaped into an n1d × n1d × n1d tensor to obtain the strong form.The intermediate weak tensor has dimensions n1 × d × n1 × d × n1 × d.
  • (1, 1)-tensor form of the Levi-Civita connection.: The strong form is obtained by multiplying the weak connection tensor by G1)−1.The resulting tensor is viewed as a map from vector fields in Rn1d to linear operators on Rn1d.
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