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Computing Diffusion Geometry

Iolo Jones, David Lanners

arXiv:2602.06006v1math.DGcs.CGmath.AT

TL;DR

Estimating calculus from finite point samples is difficult because the points are isolated, and classical Riemannian geometry applies only to manifolds. This paper computes diffusion geometry on point clouds using heat kernels and weak formulations, reporting geometric calculations that recover the ground truth in a torus example.

  • Problem

    Calculus is difficult to estimate from finite point samples, where isolated points prevent infinitesimal measurements, while classical Riemannian geometry applies only to manifolds.

    Source: §1.1 · p. 3

  • Method

    The framework computes a carré du champ operator on point clouds using a heat kernel and estimates differential operators with weak formulations.

    Source: §11 · p. 75

  • Results

    Covariant derivatives of ∇x and ∇y vanish in flat space, while covariant derivatives of an orthonormal frame on a torus recover ground truth.

    Source: §6 · p. 54

  • Takeaways & Limitations

    The framework translates vector calculus, Riemannian geometry, geometric analysis, and differential topology into simple, scalable data-analysis tools.

    Source: §11 · p. 75

    The earlier SEC approach trades multiplication accuracy against cost: low truncation can make geometric expressions inaccurate and numerically unstable, while full truncation costs O(n^4).

    Source: §10.3.3 · p. 74

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

The introduction presents diffusion geometry as a heat-diffusion reformulation for computing calculus and geometry from finite data, including spaces that are not manifolds. It introduces a data-driven computational framework and applications in analysis, geometry, and topology.

1.1 The problem of data-driven calculus

  • Finite samples prevent infinitesimal derivative measurements, and most real data are not manifolds; diffusion geometry uses heat diffusion to compute calculus on general spaces from data.The heat kernel provides a discrete approximation of diffusion that is computationally accessible and robust to noise.Source: §1.1 · p. 3

1.2 Why the “manifold hypothesis” has the wrong name

  • Low-dimensional structure does not imply manifold structure; diffusion geometry aims to extend Riemannian methods to the nonmanifold spaces commonly meant by “data manifolds.”Source: §1.2 · p. 3

1.3 Paper outline

  • On a topological data analysis task, the framework outperforms comparable methods like persistent homology by several orders of magnitude.Source: §1.3 · p. 4
  • The framework uses embedding or immersion coordinates as generators to represent vector fields, differential forms, and tensors, and can be computed from any Markov process.Its recipe directly involves the carré du champ operator and offers a heat-kernel-based data-driven solution.Source: §1.3 · p. 4
  • The paper represents major differential operators through weak formulations and studies their stability.The framework covers central calculus and geometry objects, including functions, vector fields, differential forms, and tensors.Source: §1.3 · p. 4
  • The authors develop a meshless system for differential equations and geometric analysis on arbitrary data geometries.Applications include solving heat and wave equations and computing vector field flows.Source: §1.3 · p. 4
  • The framework recovers intrinsic distances by solving the geodesic distance equation and estimates sectional curvature from non-manifold data using the Levi-Civita connection.Source: §1.3 · p. 4
  • The paper introduces topological data analysis methods for de Rham cohomology, circular coordinates, and Morse theory, with improved computational cost and robustness to noise and outliers versus Vietoris-Rips persistent homology.The methods are reported to require several orders of magnitude less time and space.Source: §1.3 · p. 4

2 Overview and main concepts

Diffusion geometry reformulates classical calculus and geometry in terms of diffusion, extending them beyond manifolds and making them computable from data.

2.1 Vector calculus on Rn and manifolds

  • On manifolds, vector fields are formed by projecting ambient directions onto tangent spaces, while coefficient functions specify their contributions.The same construction applies in R^d, where the coordinate directions are constant unit flows.Source: §2.1 · p. 5, Figure 3 · p. 6

2.2 Diffusion geometry on Rn and manifolds

  • Diffusion geometry generalizes the Riemannian metric using the carré du champ Γ, which can be defined through more general diffusions and Markov processes.The carré du champ captures the inner products of gradient fields and is expressible as the infinitesimal covariance of a diffusion.Source: §2.2 · pp. 5, 7

2.3 Computing the carr´e du champ

  • Given data points, evaluate the diffusion kernel pairwise, row-normalize it into a Markov-chain matrix, and apply the covariance formula to compute Γ.Functions on the data are represented as vectors of their values at the sampled points.Source: §2.3 · p. 7

2.4 Gradient vector fields

  • Diffusion-geometry gradients match ground truth on sampled manifolds and remain meaningful on non-manifold data with noise, variable density and dimension, and singularities.The non-manifold example includes regions of different dimensions and a branching point.Source: §2.4 · p. 7, Figure 4 · p. 8

2.5 General recipe for computing diffusion geometry

  • The general recipe is to define an object using functions and Γ, compute Γ as a covariance, and substitute it into the object's formula.The authors use this procedure to introduce computational methods for calculus, geometry, and topology.Source: §2.5 · p. 7

3 Functions, vector fields, forms, and tensors

The section develops data-driven calculus and geometry within diffusion geometry, using a Markov chain, a measure, and its carré du champ to represent functions, vector fields, forms, and tensors in computationally tractable spaces. It covers their geometric operations, metrics, products, tensor actions on vector fields, and visualisations.

3.1.1 Constructing a Markov chain from a kernel

  • Variable bandwidths adapt the kernel’s length scale to local density, while bandwidth selection must avoid transitions that become either identity-like or uniform.A k-nearest-neighbor distance is a heuristic for the bandwidth function; the implementation estimates density and sets ρ(x) = q(x)^b, following.Source: §3.1.1 · pp. 8–9
  • A kernel on a finite point cloud yields a Markov chain by row-normalizing its affinity matrix, which can be sparsified to reduce computation and storage.The full kernel matrix costs O(n^2d) time and O(n^2) space; retaining only k nearest neighbors gives O(kn) nonzeros and, using a KD-tree, costs O(n log n + d) time and O(kn) space.Source: §3.1.1 · p. 9

3.1.2 Measure

  • The Markov-chain row sums estimate the sampled measure’s density, and their normalization supplies the probability measure used in subsequent computations.For data sampled from a distribution with density q, the row sums yield a kernel density estimate of q.Source: §3.1.2 · p. 9

3.1.3 Carré du champ

  • The covariance-form carré du champ is computed on data-point functions and gives scalable, noise-robust results; mean-centering is reported as more robust and precise than the graph-Laplacian alternative.The covariance formulation was found scalable and robust on high-dimensional data [10]; the comparison reports slightly better noise robustness and more precise manifold-data results for mean-centred covariance than for the graph-Laplacian approach [51].Source: §3.1.3 · pp. 9–10

3.2 Functions; 3.2.1 Compressed function space

  • To avoid the O(n)-dimensional cost of representing every data-point function, the method uses the first n0 eigenvectors of the Markov chain as a smooth, orthonormal compressed basis.These eigenvectors maximize the Markov-chain smoothness energy; directly diagonalizing the chain incorporates sampling density into the basis choice rather than using the alternative variable-bandwidth normalization.Source: §3.2 · p. 10, §3.2.1 · pp. 10–11

3.2.2 Projection into the subspace

  • Functions project into the compressed space by an L2(Rn, µ)-best approximation, but products generally leave that space and must be projected back, potentially losing information.The method computes products directly at cost O(nn0m); the alternative precomputed product tensor used in Spectral Exterior Calculus can be faster for sufficiently many products but introduces compounding projection error in backend expansions.Source: §3.2.2 · p. 11

3.3 Vector fields; 3.3.1 Riemannian metric; 3.3.2 Inner product

  • Vector fields are represented by coefficient functions multiplying gradients of immersion coordinates, and the carré du champ defines their pointwise Riemannian metric.The metric relates gradient inner products to Γ; the global Gram matrix can be computed by a single tensor contraction rather than materializing the full pointwise metric tensor.Source: §3.3 · pp. 11–12, §3.3.1 · p. 13, §3.3.2 · pp. 13, 15

3.4 Differential forms; 3.4.1 1-forms; 3.4.2 Visualising 1-forms; 3.4.3 k-forms

  • One-forms use the same coefficient representation as vector fields through an isometric duality, while higher-degree forms represent volumes and include forms that detect local dimension.One-forms can be visualized through their dual vector fields; k-forms are built from coordinate differentials, and 0-forms are functions, though the method avoids directly working with them when the function and form representations differ.Source: §3.4.1 · p. 16, §3.4.2 · p. 16, §3.4.3 · pp. 16–17

3.4.3 k-forms

  • In 3d, 3-forms measure volume, so they vanish at the boundary of a dense ball where the space loses volume and have more mass at its centre.Source: Figure 9 · p. 17

3.4.4 Riemannian metric

  • The metric on k-forms uses determinant-based signed volumes and compound matrices, extending naturally to the metric of functions; inner products are computed with a Gram matrix.For k = 1, the construction gives the identity matrix as the metric of functions.Source: §3.4.4 · pp. 17, 19, §3.4.5 · p. 19

3.4.6 Wedge product

  • For d = 10 and k = l = 3, roughly 70% of possible wedge-product pairs are zero because only disjoint multi-indices contribute.The wedge product extends multiplication to forms of different degrees and can encode geometric and topological information.Source: §3.4.6 · pp. 19–20

3.4.7 Visualising k-forms

  • A k-form is visualised as a k-dimensional subspace with signed magnitude, computed at each point from a skew-symmetric k-tensor.For example, 2-forms appear as oriented disks and 3-forms as coloured spheres whose size and colour encode magnitude and sign.Source: §3.4.7 · p. 22

3.5 Other tensors

  • Functions, vector fields, and forms are tensor fields; the framework can compute tensors of any degree, with this section focusing on 2-tensors represented by coefficient arrays.General 2-tensors are flattened into vectors of length n1d^2.Source: §3.5 · pp. 22–23

3.5.2 Inner product

  • Symmetric 2-tensors use n1d(d + 1)/2 coordinates, with doubled off-diagonal metric entries; their inner product uses the 2-tensor Gram matrix.The pointwise metric of 2-tensors is represented by a symmetric matrix.Source: §3.5.1 · p. 23, §3.5.2 · p. 24

3.5.3 Action on vector fields

  • A (0, 2)-tensor acts on a pair of vector fields to produce a function, providing an operational interpretation of its tensor type.Source: §3.5.3 · p. 24

3.5.4 Visualising 2-tensors

  • A symmetric 2-tensor can be visualised as an ellipsoid, but this is crude because one colour cannot represent all of its axes’ signed magnitudes.Source: §3.5.4 · p. 24

4 Frame theory and weak formulations … 4.3 Frame bounds for the discrete spanning set

This section uses frame theory and weak formulations to recover coefficients in continuous and discrete spanning sets, examining how frame bounds relate to numerical stability. It derives continuous-frame bounds from pointwise carré du champ conditions and discusses discrete bounds, including spectral pollution and lower-bound selection.

4 Frame theory and weak formulations

  • Taking inner products of the gradient equation with each spanning-set element yields a weak system G(1)X = (∇f)weak, whose load vector is computable from f.The coefficients are obtained by solving the Gram system; numerical stability motivates the subsequent frame analysis.Source: §4 · p. 25

4.1.1 Bessel sequences

  • A Bessel sequence has bounded analysis and synthesis operators, and its Gram operator is self-adjoint, positive, and injective exactly when the spanning set is linearly independent.Overcomplete sets can represent a vector with multiple coefficient sequences.Source: §4.1.1 · p. 26

4.1.2 Weak formulations in Bessel sequences

  • For an overcomplete Bessel sequence, the Gram operator may have a kernel, but its pseudo-inverse gives a solution with minimum ℓ2 norm when one exists.If x = T*a, then Tx lies in ran(G), so applying G+ solves the weak formulation optimally in coefficient norm.Source: §4.1.2 · pp. 26–27
  • For a dense Bessel spanning set, boundedness of G+ is equivalent to closed range and a positive lower frame bound, resolving both instability and existence concerns.A bounded pseudo-inverse prevents perturbations from being magnified uncontrollably and ensures every x has a coefficient representation.Source: §4.1.2 · pp. 27–28

4.1.3 Frames

  • Frame bounds A and B make weak recovery possible and stable for every x, with ∥G+∥≤1/A and condition number B/A; tight frames achieve B/A = 1.The lower bound ensures every x has a coefficient sequence, while the upper bound controls the Gram operator.Source: §4.1.3 · p. 28

4.2 Frame bounds for the continuous spanning set

  • For continuous 1-forms, global frame bounds follow by integrating pointwise bounds for {dxj}, and bounds A, B extend to k-forms as Ak, Bk.The pointwise bounds are determined by the smallest and largest non-zero eigenvalues of the carré du champ matrix.Source: §4.2 · p. 29

4.2.1 Upper frame condition

  • The continuous upper frame bound is controlled by the essential supremum of the sum of the diagonal carré du champ terms.The bound follows because each positive semidefinite matrix’s eigenvalues are bounded above by its trace and each Γ(xi, xi) is bounded.Source: §4.2.1 · p. 30

4.2.2 Lower frame condition

  • A uniform positive lower bound can fail if A(p) approaches zero, but it holds for isometric embeddings and for compact settings with continuous Γ(xi, xj).For isometric embeddings the frame is Parseval; on a compact space, continuity and positivity of A(p) imply infp A(p) > 0.Source: §4.2.2 · p. 30
  • When the exact carré du champ operator is available, the stated conditions suffice to guarantee stable weak-formulation solutions using the spanning set.The passage gives a known manifold as an example where an exact operator may be available.Source: §4.2.2 · p. 30

4.3 Frame bounds for the discrete spanning set

  • Finite-dimensionality makes any spanning set a frame, so the discrete analysis focuses on how its bounds depend on n and whether weak-formulation solutions remain stable.The discrete analogue of the preceding analysis swaps integrals for sums.Source: §4.3 · p. 30
  • The discrete frame calculation relates the global inner product on 1-forms to a measure-weighted sum of pointwise inner products under the Riemannian metric.The lemma's frame inequality has equality when n1 = n.Source: §4.3 · pp. 30–31

4.3.1 Upper bound

  • With fixed bandwidth, the empirical-variance upper-bound estimate is expected to remain stable as n increases, stabilizing the Gram-matrix spectrum and weak-formulation vector norms.The passage links this stability to equations such as (20) remaining stable for all n.Source: §4.3.1 · p. 32
  • Self-tuned bandwidths depend on data density and n, making the stability claim difficult to formalize; restricting each kernel to k nearest neighbours yields a strict upper bound.The strict bound uses the maximum distance from each point to its k nearest neighbours.Source: §4.3.1 · p. 32
  • The implemented upper frame bound stays between 1 and 2 in every case, though it depends on bandwidth selection and the worst outlier point.The passage notes that the bandwidth-selection method is arbitrary, so this is an empirical rather than precise guarantee.Source: §4.3.1 · p. 32

4.3.2 Lower bound

  • A lower frame bound is difficult because finite truncation causes spectral pollution, and finite-data carré du champ matrices generally lack exact zero eigenvalues.Even dense samples from a compact manifold can have eigenvalues on a continuous spectrum, obscuring a clear zero/non-zero distinction.Source: §4.3.2 · p. 32
  • The lower-bound choice has little apparent effect even when the condition number ranges from 5 to 1012, because excluded low-eigenvalue vectors have little effect if retained.A larger lower bound excludes more vectors from the space, but their norms are low when their eigenvalues are low.Source: §4.3.2 · p. 33

4.4 Weak formulations of operators · 4.5 Convergence to continuous differential operators

The section develops weak formulations for computing operators and analyzes when their finite-data, truncated, and regularized solutions converge to continuous solutions.

4.4 Weak formulations of operators

  • Weak formulations recover discrete linear operators from Gram matrices and load vectors using a pseudoinverse, and yield adjoints by transposing the weak matrix and applying the domain Gram pseudoinverse.Source: §4.4 · p. 33

4.5 Convergence to continuous differential operators

  • Convergence is analyzed as data size grows, bandwidth shrinks, and the finite spanning set expands, reducing approximation error while increasing truncation size.Source: §4.5 · p. 33

4.5.1 Convergence of weak formulations in truncated frames

  • In the noise-free setting, each truncated weak-formulation solution is the orthogonal projection onto its frame subspace, and converges to the true solution as those subspaces become dense.The frame property provides numerical stability for each finite system and density of the subspaces for convergence.Source: §4.5.1 · pp. 33–34

4.5.2 The problem of stable convergence in noisy truncated frames

  • As k grows, κ(Gk) can diverge and destabilize noisy solutions; regularization controls conditioning, but Tikhonov is biased while spectral cutoff is unbiased for ε < A and lacks universal truncation convergence.Both methods have uniformly bounded pseudoinverses and vanishing regularization bias; spectral cutoff’s truncation convergence requires a strong assumption.Source: §4.5.2 · pp. 34–35

4.5.3 Bias convergence and stability of spectral regularisations

  • Under the stated conditions, λg⁺_ε(λ) converges to 1 as ε→0 for every λ>0, and is uniformly bounded by a constant C over ε and λ≥0.Source: §4.5.3 · p. 35

4.5.4 Truncation convergence of Tikhonov Regularisation

  • Truncated Tikhonov formulations converge in arbitrary frames because regularization commutes with projection, and the truncation error vanishes as the frame subspaces grow dense.The argument applies the Lax–Milgram theorem and Céa’s lemma.Source: §4.5.4 · p. 36

4.5.5 Truncation convergence of spectral cutoff

  • Without spectral pollution in the gap (0, A), spectral cutoff with ε in that gap has vanishing truncation error and converges to the true solution.The no-pollution condition is strong, and the paper notes that proving it for a given frame is highly non-trivial.Source: §4.5.5 · pp. 36–37

4.5.6 Towards overall convergence results

  • For manifold data, convergence analysis uses eigenfunction estimation, carré du champ estimation, and Monte Carlo integration, but guarantees remain limited beyond manifolds and require sufficiently smooth coordinates.The paper identifies convergence for non-manifold data as future work and notes that variable-bandwidth settings lack known eigenfunction convergence guarantees.Source: §4.5.6 · p. 38

5 Differential operators

This section develops computational representations of classical differential operators for diffusion geometry, including weak formulations, Hessians, tensor operators, and the Levi-Civita connection. It also covers the Hodge Laplacian spectrum as a tool for topological data analysis.

5.1 Gradient and divergence

  • The gradient maps functions to steepest-increase vector fields, while divergence is its negative adjoint and measures local expansion or contraction.Both operators are discretized through weak formulations; for an orthonormal function basis, divergence is represented by −(∇weak)^T.Source: §5.1 · pp. 39–40

5.2 Exterior derivative and codifferential

  • The exterior derivative extends the gradient to higher-degree forms, while the codifferential generalizes negative divergence; in 3D, the derivative of a dual 1-form yields curl.The derivative of a 1-form measures signed vorticity, and the codifferential of a 2-form defines rotational flow around its support boundary.Source: §5.2 · pp. 40–42

5.3 Directional derivatives

  • A vector field acts on functions as a directional derivative, represented by a matrix recovered from its weak formulation that works with any function basis.With an orthonormal basis, the operator matrix equals its weak formulation.Source: §5.3 · p. 43

5.4 Hodge Laplacian

  • The Hodge Laplacian combines ‘up’ and ‘down’ derivatives; a kernel trick computes its weak energy without inversion in a potentially larger space, and its spectrum enables topological data analysis.The weak eigenproblem is fully symmetric because the Hodge Laplacian is self-adjoint.Source: §5.4 · p. 44

5.5 Hodge decomposition

  • Hodge decomposition uniquely splits a form into orthogonal exact, harmonic, and coexact parts, but data uncertainty and numerical error often shift apparent harmonic behavior into the coexact part.The function’s harmonic component can still be computed as harmonic when the k-nearest-neighbour kernel makes the computational graph genuinely disconnected.Source: §5.5 · pp. 46–47

5.6 Lie bracket

  • The Lie bracket is a bilinear operator measuring the failure of vector-field flows to commute; its discretization recovers analytically expected results on a square and sphere.On manifolds, the bracket quantifies the infinitesimal discrepancy between composing the two flows in opposite orders.Source: §5.6 · p. 48

5.7 Hessian

  • The Hessian generalizes the second-derivative matrix to diffusion geometry, mapping functions to (0, 2)-tensors and using the carré du champ for discretization.Its weak formulation evaluates the Hessian on pairs of gradient vector fields.Source: §5.7 · pp. 48, 50
  • The Hessian is computed using only d(d + 1)/2 unique symmetric components and supports critical-point classification through second derivatives.Its action on vector fields represents second derivatives; the z-coordinate Hessian connects to surface curvature through the second fundamental form, used to compute curvature tensors [52].Source: §5.7 · p. 50

5.8 Operator form of (0, 2) tensors

  • The operator form turns a (0, 2)-tensor into a linear map from vector fields to vector fields, which is discretized using a weak formulation.This form has applications including defining the Levi-Civita connection.Source: §5.8 · pp. 50, 52

5.9 Levi-Civita connection

  • The Levi-Civita connection is discretized as a map from vector fields to 2-tensors, and its computations recover expected flat-space examples and an analogous torus calculation.It is the metric-compatible, torsion-free connection associated with a Riemannian metric.Source: §5.9 · pp. 52–53

5.10 Comparison between Lie bracket and Levi-Civita connection

  • The Levi-Civita connection depends on the pointwise value of its direction field, whereas the Lie derivative depends on neighboring field behavior and measures flow noncommutativity.Thus the connection generalizes the directional derivative to vector fields, while the Lie bracket cannot be determined from pointwise data alone.Source: §5.10 · p. 53

6 Differential equations

Diffusion geometry provides a mesh-free way to solve spatial PDEs on observed point-cloud data, where geometry affects dynamics and mesh-based methods are difficult to apply. The section describes first- and second-order equation solvers, vector-field flows, and integral curves.

6 Differential equations

  • Covariant derivative examples satisfy the Leibniz rule and recover expected rotational, flat-space, and torus-frame results.The rotational field’s derivative with itself points inward, while derivatives of ∇x and ∇y vanish in flat space.Source: §6 · p. 54, Figure 24 · p. 54
  • Diffusion geometry uses computational differential operators to solve spatial PDEs on point clouds, where mesh-based methods are hard to apply.The geometry of a spatial domain can affect dynamics, such as the speed of a wave travelling over a curved surface.Source: §6 · pp. 53–54

6.1 First-order differential equations

  • The discretized first-order system represents functions as vectors and operators as matrices, then uses diagonalization and a matrix exponential to obtain its solution.The operator is represented as T = Q^-1 diag(W)Q, giving exp(tT) = Q^-1 diag(exp(tW))Q.Source: §6.1 · pp. 54–55

6.1.1 Heat equation

  • The heat equation is solved by setting T = -∆, with solutions visualized over time on the data and as a graph.The initial condition is u0 = f.Source: §6.1.1 · p. 55, Figure 26 · p. 55

6.1.2 Vector field flows

  • A vector field’s action on functions is represented by Xop, and setting T = Xop solves their flow, corresponding on manifolds to pushforward along the induced diffeomorphisms.Xop is a square matrix describing the action of X on functions.Source: §6.1.2 · p. 56

6.2 Second-order differential equations

  • A second-order equation is converted to a first-order system on 2n0 coordinates, solved with initial conditions (f, h), and projected onto its first n0 coordinates.The conversion uses a 2n0 × 2n0 block matrix.Source: §6.2 · p. 56

6.2.1 Wave equation

  • The wave equation uses T = -∆ and S = 0, while damping adds friction γ > 0 and sets S = -γIn0.The solutions are visualized in Figures 27 and 28.Source: §6.2.1 · p. 56

6.3 Integral curves

  • Integral curves for all points are computed together by evolving coordinate functions as vector-field flows, a global approach that regularizes the calculation.Solving locally with Dirac delta functions would be extremely unstable and inaccurate; smoother coordinate functions have lower energy and reduce the Laplacian estimation error.Source: §6.3 · p. 57

7 Geometric data analysis

The section develops diffusion-geometry methods for estimating intrinsic distances and curvature from data. It formulates geodesic-distance recovery as a Lipschitz-constrained optimization and presents sectional curvature computation without a manifold assumption.

7.1 Geodesic distances

  • Geodesic distance from a point is characterized by the 1-Lipschitz function that vanishes there and has the largest mean value.The Lipschitz condition is expressed through Γ(f, f) ≤ 1 almost everywhere.Source: §7.1 · p. 57
  • In the eigenfunction basis, the distance approximation is a convex optimization with one quadratic carrée-du-champ constraint per data point.The constraints are fᵀΓⱼf ≤ 1 for j = 1, …, n; standard conic programming methods can solve the problem efficiently.Source: §7.1 · p. 59
  • Because geodesic distance is nonsmooth at its source, the method models it as ambient distance plus a non-negative, smooth low-frequency correction.The correction belongs to the compressed function space A, avoiding direct approximation of the cone-shaped distance function near its tip.Source: §7.1 · p. 59

7.2 Curvature

  • Prior curvature methods used the manifold assumption [52], local quadratic regression, or intrinsic volume comparison [49].The volume-comparison approach supports very high ambient dimensions but is less robust and sample efficient.Source: §7.2 · pp. 60–61
  • The presented sectional curvature is, to the authors’ knowledge, the first classical curvature tensor computed without assuming the data lie on a manifold.Earlier work computed scalar, Ricci, and Riemann curvature using a manifold assumption, or scalar curvature through intrinsic volume comparison, which was less robust and sample efficient.Source: §7.2 · pp. 60–61

8 Topological data analysis

The section develops diffusion-geometry tools for recovering and interpreting topology from data, including cohomology, circular coordinates, and Morse indices. It also compares these tools with persistent homology and discusses their stability and robustness.

8.1 de Rham cohomology via Hodge theory

  • Harmonic forms in the Hodge Laplacian’s kernel represent cohomology classes; eigenvalues nearer zero indicate clearer or larger classes, although the L2 method can miss classes represented by unbounded-norm forms.For example, the method will not approximate the punctured disk’s class for its missing origin because its harmonic form is not in L2.Source: §8.1 · p. 61
  • 0.54 for the torus versus 0.002 for the sphere with two circles: the cup-product inner product distinguishes these spaces despite their identical Betti numbers.The cup product is represented by projecting the wedge product of harmonic forms onto the harmonic forms of the combined degree.Source: Figure 33 · p. 63, §8.1 · p. 63

8.2 Comparison with persistent homology

  • Diffusion geometry estimates properties of the underlying probability space, whereas persistent homology describes a point-cloud filtration; its harmonic forms are canonical and vectorisable, but do not measure topological scale in ambient units.The paper also reports that diffusion geometry is several orders of magnitude more efficient in time and memory, while persistent homology is rarely applied beyond approximately 10^3 points or degree 1.Source: §8.2 · pp. 64–65

8.2.1 Stability and robustness

  • Harmonic-form eigenvalues vary continuously, while eigenvectors can jump when eigenvalues switch order; when eigenvalues are distinct, both vary smoothly under sufficiently small perturbations.The heat-kernel computation makes harmonic forms robust to large perturbations and outliers, unlike Vietoris–Rips persistence, whose features can be lost with outliers.Source: §8.2.1 · p. 65, Figure 35 · p. 66

8.3 Circular coordinates for 1-cohomology classes

  • Circular coordinates are heuristic candidates derived from eigenfunctions of the regularised flow of a harmonic 1-form; the method uses ϵ = 1 but offers no formal guarantees.It selects the complex eigenfunction with the smallest eigenvalue magnitude; too much smoothing makes eigenvalues real, while too little makes eigenfunctions irregular.Source: §8.3 · p. 67

8.4 Morse theory

  • The framework finds critical points where the gradient norm is approximately zero and computes Morse indices by counting negative eigenvalues of the Hessian’s symmetric generalised eigenproblem.The eigenproblem is independent of the chosen linearly independent vector fields used to represent the Hessian.Source: §8.4 · p. 67

9 Computational complexity and scaling

The section analyzes computational complexity in terms of data size and immersion dimension, then benchmarks diffusion geometry against persistent homology across dataset sizes and dimensions. The benchmarks report better scaling with dataset size and fast computation at extremal degrees.

9.1 Complexity

  • The main operations include a kernel matrix at O(n log(n) + d) and carré du champ at O(kn); complexity can be tuned using k, n0, and n1.The framework treats data size n and immersion dimension d as essential attributes, with d assumed to be of the order of intrinsic dimension.Source: §9.1 · pp. 68–69

9.2 Scalability

  • Persistent homology requires computing all lower degrees and has worst-case complexity O(n3k+3), whereas diffusion geometry can isolate degrees.The section describes diffusion geometry’s Hodge Laplacian as an n1-dimensional matrix; the associated complexity expression is incomplete in the supplied text.Source: §9.2 · p. 69

9.2.1 Scaling n

  • At n = 12,000, diffusion geometry computes homology in all degrees in approximately 1 second and has better scaling slopes than Ripser, independent of degree k.The comparison measures cohomology up to k = 0, 1, and 2 on torus samples; the diffusion geometry Hodge Laplacian uses fixed parameters k = 32 and n0 = n1 = 50.Source: §9.2.1 · p. 69, Figure 38 · p. 70

9.2.2 Scaling d

  • At n = 5,000, diffusion geometry computes through degree 9 in approximately 10 seconds, with exceptionally fast inference at extremal degrees.The k-form space’s combinatorial symmetry reduces computation at extremal degrees; Figure 39 also compares cohomology across dimensions with Ripser at n = 100.Source: §9.2.2 · pp. 70–71, Figure 39 · p. 70

10 Background and context

The section situates diffusion geometry among combinatorial and geometric models for data, then explains how the paper uses a diffusion-based carr0e du champ to compute geometric structures on point clouds.

10.1 Theory

  • Diffusion geometry uses Bakry-0cEmery 93-calculus, interpreting its carr0e du champ as a generalized Riemannian metric applicable to any space with a diffusion process.The passage identifies computable Hodge theory on metric-measure spaces as a notable alternative.Source: §10.1 · p. 71

10.2 Topological models

  • Combinatorial models represent functions on vertices and fields or forms on edges, enabling discrete differential operators and meaningful topological structures, but they are sensitive to complex construction, costly, and cannot support a Riemannian metric.Their multiplication fails the associative and commutative requirements for a pointwise metric inner product.Source: §10.2 · pp. 71–72

10.3 Geometric models

  • Geometric models admit a Riemannian metric through associative, commutative multiplication, but this prevents their fields, forms, and tensors from obeying the Leibniz rule, weakening their topological representation.Source: §10.3 · p. 72

10.3.1 Geometric models on a combinatorial complex

  • On combinatorial complexes, geometric models represent fields using aligned tangent spaces or operator forms; the latter avoids tangent estimation and enables operators such as the Lie bracket.Tangent-space alignment has also supported discrete parallel transport, vector-field diffusion, divergence, and curl.Source: §10.3.1 · p. 72

10.3.2 Diffusion methods on point clouds

  • Kernel diffusion methods approximate heat diffusion and, on manifolds, the Laplacian; they are statistically robust and efficient on point clouds, including non-manifold data, though theory focuses on manifold functions.They are widely used for dimensionality reduction through Laplacian eigenfunction embeddings.Source: §10.3.2 · p. 72

10.3.3 Geometric models on point clouds

  • SEC [16] showed that diffusion-derived metrics and weak formulations can support manifold geometry without tangent-space bases, but its spectral tensor, generators, and operator access limit accuracy, cost, and flexibility.Truncating the multiplication tensor compounds approximation error and can destabilize inner products; this work instead treats the carr0e du champ as a composable bilinear operator that can come from other kernels or deep learning.Source: §10.3.3 · pp. 73–74

10.4 Where does diffusion geometry fit?

  • The proposed point-cloud model combines a positive-definite diffusion metric, overcomplete feature spans, and weak formulations, with selectable immersions and Bakry-0cEmery formulas for higher geometric quantities.By default, it uses the data coordinates as the immersion; the authors report this choice improves robustness to noise and curvature estimation [52].Source: §10.4 · p. 74

11 Conclusions

Diffusion geometry reformulates Riemannian geometry through a Markov process and extends that principle to scalable computation on point clouds.

11 Conclusions

  • Diffusion geometry recasts classical Riemannian geometry through a Markov process and its carré du champ operator.The paper presents this as a simpler, more general theory.Source: §11 · p. 75
  • A heat kernel computes the carré du champ operator on point clouds, while weak formulations estimate differential operators.Source: §11 · p. 75
  • These methods translate vector calculus, Riemannian geometry, geometric analysis, and differential topology into simple, scalable data-analysis tools.Source: §11 · p. 75

A Markov chain properties · Carr´e du champ as infinitesimal covariance

The carré du champ is characterized by the infinitesimal covariance of functions under a Markov semigroup. For heat diffusion on R^d, the proof uses Taylor expansion and Gaussian transition expectations.

A Markov chain properties

  • For a Markov semigroup with carré du champ Γ and transition kernel p_t(x, dy), the infinitesimal covariance of f and h gives Γ.The expression is the covariance under X ∼ p_t(p, ·), scaled by 1/(2t); a proof of the general case is cited in.Source: Carr´e du champ as infinitesimal… · pp. 79–80

Carré du champ as infinitesimal covariance

  • For differentiable f, h : R^d → R, Proposition 2.1 establishes the carré du champ result for heat diffusion.The proof considers X ∼ N(p, 2tI).Source: Carr´e du champ as infinitesimal… · pp. 79–80
  • Taylor expansion approximates f(y) and h(y) at x by their values plus gradient terms, with O(∥y − x∥^2) remainders.The proof uses these expansions to evaluate the infinitesimal covariance.Source: Carr´e du champ as infinitesimal… · p. 80
  • Under the heat transition, the expected product of the centered function increments equals the corresponding expectation for X ∼ N(p, 2tI).This connects integration against p_t(p, y) to the Gaussian expectation.Source: Carr´e du champ as infinitesimal… · p. 80
  • Dividing the centered-product expectation by 2t yields the carré du champ result.The proof obtains the scaled limit from the Gaussian expectation.Source: Carr´e du champ as infinitesimal… · p. 80
  • Because E[f(X)] − f(p) and E[h(X)] − h(p) are O(t), replacing the centered-product expectation by the covariance preserves the limit.The same-limit argument applies to the two expectations separately.Source: Carr´e du champ as infinitesimal… · p. 80

Symmetric kernels are self-adjoint Markov chains

Row-normalizing a symmetric kernel yields a Markov chain that is reversible and self-adjoint, and these conditions are equivalent; its measure is stationary.

Symmetric kernels are self-adjoint Markov chains

  • A Markov chain is a row-normalization of a symmetric kernel if and only if it is reversible with respect to μ and self-adjoint on L2(A, μ).For a reversible chain, K_ij = μ_iP_ij is symmetric and row-normalizes back to P.Source: Symmetric kernels are self-adjoint… · pp. 80–81
  • When these equivalent conditions hold, μ is a stationary distribution for P.The proof also derives reversibility from self-adjointness and then establishes stationarity.Source: Symmetric kernels are self-adjoint… · pp. 80–81
  • The symmetric-kernel approach preserves the theoretically expected self-adjoint heat operator and speeds up finding its eigenvectors.A general Markov chain could use its stationary distribution as the measure for an L2 function space, but its heat diffusion operator would not be self-adjoint.Source: Symmetric kernels are self-adjoint… · p. 81

B Operator forms of tensors · Operator form of the Hessian

The Hessian is represented as a bilinear operator on vector fields, and its operator form can be discretized as a tensor that computes directional Hessians from functions and vector fields.

Operator form of the Hessian

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

The Levi-Civita connection is viewed as a (1,1)-tensor that maps a vector field X to a linear operator on vector fields. Its discretization represents that map with a three-dimensional tensor and obtains the strong form from the weak form.

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

  • The connection can be interpreted as the linear operator ∇X on vector fields for a fixed vector field X.Equivalently, the (1,1)-tensor maps X to a linear map on vector fields.Source: (1, 1)-tensor form of the Levi-Civita… · p. 82
  • The discretization represents the connection as an n1d × n1d × n1d tensor mapping vector fields in Rn1d to n1d × n1d linear operators.The tensor is obtained by reshaping the weak connection representation.Source: (1, 1)-tensor form of the Levi-Civita… · p. 82
  • The strong connection form is obtained by applying (G1)^-1 to the weak form, yielding a tensor that maps each vector field to a linear operator.For vector fields X and Y, the resulting operator ∇(1,1)X acts on Y.Source: (1, 1)-tensor form of the Levi-Civita… · p. 82

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