Finsler Geometry, Graph Neural Networks, and You

# Finsler Geometry, Graph Neural Networks, and You
Source: [https://arxiv.org/html/2606.17185](https://arxiv.org/html/2606.17185)
\\theorembodyfont\\theoremheaderfont\\theorempostheader

:\\theoremsep \\jmlrvolume334\\jmlryear2026\\jmlrworkshopTopology, Algebra, and Geometry in Data Science

\\NameT\. Mitchell Roddenberry\\Emailmitch@rice\.edu \\NameRichard G\. Baraniuk\\Emailrichb@rice\.edu \\addrRice UniversityTXUSA

###### Abstract

Graph neural network architectures based on the graph Laplacian approximate the Laplace–Beltrami operator, thus limiting their application to isotropic operators\. As a nonlinear alternative to the Laplace–Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold\. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows\. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry\. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice\.

## 1Introduction

Graph neural networks operate on discrete collections of points with specified pairwise relations\. These methods incorporate the relations into the network parameterization, often inspired by the “right” operator on a limiting object that the graph approximates\. For example, the graph Laplacian is understood as a discrete approximation of the Laplace–Beltrami operator, so that graph neural networks using the Laplacian implicitly treat the graph as being sampled from a Riemannian manifold\.

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/finsler-tangent-space.png)Figure 1:Riemannian \(dashed\) and Finsler \(solid\) unit balls on the tangent bundle of a manifold\. The Finsler metric exhibits anisotropy, non\-ellipticity, and asymmetry\.Using a basic operator, such as the Laplacian, graph neural networks are constructed by interleaving the operator and local nonlinearities, mimicking the composition of affine transforms and nonlinearities in feedforward neural networks\. Despite the nonlinearity, these neural networks are essentially isotropic\.

The natural generalization of Riemannian geometry to allow for anisotropy is*Finsler geometry*, which equips the tangent spaces of a smooth manifold with a general Minkowski norm, rather than an inner product\. This allows for asymmetric, non\-elliptical geometry, as pictured in\\Creffig:minkowski\. Finsler manifolds share many properties with their Riemannian counterparts, including a Laplace operator as the Fréchet derivative of an energy functional\. Defining the Laplacian on a Finsler manifold in this way yields a*nonlinear operator*, making it a viable candidate for nonlinear learning techniques that use interpretable geometric structures from the outset\.

Extending the approximation of the Laplace–Beltrami operator on submanifolds by the Laplacian of a sampled graph, we consider the problem of discretely approximating nonlinear Laplacians that inherit their structure from Finsler geometries\. To wit, we

1. 1\.Define the empirical Finsler Laplacian for point clouds \(\\Crefsec:background:empirical\)
2. 2\.Prove that the empirical Finsler Laplacian almost surely converges to the continuüm operator \(\\Crefthm:graph\-uniform\-convergence\)
3. 3\.Express the empirical Finsler Laplacian as a graph neural network layer \(\\Crefsec:gnn\), and define a class of Finslerian graph neural networks \(\\Crefsec:gnn:finslerian\)
4. 4\.Demonstrate the application of the Finslerian graph neural network to the inverse problem of recovering a Finsler metric from observed heat diffusion \(\\Crefsec:experiments\)\.

Related Work\.The convergence of graph Laplacians to the Laplace–Beltrami operator on manifolds has been studied from multiple angles\(belkin2004;belkin2008;coifman2006;garcia2020\)and application settings, particularly in semi\-supervised learning\(trillos2018;calder2023\)\. The convergence ofpp\-Laplacians has also been studied\(slepcev2019\), yielding a class of methods for semi\-supervised learning that are nonlinear, but still isotropic\. Additionally, convergence of graph neural networks to manifold neural networks has been considered\(wang2025\)using convolution operators based on the Laplace–Beltrami operator\.

There have been recent works on applications of Finsler geometry in computer vision and data science, with emphasis on the use of Randers metrics in particular\(weber2024;dages2025;gahtan2026\)\. Our discretization of the Finsler Laplacian draws inspiration from methods for solving partial differential equations on point clouds\(liang2013\)\. Similar to Randers metrics, “magnetic Laplacians” have been used in graph signal processing\(furutani2019\)and subsequently for the construction of graph neural networks\(zhang2021;he2022\), with the goal of modeling asymmetric systems on graphs\.

## 2Finsler Laplacians

LetFFbe a Minkowski norm onℝD\\mathbb\{R\}^\{D\}, and let a probability measureμ\\musupported on a closed, compact, smooth,dd\-dimensional submanifoldℳ⊂ℝD\\mathcal\{M\}\\subset\\mathbb\{R\}^\{D\}be given\. Letppbe the density ofμ\\muwith respect to the Hausdorff measure of the manifold\. Under this structure,ℳ\\mathcal\{M\}is a weighted Finsler manifold\(bao2000introduction\)with metric pulled back fromFFvia the inclusion map, with the co\-Finsler metric given by the dual normF∗F^\{\*\}\. For technical background, we refer to\\Crefsec:finsler\.

Define an energy functional for differentiable functions onℳ\\mathcal\{M\}:

E​\[f\]=∫12​\[F∗​\(∇f​\(x\)\)\]2​𝑑μ​\(x\)\.E\[f\]=\\int\\frac\{1\}\{2\}\\left\[F^\{\*\}\(\\nabla f\(x\)\)\\right\]^\{2\}d\\mu\(x\)\.\(2\)The*Finsler Laplacian*\(ge2000;ohta2009\)is a nonlinear operator defined as the Fréchet derivative ofEE, or equivalently as the divergence of the gradient of the squared \(co\-\)norm:

Δ​\[f\]=1p​D​\{E​\[f\]\}=1p​div​\(p​J​\(∇f\)\),\\Delta\[f\]=\\frac\{1\}\{p\}D\\\{E\[f\]\\\}=\\frac\{1\}\{p\}\\mathrm\{div\}\\left\(pJ\(\\nabla f\)\\right\),\(3\)where

J​\(ξ\)=∇ξ12​\[F∗​\(ξ\)\]2=F∗​\(ξ\)​∇ξF∗​\(ξ\)\.J\(\\xi\)=\\nabla\_\{\\xi\}\\frac\{1\}\{2\}\\left\[F^\{\*\}\(\\xi\)\\right\]^\{2\}=F^\{\*\}\(\\xi\)\\nabla\_\{\\xi\}F^\{\*\}\(\\xi\)\.\(4\)Observe that takingFFas the Euclidean norm recovers the usual density\-weighted Laplace–Beltrami operator\.

###### Example 2\.2\.

Letℳ⊂ℝ3\\mathcal\{M\}\\subset\\mathbb\{R\}^\{3\}be a cylindrical surface\. We consider the heat equationf˙​\(t\)=−Δ​\[f​\(t\)\]\\dot\{f\}\(t\)=\-\\Delta\[f\(t\)\], wheref​\(0\)f\(0\)is the sum of two Dirac delta functions onℳ\\mathcal\{M\}, andΔ\\Deltais either the Laplace–Beltrami operator, or the Finsler Laplacian induced by a Minkowski norm on the ambient space\. In particular, for some vectorvvwith norm less than one, letF∗​\(ξ\)=‖ξ‖\+⟨ξ,v⟩F^\{\*\}\(\\xi\)=\\\|\\xi\\\|\+\\langle\\xi,v\\rangle, so thatF∗F^\{\*\}is a Randers metric on the cotangent bundle of the ambient space\. As shown in\\Creffig:heat, the Randers metric causes the diffusion of heat to “drift” according to the vectorvv\.

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/beltrami-heat.png)

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/randers-heat.png)

Figure 2:Heat diffusion on a cylinder \(identified edges marked by arrows\) with two point sources \(marked×\\times\) using the isotropic Laplace–Beltrami operator \(left\) and the Finsler Laplacian derived from an ambient Randers metric \(right\)\. Panels show Tissot’s indicatrices for the respective metrics\. The Randers heat equation exhibits “drift” in accordance with the off\-center geometry\.### 2\.1The Finsler Graph Laplacian

Let𝒳=\{xi\}i=1n\\mathcal\{X\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}be sampledi\.i\.d\.from the distributionμ\\mu, and suppose we have access to the function valuesf​\(xi\)f\(x\_\{i\}\)for eachii\. We wish to estimate the gradients∇f\\nabla fbased on these samples, given a translation\-invariant nonnegative kernelκ\\kappaandϵ\>0\\epsilon\>0\. Assume thatκ:ℝD→ℝ\\kappa:\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}is smooth, compactly supported, and radially symmetric\. Defineκϵ​\(z\):=ϵ−d​κ​\(z/ϵ\)\\kappa\_\{\\epsilon\}\(z\):=\\epsilon^\{\-d\}\\kappa\(z/\\epsilon\)\.

We use a local principal component analysis method for estimating the gradient offf\. Namely, we define

∇ϵ,nf​\(x\)=Cϵ,n†​\(x\)​bϵ,n​\[f\]​\(x\),\\nabla\_\{\\epsilon,n\}f\(x\)=C\_\{\\epsilon,n\}^\{\\dagger\}\(x\)b\_\{\\epsilon,n\}\[f\]\(x\),\(5\)whereCϵ,n†C\_\{\\epsilon,n\}^\{\\dagger\}is \(almost\) a pseudoinverse of the kernel\-weighted covariance centered atxx, andbϵ,n​\[f\]​\(x\)b\_\{\\epsilon,n\}\[f\]\(x\)is the kernel\-weighted local variation offfnearxx\.111We offer more precise definitions of these objects in\\Crefsec:gradient\.Using the empirical gradient, we define the empirical energy functional

Eϵ,n​\[f\]=1n​∑i=1n12​\[F∗​\(∇ϵ,nf​\(xi\)\)\]2,E\_\{\\epsilon,n\}\[f\]=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\frac\{1\}\{2\}\\left\[F^\{\*\}\(\\nabla\_\{\\epsilon,n\}f\(x\_\{i\}\)\)\\right\]^\{2\},\(6\)and the*empirical Finsler Laplacian*via the Fréchet derivative:Δϵ,n​\[f\]=n​D​\{E​\[f\]\}\\Delta\_\{\\epsilon,n\}\[f\]=nD\\\{E\[f\]\\\}\. For convenience, define the \(co\)vectorξ​\(x\)=Cϵ,n†​\(x\)​J​\(∇ϵ,nf​\(x\)\)\\xi\(x\)=C\_\{\\epsilon,n\}^\{\\dagger\}\(x\)J\(\\nabla\_\{\\epsilon,n\}f\(x\)\)\. This yields the formula

Δϵ,n​\[f\]​\(x\)=1n​ϵ2​∑i=1nκϵ​\(x−xi\)​⟨ξ​\(x\)\+ξ​\(xi\),x−xi⟩\.\\Delta\_\{\\epsilon,n\}\[f\]\(x\)=\\frac\{1\}\{n\\epsilon^\{2\}\}\\sum\_\{i=1\}^\{n\}\\kappa\_\{\\epsilon\}\\left\(x\-x\_\{i\}\\right\)\\langle\\xi\(x\)\+\\xi\(x\_\{i\}\),x\-x\_\{i\}\\rangle\.\(7\)We define the*Finsler graph Laplacian*ℒ\\mathcal\{L\}as the restriction of the empirical Finsler Laplacian to the point cloud𝒳\\mathcal\{X\}\.

Although the Finsler Laplacian and its graph counterpart are derived from similar energy functionals, the capacity of the Finsler graph Laplacian to approximate the underlying operator does not follow immediately\. Sincen→∞n\\to\\infty, ifϵ→0\\epsilon\\to 0at a suitable rate, the Finsler graph Laplacian converges to the continuüm Finsler Laplacian in the following sense:

###### Theorem 2\.3\.

Letμ\\mube a probability measure supported on a smooth, closed, and compactdd\-dimensional submanifoldℳ⊂ℝD\\mathcal\{M\}\\subset\\mathbb\{R\}^\{D\}, with densityppbounded away from zero on its support\. Assume thatffis aC3C^\{3\}function on this submanifold\. For eachn≥1n\\geq 1, let\{xi\}i=1n\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}be a set of points sampled i\.i\.d\. according to the measureμ\\mu\. Then, forϵ=O​\(log⁡n/n\)1/\(3​d\+4\)\\epsilon=O\\left\(\\log n/n\\right\)^\{1/\(3d\+4\)\},

limn→∞maxj∈\[1,n\]⁡\|Δ​\[f\]​\(xj\)−ℒ​\[f\]​\(xj\)\|=0almost surely\.\\lim\_\{n\\to\\infty\}\\max\_\{j\\in\[1,n\]\}\\Big\|\\Delta\[f\]\(x\_\{j\}\)\-\\mathcal\{L\}\[f\]\(x\_\{j\}\)\\Big\|=0\\quad\\text\{almost surely\}\.\(8\)

We prove this in\\Crefsec:proof\-ptwise\.

## 3The Finsler Graph Laplacian and Neural Networks

Cellular sheaves are a powerful tool for organizing operators on graphs that transform data between vector spaces\(curry2014;hansen2020sheaf;barbero2022sheaf\)\. We cast the Finsler graph Laplacian in this framework, which motivates a new graph neural network architecture\.

### 3\.1Cellular Sheaves for the Finsler Laplacian

Let𝒳=\{xi\}i=1n\\mathcal\{X\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}be a point cloud, with an observed functionf:𝒳→ℝf:\\mathcal\{X\}\\to\\mathbb\{R\}\. After constructing a graph using𝒳\\mathcal\{X\}, we construct sheaf structures over the graph such that appropriate composition of restriction maps, morphisms, and their adjoints computes the Finsler graph Laplacian\. Construct a graph𝒢=\(𝒳,ℰ\)\\mathcal\{G\}=\(\\mathcal\{X\},\\mathcal\{E\}\)so that𝒳\\mathcal\{X\}is the set of nodes andℰ\\mathcal\{E\}consists of unordered pairs of nodes\(xi,xj\)\(x\_\{i\},x\_\{j\}\)such thatκϵ​\(xi−xj\)≠0\\kappa\_\{\\epsilon\}\(x\_\{i\}\-x\_\{j\}\)\\neq 0\. We define two cellular sheaves222See\\Crefsec:app\-sheaf for background information on cellular \(co\)sheaves\.on𝒢\\mathcal\{G\}\. The first we denote byℱ\\mathcal\{F\}, such that all vector spacesℱ​\(x\)\\mathcal\{F\}\(x\)andℱ​\(e\)\\mathcal\{F\}\(e\)are the real lineℝ\\mathbb\{R\}\. For any incident node\-edge pair, the restriction map is defined asℱx→e​fx=fx\\mathcal\{F\}\_\{x\\to e\}f\_\{x\}=f\_\{x\}forfx∈ℱ​\(x\)≃ℝf\_\{x\}\\in\\mathcal\{F\}\(x\)\\simeq\\mathbb\{R\}\. The second sheaf is denoted by𝒯\\mathcal\{T\}, and assigns a copy of the tangent spaceTx​ℝDT\_\{x\}\\mathbb\{R\}^\{D\}to each nodex∈𝒳x\\in\\mathcal\{X\}, while still assigning the real line to each edgee∈ℰe\\in\\mathcal\{E\}\. For each nodex∈𝒳x\\in\\mathcal\{X\}and any incident edge\(x,y\)∈ℰ\(x,y\)\\in\\mathcal\{E\}, the restriction map is

𝒯x→\(x,y\)​v=1n​ϵ2​κϵ​\(x−y\)​⟨Cϵ,n†​\(x\)​\(x−y\),v⟩,\\mathcal\{T\}\_\{x\\to\(x,y\)\}v=\\frac\{1\}\{n\\epsilon^\{2\}\}\\kappa\_\{\\epsilon\}\\left\(x\-y\\right\)\\langle C\_\{\\epsilon,n\}^\{\\dagger\}\(x\)\(x\-y\),v\\rangle,\(9\)forv∈Tx​ℝD≃ℝDv\\in T\_\{x\}\\mathbb\{R\}^\{D\}\\simeq\\mathbb\{R\}^\{D\}\. We identify the tangent spaces with the cotangent spacesTx​ℝD≃Tx∗​ℝDT\_\{x\}\\mathbb\{R\}^\{D\}\\simeq T\_\{x\}^\{\*\}\\mathbb\{R\}^\{D\}using the ambient Euclidean metric\. We define the following maps on the spaces of edge/node data:

- •id:C1​\(ℱ\)→C1​\(𝒯\)\\mathrm\{id\}:C^\{1\}\(\\mathcal\{F\}\)\\to C^\{1\}\(\\mathcal\{T\}\)andid∗:C1​\(𝒯\)→C1​\(ℱ\)\\mathrm\{id\}^\{\*\}:C^\{1\}\(\\mathcal\{T\}\)\\to C^\{1\}\(\\mathcal\{F\}\)are both identity maps under the obvious identificationC1​\(ℱ\)≃C1​\(𝒯\)C^\{1\}\(\\mathcal\{F\}\)\\simeq C^\{1\}\(\\mathcal\{T\}\),
- •J:C0​\(𝒯\)→C0​\(𝒯\)J:C^\{0\}\(\\mathcal\{T\}\)\\to C^\{0\}\(\\mathcal\{T\}\)passes \(co\)vector data on the nodes through the mapJJas defined in \([4](https://arxiv.org/html/2606.17185#S2.E4)\)\.

The vector spaces constituting both sheaves are endowed with the usual inner product structure; the restriction maps and their adjoints induce a differentiald:C0→C1d:C^\{0\}\\to C^\{1\}and a codifferentialδ:C1→C0\\delta:C^\{1\}\\to C^\{0\}for both sheavesℱ,𝒯\\mathcal\{F\},\\mathcal\{T\}\. We model the function on the nodes as a0\-cochainf∈C0​\(ℱ\)f\\in C^\{0\}\(\\mathcal\{F\}\)\. The Finsler graph Laplacian is represented as

ℒ​\[f\]=\(δ∘id∗∘d∘J∘δ∘id∘d\)​\[f\],\\mathcal\{L\}\[f\]=\(\\delta\\circ\\mathrm\{id\}^\{\*\}\\circ d\\circ J\\circ\\delta\\circ\\mathrm\{id\}\\circ d\)\[f\],\(10\)whered:C0→C1d:C^\{0\}\\to C^\{1\}generically indicates the differential andδ:C1→C0\\delta:C^\{1\}\\to C^\{0\}the codifferential\. Perhaps more legibly, the Finsler graph Laplacian traverses the following \(non\-commutative\) diagram:

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\}\\pgfsys@endscope\}\}\} \{\{\}\}\{\{\{\}\}\{\{\}\}\}\{\}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{\{\}\}\{\}\{\{\}\}\{\} \{\{\}\{\}\}\{\{\}\{\}\}\{\{\}\{\}\}\{\{\}\} \{\{\}\{\}\}\{\{\{\}\}\{\{\}\}\}\{\{\}\}\{\{\}\{\}\}\{\{\{\}\}\{\{\}\}\}\{\{\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{105\.21962pt\}\{9\.05952pt\}\\pgfsys@curveto\{128\.52676pt\}\{25\.3794pt\}\{127\.61034pt\}\{\-25\.61978pt\}\{104\.63081pt\}\{\-9\.52931pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{\-0\.81915\}\{0\.57358\}\{\-0\.57358\}\{\-0\.81915\}\{104\.467pt\}\{\-9\.41463pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{124\.59451pt\}\{\-2\.51186pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{J\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\.\(11\)
Considering the mapJJas an activation function, the Finsler graph Laplacian can be thought of as a layer in a sheaf neural network\(hansen2020sheaf;barbero2022sheaf\), where the nonlinearity is applied in a “lifted” space \(namely, the cotangent space at each point\)\. In contrast, the nonlinearity for a typically defined sheaf neural network is applied in the same domain as the input space, that is, as a mapσ:C0​\(ℱ\)→C0​\(ℱ\)\\sigma:C^\{0\}\(\\mathcal\{F\}\)\\to C^\{0\}\(\\mathcal\{F\}\)\.

### 3\.2Finslerian Graph Neural Networks

We now consider how a graph neural network may be designed to parameterize a Finsler metric\. Specifically, we will replaceJJwith a simple neural network\. ForW,V∈ℝE×DW,V\\in\\mathbb\{R\}^\{E\\times D\}, and an activation functionσ:ℝ→ℝ\\sigma:\\mathbb\{R\}\\to\\mathbb\{R\}that is positive homogeneous of degree11withσ′≥0\\sigma^\{\\prime\}\\geq 0, define the approximation

J^​\(ξ\)=V⊤​σ​\(W​ξ\),\\widehat\{J\}\(\\xi\)=V^\{\\top\}\\sigma\(W\\xi\),\(12\)whereσ\\sigmais applied elementwise in the standard basis\. The following result gives a condition that guaranteesJ^\\widehat\{J\}to be “Finslerian,” in the sense of being the gradient of a squared\-norm:

###### Proposition 3\.1\.

IfV=WV=Win \([12](https://arxiv.org/html/2606.17185#S3.E12)\),rank\(W\)=D\\operatorname\*\{rank\}\(W\)=D, andσ′\>0\\sigma^\{\\prime\}\>0, thenJ^​\(ξ\)=∇ξ12​\[F^∗​\(ξ\)\]2\\widehat\{J\}\(\\xi\)=\\nabla\_\{\\xi\}\\frac\{1\}\{2\}\\left\[\\widehat\{F\}^\{\*\}\(\\xi\)\\right\]^\{2\}where

F^∗​\(ξ\)=2​⟨ξ,J^​\(ξ\)⟩\\widehat\{F\}^\{\*\}\(\\xi\)=\\sqrt\{2\\langle\\xi,\\widehat\{J\}\(\\xi\)\\rangle\}\(13\)is an \(almost\) Minkowski norm\. Under the further assumption thatσ​\(x\)=α​x\\sigma\(x\)=\\alpha xforα\>0\\alpha\>0, the normF^∗\\widehat\{F\}^\{\*\}is Riemannian\.

We prove\\Crefprop:symmetry in\\Crefsec:finsler:symmetry\. Based on this, for a \(potentially constant\) learnable matrixW​\(t\)W\(t\), we define a*Finslerian graph neural network*under the framework of graph neural diffusion\(chamberlain2021;thorpe2022;bodnar2022neural\); that is, an ordinary differential equation of the form

f˙​\(t\)=−ℒ^W​\(t\)​\[f​\(t\)\],\\dot\{f\}\(t\)=\-\\widehat\{\\mathcal\{L\}\}\_\{W\(t\)\}\[f\(t\)\],\(14\)whereℒ^W​\(t\)\\widehat\{\\mathcal\{L\}\}\_\{W\(t\)\}is the Finsler graph Laplacian defined usingJ^​\(ξ\)=W⊤​\(t\)​σ​\(W​\(t\)​ξ\)\\widehat\{J\}\(\\xi\)=W^\{\\top\}\(t\)\\sigma\(W\(t\)\\xi\)\.

## 4Numerical Experiments

We demonstrate the Finsler graph Laplacian and Finslerian graph neural networks on a collection of numerical examples, corroborating our convergence result \(\\Crefthm:graph\-uniform\-convergence\) and the utility of a symmetric nonlinear network for representing Minkowski norms \(\\Crefprop:symmetry\)\. Implementation details can be found in\\Crefsec:methods, and source code is available[\[here\]](https://github.com/tmrod/finsler-gnn)\.

### 4\.1Convergence of the Finsler Graph Laplacian

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/exp11_f.png)

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/exp11_Lf.png)

Figure 3:The functionf​\(x,y,z\)=cos⁡\(x\)\+cos⁡\(y\)\+zf\(x,y,z\)=\\cos\(x\)\+\\cos\(y\)\+zrestricted to the torus \(left\) and the Finsler LaplacianΔ​\[f\]\\Delta\[f\]defined so thatF∗​\(ξ\)=‖ξ‖3F^\{\*\}\(\\xi\)=\\\|\\xi\\\|\_\{3\}\(right\)\. Identified edges of torus marked by arrows\.We first validate the convergence result stated in\\Crefthm:graph\-uniform\-convergence\. Letℳ⊂ℝ3\\mathcal\{M\}\\subset\\mathbb\{R\}^\{3\}be a torus with major radius22and minor radius11oriented along thex​yxy\-plane\. We consider the restriction of a functionf​\(x,y,z\)=cos⁡\(x\)\+cos⁡\(y\)\+zf\(x,y,z\)=\\cos\(x\)\+\\cos\(y\)\+zon the ambient space toℳ\\mathcal\{M\}, and consider the Finsler Laplacian defined by theℓ3\\ell\_\{3\}\-norm, that is, whereF∗​\(ξ\)=\(∑i\|ξi\|3\)1/3F^\{\*\}\(\\xi\)=\\left\(\\sum\_\{i\}\|\\xi\_\{i\}\|^\{3\}\\right\)^\{1/3\}\. The functionffand the application of the Finsler LaplacianΔ​\[f\]\\Delta\[f\]are pictured in\\Creffig:convergence\-functions\.

![Refer to caption](https://arxiv.org/html/2606.17185v1/pics/exp11.png)

00\.50\.511G\.T\.ReLULinear

Figure 4:\(Bottom left\) Similarity between the Finsler graph Laplacian and the true Finsler Laplacian as a function ofnnandϵ\\epsilon\. \(Top left\) Cosine similarity for “best”ϵ​\(n\)\\epsilon\(n\)\. \(Right\) Recovery of a Randers metric by a Finslerian graph neural network\.We sample points on the torus uniformly according to the ambient \(Hausdorff\) measure, and evaluate the Finsler graph Laplacian constructed using varying numbers of sampled pointsnnand bandwidthsϵ\\epsilon\. We then measure the cosine similarity between the true Finsler Laplacian evaluated at the sampled points and the Finsler graph Laplacian, to account for scaling\. The results of this are shown in\\Creffig:experiments \(left\)\. As expected, the similarity between the true and graph Laplacian map increases asnngrows from250250to95009500, with the corresponding “best” value ofϵ\\epsilondecreasing at a modest rate as the point cloud becomes denser\. This corroborates the convergence result established by\\Crefthm:graph\-uniform\-convergence\.

### 4\.2Learning Geometry using Finslerian Graph Neural Networks

We consider a semisupervised geometric inverse problem of recovering a Finsler metric from an observed instance of heat diffusion\. Namely, we sample a point cloud ofn=600n=600points uniformly on the unit square, and construct a graph using a gaussian kernel with a bandwidth ofϵ=0\.2\\epsilon=0\.2\. In the graph construction, we periodize the domain, giving the manifold the topology of a torus\. We generate a functionf​\(0\)f\(0\)on the domain as a superposition of55Fourier modes of varying frequency, phase, and amplitude, then sample it to yield a function on the point cloud\. Then, we run the heat equation using a Finsler Laplacian defined using a Randers \(co\)metric \(see\\Crefexam:randers\) with a drift vectorv=\[0\.25,0\.25\]v=\[0\.25,0\.25\]\. We observe the statef​\(T\)f\(T\)forT=1T=1, and use20%20\\%of the labeled nodes for training\.

Given this single input\-output sample of the heat equation over one time unit, we apply multiple techniques to model the heat diffusion\. The first two are Finslerian graph neural networks \(FGNNs\), both with hidden dimensionE=64E=64, where one uses the ReLU activation and the other uses a linear activation\. The third and fourth learn maps on the node signal space that more closely resembles a graph neural network \(GNN\)\. The initial inputf​\(0\)f\(0\)is linearly lifted toℝE\\mathbb\{R\}^\{E\}, and heat diffusion is performed for node features inℝE\\mathbb\{R\}^\{E\}wheref˙​\(t\)=MLP​\(L​f​\(t\)\)\\dot\{f\}\(t\)=\\mathrm\{MLP\}\(Lf\(t\)\)withLLdenoting the standard graph Laplacian, andMLP\\mathrm\{MLP\}a single\-hidden\-layer neural network with either ReLU or linear activations, mappingℝE\\mathbb\{R\}^\{E\}toℝE\\mathbb\{R\}^\{E\}on each node\. The output of the diffusion is then linearly projected back toℝ\\mathbb\{R\}\.

We train all methods to map the initial inputf​\(0\)f\(0\)to predict the labeled20%20\\%of nodes atf​\(T\)f\(T\)\. We then measure the relative mean\-squared error \(MSE\) over the remaining80%80\\%of nodes, recorded under the column “Train Graph” in\\Creftab:mse\. The Finslerian graph neural networks strongly outperform the standard graph neural networks\.

Table 1:MSE of learned approximations to Finsler heat flow\.Moreover, the nonlinear \(ReLU\) Finslerian graph neural network performs an order of magnitude better than the linear one, because the linear method is only capable of*Riemannian*diffusion, by\\Crefprop:symmetry\. To test generalization, we resample the graph and generate a new initial conditionf​\(0\)f\(0\), and compare the trained models to the true Finsler heat equation on the new, unseen graph\. As shown in\\Creftab:mse under the column “Test Graph,” the Finslerian graph neural networks suffer no performance loss, reflecting the generalization suggested by\\Crefthm:graph\-uniform\-convergence\. On the other hand, the graph neural networks appear to overfit to the training graph, performing much worse on the test graph\.

We also examine the qualitative properties of the Finslerian graph neural networks after training\. Since both methods amounts to learningJ^​\(ξ\)=∇ξ12​\[F^∗​\(ξ\)\]2\\widehat\{J\}\(\\xi\)=\\nabla\_\{\\xi\}\\frac\{1\}\{2\}\[\\widehat\{F\}^\{\*\}\(\\xi\)\]^\{2\}, we plot the unit balls of the normsF^∗\\widehat\{F\}^\{\*\}in\\Creffig:experiments \(right\)\. The only approach that yields a metric close to the ground truth \(G\.T\.\) metric is the model with ReLU activation\. The model with linear activation learns a metric whose unit ball is an ellipse, again due to its restriction to Riemannian metrics by\\Crefprop:symmetry\.

## 5Conclusion

We have developed the topological and algebraic tools for approximating Finsler geometric Laplacian operators on sampled point clouds\. The proposed empirical Finsler Laplacian is shown to converge in a uniform pointwise sense, demonstrating its utility both for computations in PDEs and in semisupervised learning problems arising in data science\. Beyond the applications, the empirical Finsler Laplacian demonstrates the limitations of graph signal processing and graph neural network methods that rely too heavily on isotropy\. By linking the structure of the Finsler Laplacian to a class of nonlinear graph neural networks, we hope to promote study into how nonlinearities in neural architectures implicitly impose geometric structures on the domain\. By tying the neural architecture design to a known geometric structure, such as a Minkowski norm, solutions found by learning algorithms become interpretable via the induced geometry\.

\\acks

This work was supported by ONR grant N00014\-23\-1\-2714, DOE grant DE\-SC0020345, DOI grant 140D0423C0076, and a Google Cloud Computing Award\.

## References

## Appendix AFinsler Geometry

We treat the ambient spaceℝD\\mathbb\{R\}^\{D\}as an affine space, so that the tangent spacesTx​ℳT\_\{x\}\\mathcal\{M\}are all isomorphic toℝD\\mathbb\{R\}^\{D\}, and the ambient space has no curvature\. A*Minkowski norm*onℝD\\mathbb\{R\}^\{D\}is a continuous functionF:ℝD→ℝ≥0F:\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{\\geq 0\}with the following properties for allv,w∈ℝD,λ≥0v,w\\in\\mathbb\{R\}^\{D\},\\lambda\\geq 0:

Smoothness:FFis smooth away from zero\.

Strong convexity:∇2\[F2​\(v\)\]≻0\\nabla^\{2\}\[F^\{2\}\(v\)\]\\succ 0forv≠0v\\neq 0\.

Positive homogeneity:F​\(λ​v\)=λ​F​\(v\)F\(\\lambda v\)=\\lambda F\(v\)\.

Positive definiteness:F​\(v\)\>0F\(v\)\>0unlessv=0v=0\.

Unlike a typical norm, we only require homogeneity to hold forλ≥0\\lambda\\geq 0, rather than for allλ∈ℝ\\lambda\\in\\mathbb\{R\}; this allows forFFto be asymmetric\.

If the conditions for the Minkowski normFFonly hold on a subsetℝD∖S\\mathbb\{R\}^\{D\}\\setminus SwhereSSis a closed, conic subset ofℝD\\mathbb\{R\}^\{D\}, we say thatFFis an*almost Minkowski norm*\(davis2026, Definition 4\)\.

For a linear subspaceU⊂ℝDU\\subset\\mathbb\{R\}^\{D\}, we pullback the Minkowski norm toUUvia the inclusion mapid:U↪ℝD\\mathrm\{id\}:U\\hookrightarrow\\mathbb\{R\}^\{D\}asF​\(u\):=\(F∘id\)​\(u\)F\(u\):=\(F\\circ\\mathrm\{id\}\)\(u\)foru∈Uu\\in U\. One can check that this defines a Minkowski norm onUU\. For a smooth submanifoldℳ⊂ℝD\\mathcal\{M\}\\subset\\mathbb\{R\}^\{D\}, endowing the tangent bundle ofℳ\\mathcal\{M\}with the pullback of the Minkowski norm onℝD\\mathbb\{R\}^\{D\}defines a*Finsler manifold*\(bao2000introduction\)\.

Notably, as shown byburago1993, if the Minkowski normFFis odd andℳ\\mathcal\{M\}is compact, then there exists a Minkowski normF′F^\{\\prime\}onℝ2​d\\mathbb\{R\}^\{2d\}such thatℳ\\mathcal\{M\}can be isometrically embedded inℝ2​d\\mathbb\{R\}^\{2d\}\.

### A\.1Proof of\\Crefprop:symmetry

ForJ^\\widehat\{J\}to be the gradient of a scalar\-valued energy functional on the cotangent bundle, the Jacobian ofJ^\\widehat\{J\}must be a symmetric tensor, since that computes the Hessian of the functional\. Specifically,

∇J^​\(ξ\)=W⊤​diag\(σ′​\(W​ξ\)\)⁡W\.\\nabla\\widehat\{J\}\(\\xi\)=W^\{\\top\}\\operatorname\*\{diag\}\(\\sigma^\{\\prime\}\(W\\xi\)\)W\.\(15\)Observe that enforcingV=W⊤V=W^\{\\top\}guarantees that∇J^​\(ξ\)\\nabla\\widehat\{J\}\(\\xi\)is symmetric\. Moreover, ifrank\(W\)=D\\operatorname\*\{rank\}\(W\)=Dandσ′\>0\\sigma^\{\\prime\}\>0, it also holds that the Jacobian ofJ^\\widehat\{J\}is positive definite\. Moreover, sinceσ\\sigmais positive homogeneous of degree11, the norm \([13](https://arxiv.org/html/2606.17185#S3.E13)\) is also positive homogeneous of degree11\. Ifσ\\sigmais nonsmooth at zero \(for instance, ifσ\\sigmais the Leaky ReLU\), then we only have smoothness ofFFat points where none of the entries ofW​ξW\\xiare equal to zero: the set of suchξ\\xiis a closed, conic subset ofℝD\\mathbb\{R\}^\{D\}with measure zero\. Hence, we have established the conditions forF^∗\\widehat\{F\}^\{\*\}to be an almost Minkowski norm, as desired\.

## Appendix BEmpirical Gradient Estimation

We develop a suitable estimate of the gradient of a functionffgiven samples over a finite set of points\{xi\}i=1n\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}\. A first approach is the moving least\-squares estimate:

∇ϵ,nf​\(x\)=arg​minξ​∑i=1nκϵ​\(x−xi\)​\(f​\(x\)−f​\(xi\)−⟨ξ,x−xi⟩\)2\.\\nabla\_\{\\epsilon,n\}f\(x\)=\\operatorname\*\{arg\\,min\}\_\{\\xi\}\\sum\_\{i=1\}^\{n\}\\kappa\_\{\\epsilon\}\\left\(x\-x\_\{i\}\\right\)\\left\(f\(x\)\-f\(x\_\{i\}\)\-\\langle\\xi,x\-x\_\{i\}\\rangle\\right\)^\{2\}\.\(16\)In other words,

∇ϵ,nf​\(x\)=arg​minξ⁡Gϵ,n​\(x,ξ\),\\nabla\_\{\\epsilon,n\}f\(x\)=\\operatorname\*\{arg\\,min\}\_\{\\xi\}G\_\{\\epsilon,n\}\(x,\\xi\),\(17\)where

Gϵ,n​\(x,ξ\)=1ϵ2​∑i=1nκϵ​\(x−xi\)​\(f​\(x\)−f​\(xi\)−⟨ξ,x−xi⟩\)2\.G\_\{\\epsilon,n\}\(x,\\xi\)=\\frac\{1\}\{\\epsilon^\{2\}\}\\sum\_\{i=1\}^\{n\}\\kappa\_\{\\epsilon\}\\left\(x\-x\_\{i\}\\right\)\\left\(f\(x\)\-f\(x\_\{i\}\)\-\\langle\\xi,x\-x\_\{i\}\\rangle\\right\)^\{2\}\.\(18\)Setting∇ξGϵ,n​\(x,ξ\)=0\\nabla\_\{\\xi\}G\_\{\\epsilon,n\}\(x,\\xi\)=0yields the solution

∇ϵ,nf​\(x\)=\(Cϵ,n​\(x\)\)−1​bϵ,n​\[f\]​\(x\)Cϵ,n​\(x\)=1n​ϵ2​∑i=1nκϵ​\(x−xi\)​\(x−xi\)⊗\(x−xi\)bϵ,n​\[f\]​\(x\)=1n​ϵ2​∑i=1nκϵ​\(x−xi\)​\(f​\(x\)−f​\(xi\)\)​\(x−xi\)\.\\begin\{gathered\}\\nabla\_\{\\epsilon,n\}f\(x\)=\\left\(C\_\{\\epsilon,n\}\(x\)\\right\)^\{\-1\}b\_\{\\epsilon,n\}\[f\]\(x\)\\\\ C\_\{\\epsilon,n\}\(x\)=\\frac\{1\}\{n\\epsilon^\{2\}\}\\sum\_\{i=1\}^\{n\}\\kappa\_\{\\epsilon\}\\left\(x\-x\_\{i\}\\right\)\(x\-x\_\{i\}\)\\otimes\(x\-x\_\{i\}\)\\\\ b\_\{\\epsilon,n\}\[f\]\(x\)=\\frac\{1\}\{n\\epsilon^\{2\}\}\\sum\_\{i=1\}^\{n\}\\kappa\_\{\\epsilon\}\\left\(x\-x\_\{i\}\\right\)\(f\(x\)\-f\(x\_\{i\}\)\)\(x\-x\_\{i\}\)\.\\end\{gathered\}\(19\)We call∇ϵ,n\\nabla\_\{\\epsilon,n\}the empirical gradient,Cϵ,nC\_\{\\epsilon,n\}the empirical covariance, andbϵ,nb\_\{\\epsilon,n\}the empirical cogradient\.

To account for this, we define

∇ϵ,nf​\(x\)=hϵ​\(Cϵ,n​\(x\)\)​bϵ,n​\[f\]​\(x\)\\nabla\_\{\\epsilon,n\}f\(x\)=h\_\{\\epsilon\}\\left\(C\_\{\\epsilon,n\}\(x\)\\right\)b\_\{\\epsilon,n\}\[f\]\(x\)\(20\)where

hϵ​\(A\)=\(A4\+ϵ4​I\)−1​A3\.h\_\{\\epsilon\}\(A\)=\(A^\{4\}\+\\epsilon^\{4\}I\)^\{\-1\}A^\{3\}\.\(21\)This choice ofhϵh\_\{\\epsilon\}approximates the matrix inverse for parts of the spectrum far from zero, while remaining close to zero for parts of the spectrum close to zero; in other words, it acts as a soft Moore\-Penrose pseudoinverse\. Hence, we denoteCϵ,n†​\(x\):=hϵ​\(Cϵ,n​\(x\)\)C\_\{\\epsilon,n\}^\{\\dagger\}\(x\):=h\_\{\\epsilon\}\(C\_\{\\epsilon,n\}\(x\)\)\.

## Appendix CCellular \(Co\)sheaf Laplacians

We provide background on cellular \(co\)sheaves\. For deeper coverage, we suggest the thesis ofcurry2014\. A*cellular sheaf*ℱ\\mathcal\{F\}on the graph𝒢\\mathcal\{G\}assigns

- •to each nodex∈𝒳x\\in\\mathcal\{X\}a vector spaceℱ​\(x\)\\mathcal\{F\}\(x\),
- •to each edgee∈ℰe\\in\\mathcal\{E\}a vector spaceℱ​\(e\)\\mathcal\{F\}\(e\), and
- •to each incident vertex\-edge pair\(x,e\)\(x,e\)a linear*restriction map*ℱx→e:ℱ​\(x\)→ℱ​\(e\)\\mathcal\{F\}\_\{x\\to e\}:\\mathcal\{F\}\(x\)\\to\\mathcal\{F\}\(e\)\.

The*dual cosheaf*ℱ∗\\mathcal\{F\}^\{\*\}on the graph𝒢\\mathcal\{G\}assigns

- •to each nodex∈𝒳x\\in\\mathcal\{X\}the dual vector spaceℱ∗​\(x\)\\mathcal\{F\}^\{\*\}\(x\),
- •to each edgee∈ℰe\\in\\mathcal\{E\}the dual vector spaceℱ∗​\(e\)\\mathcal\{F\}^\{\*\}\(e\), and
- •to each incident vertex\-edge pair\(x,e\)\(x,e\)a linear*extension map*ℱx→e∗:ℱ∗​\(e\)→ℱ∗​\(x\)\\mathcal\{F\}^\{\*\}\_\{x\\to e\}:\\mathcal\{F\}^\{\*\}\(e\)\\to\\mathcal\{F\}^\{\*\}\(x\)\.

For a given cellular \(co\)sheafℱ\\mathcal\{F\}, the space of0\-cochainsC0​\(ℱ\)C^\{0\}\(\\mathcal\{F\}\)constitutes the set of all assignments of vectors to each node according to the sheaf structure\. That is, a0\-cochainf∈C0​\(ℱ\)f\\in C^\{0\}\(\\mathcal\{F\}\)is a map sending eachx∈𝒳x\\in\\mathcal\{X\}to an element ofℱ​\(x\)\\mathcal\{F\}\(x\)\.

The space of11\-cochains is defined similarly, with additional information to account for the orientation of edges\. A11\-cochainβ∈C1​\(ℱ\)\\beta\\in C^\{1\}\(\\mathcal\{F\}\)is a map sending each*oriented edge*e→\\overrightarrow\{e\}to an element ofℱ​\(e\)\\mathcal\{F\}\(e\), with the property that reversing the orientation changes the sign:β​\(e←\)=−β​\(e→\)\\beta\(\\overleftarrow\{e\}\)=\-\\beta\(\\overrightarrow\{e\}\)\.

This yields the differential map fromC0​\(ℱ\)C^\{0\}\(\\mathcal\{F\}\)toC1​\(ℱ\)C^\{1\}\(\\mathcal\{F\}\)for a sheafℱ\\mathcal\{F\}as follows\. Lete→=\[x,y\]\\overrightarrow\{e\}=\[x,y\]be an oriented edge\. Then, forf∈C0​\(ℱ\)f\\in C^\{0\}\(\\mathcal\{F\}\), put

\(d​f\)​\(e→\)=ℱy→e​\(f​\(y\)\)−ℱx→e​\(f​\(x\)\)\.\(df\)\(\\overrightarrow\{e\}\)=\\mathcal\{F\}\_\{y\\to e\}\(f\(y\)\)\-\\mathcal\{F\}\_\{x\\to e\}\(f\(x\)\)\.\(22\)Similarly, for the dual cosheafℱ∗\\mathcal\{F\}^\{\*\}, we have the codifferential map fromC1​\(ℱ∗\)C^\{1\}\(\\mathcal\{F\}^\{\*\}\)toC0​\(ℱ∗\)C^\{0\}\(\\mathcal\{F\}^\{\*\}\)defined forβ∈C1​\(ℱ∗\)\\beta\\in C^\{1\}\(\\mathcal\{F\}^\{\*\}\)as

\(δ​β\)​\(x\)=∑y∈𝒳:\(x,y\)∈ℰℱx→e∗​\(β​\(\[x,y\]\)\)\.\(\\delta\\beta\)\(x\)=\\sum\_\{y\\in\\mathcal\{X\}:\(x,y\)\\in\\mathcal\{E\}\}\\mathcal\{F\}^\{\*\}\_\{x\\to e\}\(\\beta\(\[x,y\]\)\)\.\(23\)
We now repeat the construction of the Finsler Laplacian\. Letℱ\\mathcal\{F\}and𝒯\\mathcal\{T\}be the respective cellular sheaves of*functions*and*vectors*defined in\\crefsec:gnn, with corresponding dual cosheavesℱ∗\\mathcal\{F\}^\{\*\}and𝒯∗\\mathcal\{T\}^\{\*\}of*cofunctions*and*covectors*\. Letid:C1​\(ℱ\)→C1​\(𝒯∗\),id∗:C1​\(𝒯\)→C1​\(ℱ∗\)\\mathrm\{id\}:C^\{1\}\(\\mathcal\{F\}\)\\to C^\{1\}\(\\mathcal\{T\}^\{\*\}\),\\mathrm\{id\}^\{\*\}:C^\{1\}\(\\mathcal\{T\}\)\\to C^\{1\}\(\\mathcal\{F\}^\{\*\}\)be the identity maps under the obvious identification of the spaces\. Given thatJJmaps covectors to vectors, it can be extended to a mapJ:C0​\(𝒯∗\)→C0​\(𝒯\)J:C^\{0\}\(\\mathcal\{T\}^\{\*\}\)\\to C^\{0\}\(\\mathcal\{T\}\)\. Then, we have that the Finsler graph Laplacian is a mapℒ:C0​\(ℱ\)→C0​\(ℱ∗\)\\mathcal\{L\}:C^\{0\}\(\\mathcal\{F\}\)\\to C^\{0\}\(\\mathcal\{F\}^\{\*\}\)defined by composition

ℒ​\[f\]=\(δ∘id∗∘d∘J∘δ∘id∘d\)​\[f\],\\mathcal\{L\}\[f\]=\(\\delta\\circ\\mathrm\{id\}^\{\*\}\\circ d\\circ J\\circ\\delta\\circ\\mathrm\{id\}\\circ d\)\[f\],\(24\)or, as a diagram,

C0​\(ℱ\)C0​\(𝒯∗\)C0​\(𝒯\)C0​\(ℱ∗\)C1​\(ℱ\)C1​\(𝒯∗\)C1​\(𝒯\)C1​\(ℱ∗\)dJdidδid∗δ\.\\hbox to223\.64pt\{\\vbox to53\.92pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 111\.82219pt\\lower\-26\.95946pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\nullfont\{\}\{\}\{\}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\}\{\{\{\}\}\}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-111\.82219pt\}\{\-20\.79976pt\}\\pgfsys@invoke\{ \}\\hbox\{\\vbox\{\\halign\{\\pgf@matrix@init@row\\pgf@matrix@step@column\{\\pgf@matrix@startcell\#\\pgf@matrix@endcell\}&\#\\pgf@matrix@padding&&\\pgf@matrix@step@column\{\\pgf@matrix@startcell\#\\pgf@matrix@endcell\}&\#\\pgf@matrix@padding\\cr\\hfil\\qquad\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.3271pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{0\}\(\\mathcal\{F\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\}\}&\\qquad\\hfil&\\hfil\\hskip 44\.27844pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-15\.97293pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{0\}\(\\mathcal\{T\}^\{\*\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\}\}&\\hskip 20\.27847pt\\hfil&\\hfil\\hskip 41\.97983pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.67432pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{0\}\(\\mathcal\{T\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\}\}&\\qquad\\hfil&\\hfil\\hskip 43\.93123pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-15\.62572pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{0\}\(\\mathcal\{F\}^\{\*\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}&\\qquad\\hfil\\cr\\vskip 18\.00005pt\\cr\\hfil\\qquad\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.3271pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{1\}\(\\mathcal\{F\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}&\\qquad\\hfil&\\hfil\\hskip 44\.27844pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-15\.97293pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{1\}\(\\mathcal\{T\}^\{\*\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}&\\hskip 20\.27847pt\\hfil&\\hfil\\hskip 41\.97983pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.67432pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{1\}\(\\mathcal\{T\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}&\\qquad\\hfil&\\hfil\\hskip 43\.93123pt\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-15\.62572pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\{\{C^\{1\}\(\\mathcal\{F\}^\{\*\}\)\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}&\\qquad\\hfil\\cr\}\}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\{\{\{\{\}\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\{\{\}\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ 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\}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-2\.54236pt\}\{20\.0125pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{J\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{29\.97984pt\}\{8\.80002pt\}\\pgfsys@lineto\{29\.97984pt\}\{\-8\.40007pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{0\.0\}\{\-1\.0\}\{1\.0\}\{0\.0\}\{29\.97984pt\}\{\-8\.60005pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{32\.33261pt\}\{\-2\.43054pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{d\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{\-76\.3569pt\}\{\-18\.29976pt\}\\pgfsys@lineto\{\-53\.15689pt\}\{\-18\.29976pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-52\.95691pt\}\{\-18\.29976pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-67\.8972pt\}\{\-15\.94699pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{\\mathrm\{id\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{\-32\.27846pt\}\{\-8\.80003pt\}\\pgfsys@lineto\{\-32\.27846pt\}\{8\.40005pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{0\.0\}\{1\.0\}\{\-1\.0\}\{0\.0\}\{\-32\.27846pt\}\{8\.60004pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{\{\}\{\}\}\}\{\{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-38\.26596pt\}\{\-2\.43056pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{\\delta\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{48\.1597pt\}\{\-18\.29976pt\}\\pgfsys@lineto\{71\.35971pt\}\{\-18\.29976pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{71\.5597pt\}\{\-18\.29976pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{54\.5291pt\}\{\-15\.94699pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{\\mathrm\{id\}^\{\*\}\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{ \{\}\{\}\{\}\}\{\}\{ \{\}\{\}\{\}\} \{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{ \{\}\{\}\}\{\}\{\}\{\{\}\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\{\}\{\{\}\}\{\}\{\}\{\}\{\{\{\}\{\}\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\{\}\{\}\{\}\{\}\{\{\}\}\{\}\{\}\{\{\}\}\\pgfsys@moveto\{91\.89093pt\}\{\-8\.80003pt\}\\pgfsys@lineto\{91\.89093pt\}\{8\.40005pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\}\}\{\{\}\{\{\}\}\{\}\{\}\{\{\}\}\{\{\{\}\}\{\{\{\}\}\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{0\.0\}\{1\.0\}\{\-1\.0\}\{0\.0\}\{91\.89093pt\}\{8\.60004pt\}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\{\{\}\}\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{\{\}\{\}\}\}\{\{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{85\.90343pt\}\{\-2\.43056pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{$\\scriptstyle\{\\delta\}$\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\.\(25\)Of course, if the vector spaces for the sheaves are inner product spaces, we can identify them with their duals\. This allows for the simplification of \([25](https://arxiv.org/html/2606.17185#A3.E25)\) to the diagram \([11](https://arxiv.org/html/2606.17185#S3.E11)\)\.

## Appendix DProof of Laplacian Convergence

In this section, we prove a collection of results towards\\Crefthm:graph\-uniform\-convergence\. Throughout, we will assume thatϵ\\epsilonis sufficiently small so that it is less than the reach of the manifoldℳ⊂ℝD\\mathcal\{M\}\\subset\\mathbb\{R\}^\{D\}\. For fixedϵ\>0\\epsilon\>0, it will be useful to consider the empirical Finsler Laplacian in the limiting régimen→∞n\\to\\infty\. Under mild conditions on the kernelκ\\kappaand densitypp, and assumingffis appropriately bounded, we have

∇ϵf​\(x\)=hϵ​\(Cϵ​\(x\)\)​\(bϵ​\[f\]​\(x\)\)Cϵ​\(x\)=1ϵd\+2​∫κ​\(x−yϵ\)​\(x−y\)⊗\(x−y\)​𝑑μ​\(y\)bϵ​\[f\]​\(x\)=1ϵd\+2​∫κ​\(x−yϵ\)​\(f​\(x\)−f​\(y\)\)​\(x−y\)​𝑑μ​\(y\)\.\\begin\{gathered\}\\nabla\_\{\\epsilon\}f\(x\)=h\_\{\\epsilon\}\\left\(C\_\{\\epsilon\}\(x\)\\right\)\\left\(b\_\{\\epsilon\}\[f\]\(x\)\\right\)\\\\ C\_\{\\epsilon\}\(x\)=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\\kappa\\left\(\\frac\{x\-y\}\{\\epsilon\}\\right\)\(x\-y\)\\otimes\(x\-y\)d\\mu\(y\)\\\\ b\_\{\\epsilon\}\[f\]\(x\)=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\\kappa\\left\(\\frac\{x\-y\}\{\\epsilon\}\\right\)\(f\(x\)\-f\(y\)\)\(x\-y\)d\\mu\(y\)\.\\end\{gathered\}\(26\)
Using this smoothed estimator of the gradient∇ϵf\\nabla\_\{\\epsilon\}f, we define the smoothed asymptotic energy functional

Eϵ​\[f\]=∫12​\[F∗​\(∇ϵf​\(x\)\)\]2​𝑑μ​\(x\),E\_\{\\epsilon\}\[f\]=\\int\\frac\{1\}\{2\}\\left\[F^\{\*\}\(\\nabla\_\{\\epsilon\}f\(x\)\)\\right\]^\{2\}d\\mu\(x\),\(27\)and the smoothed Laplacian by taking the Fréchet derivative

Δϵ​\[f\]=1p​D​Eϵ​\[f\]\.\\Delta\_\{\\epsilon\}\[f\]=\\frac\{1\}\{p\}DE\_\{\\epsilon\}\[f\]\.\(28\)Under the assumption that the kernelκ\\kappais radially symmetric \(isotropic with respect to the Euclidean norm\) about the origin, we evaluate the smoothed Laplacian \([28](https://arxiv.org/html/2606.17185#A4.E28)\) to

Δϵ​\[f\]​\(x\)\\displaystyle\\Delta\_\{\\epsilon\}\[f\]\(x\)=1ϵd\+2∫κ\(x−yϵ\)\(hϵ\(Cϵ\(x\)\)J\(∇ϵf\(x\)\)\\displaystyle=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\\kappa\\left\(\\frac\{x\-y\}\{\\epsilon\}\\right\)\\Big\(h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)\(29\)\+hϵ\(Cϵ\(y\)\)J\(∇ϵf\(y\)\)\)⋅\(x−y\)dμ\(y\)\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\+h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(y\)\)J\(\\nabla\_\{\\epsilon\}f\(y\)\)\\Big\)\\cdot\(x\-y\)d\\mu\(y\)\.
The strategy will amount to a bound between the difference ofΔϵ,n​\[f\]\\Delta\_\{\\epsilon,n\}\[f\]andΔ​\[f\]\\Delta\[f\], intermediated byΔϵ​\[f\]\\Delta\_\{\\epsilon\}\[f\]via the triangle inequality\. In particular, we will bound the sum

\|Δ​\[f\]−Δϵ,n​\[f\]\|≤\|Δ​\[f\]−Δϵ​\[f\]\|\+\|Δϵ​\[f\]−Δϵ,n​\[f\]\|\.\|\\Delta\[f\]\-\\Delta\_\{\\epsilon,n\}\[f\]\|\\leq\|\\Delta\[f\]\-\\Delta\_\{\\epsilon\}\[f\]\|\+\|\\Delta\_\{\\epsilon\}\[f\]\-\\Delta\_\{\\epsilon,n\}\[f\]\|\.\(30\)The first term measures the approximation error due to the smoothing parameterϵ\\epsilon, and the second measures the error due to discretization\.

### D\.1Local Kernel Moments

Letx∈ℳx\\in\\mathcal\{M\}be given\. Through a rigid transformation, we choose coordinates such thatx=0x=0and the tangent plane atxxoccupies the firstddcoordinates, with the normal plane occupying the remainingD−dD\-dcoordinates\. Denoting the tangent coordinates byu∈Tx​ℳu\\in T\_\{x\}\\mathcal\{M\}, we representℳ\\mathcal\{M\}locally as a graphyx​\(u\)=\(u,gx​\(u\)\)∈ℝDy\_\{x\}\(u\)=\(u,g\_\{x\}\(u\)\)\\in\\mathbb\{R\}^\{D\}\. The normal coordinatesgx​\(u\)g\_\{x\}\(u\)satisfy

gx​\(u\)=Πx​\[u,u\]\+Dx​\[u,u,u\]\+O​\(u4\),g\_\{x\}\(u\)=\\Pi\_\{x\}\[u,u\]\+D\_\{x\}\[u,u,u\]\+O\(u^\{4\}\),\(31\)whereΠx\\Pi\_\{x\}is a symmetric tensor given by the second fundamental form atxx,DxD\_\{x\}is a third\-order tensor, andO​\(u4\)O\(u^\{4\}\)denotes some fourth\-order tensor expression ofuu\. Denoting the tensor product of a vectorvvwith itselfkktimes byv⊗kv^\{\\otimes k\}, we consider expressions of the form

M​\(x,k,ϵ\)=1ϵd\+2​∫κ​\(yx​\(u\)ϵ\)​yx​\(u\)⊗k​𝑑u\.M\(x,k,\\epsilon\)=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\\kappa\\left\(\\frac\{y\_\{x\}\(u\)\}\{\\epsilon\}\\right\)y\_\{x\}\(u\)^\{\\otimes k\}du\.\(32\)First, we carry out the change of variablesv=u/ϵv=u/\\epsilon, yielding

M​\(x,k,ϵ\)=1ϵ2​∫κ​\(yx​\(ϵ​v\)ϵ\)​yx​\(ϵ​v\)⊗k​𝑑v\.M\(x,k,\\epsilon\)=\\frac\{1\}\{\\epsilon^\{2\}\}\\int\\kappa\\left\(\\frac\{y\_\{x\}\(\\epsilon v\)\}\{\\epsilon\}\\right\)y\_\{x\}\(\\epsilon v\)^\{\\otimes k\}dv\.\(33\)We assume thatκ​\(z\)\\kappa\(z\)is a smooth, compactly supported function of‖z‖2\\\|z\\\|^\{2\},i\.e\.,κ​\(z\)=κ​\(‖z‖2\)\\kappa\(z\)=\\kappa\(\\\|z\\\|^\{2\}\)\. By a Taylor series expansion centered at a givenzz, forw⟂zw\\perp z, we have

κ​\(z\+w\)\\displaystyle\\kappa\(z\+w\)=κ​\(‖z‖2\+‖w‖2\)\\displaystyle=\\kappa\(\\\|z\\\|^\{2\}\+\\\|w\\\|^\{2\}\)\(34\)=κ​\(‖z‖2\)\+κ′​\(‖z‖2\)2​‖w‖2\+O​\(‖w‖4\)\.\\displaystyle=\\kappa\(\\\|z\\\|^\{2\}\)\+\\frac\{\\kappa^\{\\prime\}\(\\\|z\\\|^\{2\}\)\}\{2\}\\\|w\\\|^\{2\}\+O\(\\\|w\\\|^\{4\}\)\.
#### D\.1\.1Flattening of the Graph

With the preliminary definitions handled, we begin by considering the behavior ofy\(ϵvϵ\\frac\{y\(\\epsilon v\}\{\\epsilon\}asϵ→0\\epsilon\\to 0\. Observe thaty​\(ϵ​v\)ϵ=\(v,gx​\(ϵ​v\)/ϵ\)\\frac\{y\(\\epsilon v\)\}\{\\epsilon\}=\(v,g\_\{x\}\(\\epsilon v\)/\\epsilon\), so the main quantity of interest isgx​\(ϵ​v\)g\_\{x\}\(\\epsilon v\)\. By direct substitution into \([31](https://arxiv.org/html/2606.17185#A4.E31)\), we have

gx​\(ϵ​v\)=ϵ2​Πx​\[v,v\]\+ϵ3​Dx​\[v,v,v\]\+ϵ4​O​\(v4\)\.g\_\{x\}\(\\epsilon v\)=\\epsilon^\{2\}\\Pi\_\{x\}\[v,v\]\+\\epsilon^\{3\}D\_\{x\}\[v,v,v\]\+\\epsilon^\{4\}O\(v^\{4\}\)\.\(35\)Note thatgx​\(ϵ​v\)⟂vg\_\{x\}\(\\epsilon v\)\\perp v, and the squared\-norm satisfies

‖gx​\(ϵ​v\)‖2=ϵ4​‖Πx​\[v,v\]‖2\+2​ϵ5​⟨Πx​\[v,v\],Dx​\[v,v,v\]⟩\+ϵ6​O​\(v6\)\.\\\|g\_\{x\}\(\\epsilon v\)\\\|^\{2\}=\\epsilon^\{4\}\\\|\\Pi\_\{x\}\[v,v\]\\\|^\{2\}\+2\\epsilon^\{5\}\\langle\\Pi\_\{x\}\[v,v\],D\_\{x\}\[v,v,v\]\\rangle\+\\epsilon^\{6\}O\(v^\{6\}\)\.\(36\)We conclude, by substitution into \([34](https://arxiv.org/html/2606.17185#A4.E34)\), that

κ​\(y​\(ϵ​v\)ϵ\)=κ​\(‖v‖2\)\+κ′​\(‖v‖2\)​\(ϵ2​‖Πx​\[v,v\]‖2\+2​ϵ3​⟨Πx​\[v,v\],Dx​\[v,v,v\]⟩\)\+O​\(ϵ4\),\\kappa\\left\(\\frac\{y\(\\epsilon v\)\}\{\\epsilon\}\\right\)=\\kappa\(\\\|v\\\|^\{2\}\)\+\\kappa^\{\\prime\}\(\\\|v\\\|^\{2\}\)\\left\(\\epsilon^\{2\}\\\|\\Pi\_\{x\}\[v,v\]\\\|^\{2\}\+2\\epsilon^\{3\}\\langle\\Pi\_\{x\}\[v,v\],D\_\{x\}\[v,v,v\]\\rangle\\right\)\+O\(\\epsilon^\{4\}\),\(37\)recalling that the compact support ofκ\\kappaallows us to assumev6=O​\(1\)v^\{6\}=O\(1\)\. That is,κ​\(y​\(ϵ​v\)/ϵ\)=κ​\(‖v‖2\)\+O​\(ϵ2\)\\kappa\(y\(\\epsilon v\)/\\epsilon\)=\\kappa\(\\\|v\\\|^\{2\}\)\+O\(\\epsilon^\{2\}\), so that the normal coordinates become negligible asϵ→0\\epsilon\\to 0\.

#### D\.1\.2Parity of Local Kernel Moment Integrals

LetT:\(Tx​ℳ\)k→ℝT:\(T\_\{x\}\\mathcal\{M\}\)^\{k\}\\to\\mathbb\{R\}be a multilinear form for some odd value ofkk, and consider the integral ofκ​\(‖v‖2\)​T​\[v,…,v\]\\kappa\(\\\|v\\\|^\{2\}\)T\[v,\\ldots,v\]\. Sincekkis odd,T​\[v,…,v\]T\[v,\\ldots,v\]is odd, so the integral vanishes by radial symmetry,i\.e\.,

∫κ​\(‖v‖2\)​T​\[v,…,v\]​𝑑v=0\.\\int\\kappa\(\\\|v\\\|^\{2\}\)T\[v,\\ldots,v\]dv=0\.\(38\)A similar property holds forκ′\\kappa^\{\\prime\}, so that

∫κ′​\(‖v‖2\)​T​\[v,…,v\]​𝑑v=0\.\\int\\kappa^\{\\prime\}\(\\\|v\\\|^\{2\}\)T\[v,\\ldots,v\]dv=0\.\(39\)
We define a few quantities of interest for integrals of tensor forms with even parity\. The first is the second moment ofκ​\(‖v‖2\)\\kappa\(\\\|v\\\|^\{2\}\):

mκ​Id=∫κ​\(‖v‖2\)​v⊗2​𝑑v,m\_\{\\kappa\}I\_\{d\}=\\int\\kappa\(\\\|v\\\|^\{2\}\)v^\{\\otimes 2\}dv,\(40\)whereIdI\_\{d\}denotes thed×dd\\times didentity matrix\. The second is defined based on the curvature atxx:

mκ,x=∫κ​\(‖v‖2\)​Πx​\[v,v\]​𝑑v\.m\_\{\\kappa,x\}=\\int\\kappa\(\\\|v\\\|^\{2\}\)\\Pi\_\{x\}\[v,v\]dv\.\(41\)Note thatmκm\_\{\\kappa\}is a scalar, butmκ,xm\_\{\\kappa,x\}is a vector normal toxx\.

#### D\.1\.3Computation of Local Kernel Moments

With these tools in place, computing the momentsM​\(x,k,ϵ\)M\(x,k,\\epsilon\)becomes a trivial exercise\. Whenk=1k=1, substituting \([37](https://arxiv.org/html/2606.17185#A4.E37)\) and \([35](https://arxiv.org/html/2606.17185#A4.E35)\) into \([33](https://arxiv.org/html/2606.17185#A4.E33)\) and zeroing all terms with odd parity, we have

M​\(x,1,ϵ\)=\(0,mκ,x\)\+O​\(ϵ2\)\.M\(x,1,\\epsilon\)=\(0,m\_\{\\kappa,x\}\)\+O\(\\epsilon^\{2\}\)\.\(42\)Similarly, whenk=2k=2, we have

M​\(x,2,ϵ\)=\[mκ​Id000\]\+O​\(ϵ2\)\.M\(x,2,\\epsilon\)=\\begin\{bmatrix\}m\_\{\\kappa\}I\_\{d\}&0\\\\ 0&0\\end\{bmatrix\}\+O\(\\epsilon^\{2\}\)\.\(43\)

### D\.2Smoothed Gradient

We consider now the deviation between the smoothed gradient∇ϵf\\nabla\_\{\\epsilon\}fand the true gradient∇f\\nabla f\. As before, letx∈ℳx\\in\\mathcal\{M\}be given, and apply a rigid transformation so thatx=0x=0and the tangent plane occupies the firstddcoordinates\. We pull back the densityppand the functionfftoTx​ℳT\_\{x\}\\mathcal\{M\}so that the relevant integrals may locally be defined as integrals over the tangent plane\. We rewrite \([26](https://arxiv.org/html/2606.17185#A4.E26)\) as

∇ϵf​\(x\)=hϵ​\(Cϵ​\(x\)\)​bϵ​\[f\]​\(x\)Cϵ​\(x\)=1ϵd\+2​∫Tx​ℳκ​\(yx​\(u\)ϵ\)​yx​\(u\)⊗2​px​\(u\)​𝑑ubϵ​\[f\]​\(x\)=1ϵd\+2​∫Tx​ℳκ​\(yx​\(u\)ϵ\)​yx​\(u\)​\(fx​\(u\)−fx​\(0\)\)​px​\(u\)​𝑑u\.\\begin\{gathered\}\\nabla\_\{\\epsilon\}f\(x\)=h\_\{\\epsilon\}\\left\(C\_\{\\epsilon\}\(x\)\\right\)b\_\{\\epsilon\}\[f\]\(x\)\\\\ C\_\{\\epsilon\}\(x\)=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\_\{T\_\{x\}\\mathcal\{M\}\}\\kappa\\left\(\\frac\{y\_\{x\}\(u\)\}\{\\epsilon\}\\right\)y\_\{x\}\(u\)^\{\\otimes 2\}p\_\{x\}\(u\)du\\\\ b\_\{\\epsilon\}\[f\]\(x\)=\\frac\{1\}\{\\epsilon^\{d\+2\}\}\\int\_\{T\_\{x\}\\mathcal\{M\}\}\\kappa\\left\(\\frac\{y\_\{x\}\(u\)\}\{\\epsilon\}\\right\)y\_\{x\}\(u\)\(f\_\{x\}\(u\)\-f\_\{x\}\(0\)\)p\_\{x\}\(u\)du\.\\end\{gathered\}\(44\)
#### D\.2\.1Local Taylor Series

It will be useful to gather some local approximations ofpxp\_\{x\}andfxf\_\{x\}by their Taylor series representations\. Taking a Taylor expansion ofpxp\_\{x\}, we have

px​\(u\)=px​\(0\)\+⟨∇px​\(0\),u⟩\+12​∇2px​\(0\)​\[u,u\]\+16​∇3px​\(0\)​\[u,u,u\]\+O​\(u4\)\.p\_\{x\}\(u\)=p\_\{x\}\(0\)\+\\langle\\nabla p\_\{x\}\(0\),u\\rangle\+\\frac\{1\}\{2\}\\nabla^\{2\}p\_\{x\}\(0\)\[u,u\]\+\\frac\{1\}\{6\}\\nabla^\{3\}p\_\{x\}\(0\)\[u,u,u\]\+O\(u^\{4\}\)\.\(45\)Similarly, carrying out a Taylor series expansion offx​\(u\)−fx​\(0\)f\_\{x\}\(u\)\-f\_\{x\}\(0\)centered at zero, we have

fx​\(u\)−fx​\(0\)=∇fx​\(0\)​\[u\]\+12​∇2fx​\(0\)​\[u,u\]\+16​∇3fx​\(0\)​\[u,u,u\]\+O​\(u4\)\.f\_\{x\}\(u\)\-f\_\{x\}\(0\)=\\nabla f\_\{x\}\(0\)\[u\]\+\\frac\{1\}\{2\}\\nabla^\{2\}f\_\{x\}\(0\)\[u,u\]\+\\frac\{1\}\{6\}\\nabla^\{3\}f\_\{x\}\(0\)\[u,u,u\]\+O\(u^\{4\}\)\.\(46\)

#### D\.2\.2Smoothed Covariance

Performing the substitutionv=u/ϵv=u/\\epsilonin \([44](https://arxiv.org/html/2606.17185#A4.E44)\) yields

Cϵ​\(x\)=1ϵ2​∫Tx​ℳκ​\(yx​\(ϵ​v\)ϵ\)​yx​\(ϵ​v\)⊗2​px​\(ϵ​v\)​𝑑v\.C\_\{\\epsilon\}\(x\)=\\frac\{1\}\{\\epsilon^\{2\}\}\\int\_\{T\_\{x\}\\mathcal\{M\}\}\\kappa\\left\(\\frac\{y\_\{x\}\(\\epsilon v\)\}\{\\epsilon\}\\right\)y\_\{x\}\(\\epsilon v\)^\{\\otimes 2\}p\_\{x\}\(\\epsilon v\)dv\.\(47\)Substituting in the Taylor series expansion of the density \([45](https://arxiv.org/html/2606.17185#A4.E45)\) yields

Cϵ​\(x\)=\[px​\(0\)​mκ​Id000\]\+O​\(ϵ2\)\.C\_\{\\epsilon\}\(x\)=\\begin\{bmatrix\}p\_\{x\}\(0\)m\_\{\\kappa\}I\_\{d\}&0\\\\ 0&0\\end\{bmatrix\}\+O\(\\epsilon^\{2\}\)\.\(48\)That is,Cϵ​\(x\)C\_\{\\epsilon\}\(x\)is approximatelyp​\(x\)​mκ​Idp\(x\)m\_\{\\kappa\}I\_\{d\}in the tangent coordinates, apart from anO​\(ϵ2\)O\(\\epsilon^\{2\}\)error term\.

#### D\.2\.3Smooth Cogradient

We carry out the same substitution as before,v=u/ϵv=u/\\epsilon, yielding

bϵ​\[f\]​\(0\)=1ϵ2​∫κ​\(y​\(ϵ​v\)ϵ\)​y​\(ϵ​v\)​\(fx​\(ϵ​v\)−fx​\(0\)\)​px​\(ϵ​v\)​𝑑v\.b\_\{\\epsilon\}\[f\]\(0\)=\\frac\{1\}\{\\epsilon^\{2\}\}\\int\\kappa\\left\(\\frac\{y\(\\epsilon v\)\}\{\\epsilon\}\\right\)y\(\\epsilon v\)\(f\_\{x\}\(\\epsilon v\)\-f\_\{x\}\(0\)\)p\_\{x\}\(\\epsilon v\)dv\.\(49\)Substituting the Taylor series expansions \([45](https://arxiv.org/html/2606.17185#A4.E45)\) and \([46](https://arxiv.org/html/2606.17185#A4.E46)\), we have

bϵ​\[f\]​\(x\)=Cϵ​\(x\)​∇f​\(x\)\+O​\(ϵ2\)\.b\_\{\\epsilon\}\[f\]\(x\)=C\_\{\\epsilon\}\(x\)\\nabla f\(x\)\+O\(\\epsilon^\{2\}\)\.\(50\)

#### D\.2\.4Combined Error

By \([50](https://arxiv.org/html/2606.17185#A4.E50)\), and usinghϵh\_\{\\epsilon\}from \([21](https://arxiv.org/html/2606.17185#A2.E21)\), we have

∇ϵf​\(x\)\\displaystyle\\nabla\_\{\\epsilon\}f\(x\)=hϵ​\(Cϵ​\(x\)\)​Cϵ​\(x\)​∇f​\(x\)\+O​\(ϵ2\)​hϵ​\(Cϵ​\(x\)\)\\displaystyle=h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)C\_\{\\epsilon\}\(x\)\\nabla f\(x\)\+O\(\\epsilon^\{2\}\)h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)\(51\)=∇f​\(x\)\+O​\(ϵ\),\\displaystyle=\\nabla f\(x\)\+O\(\\epsilon\),where∇f​\(x\)∈ℝD\\nabla f\(x\)\\in\\mathbb\{R\}^\{D\}by the obvious embedding of the tangent plane\. It follows thatJ​\(∇ϵf​\(x\)\)=J​\(∇f​\(x\)\)\+O​\(ϵ\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)=J\(\\nabla f\(x\)\)\+O\(\\epsilon\)\.

Sincep​\(x\)p\(x\)is bounded away from zero, for sufficiently smallϵ\>0\\epsilon\>0we havehϵ​\(p​\(x\)​mκ​Id\)≈1p​\(x\)​mκ​Idh\_\{\\epsilon\}\(p\(x\)m\_\{\\kappa\}I\_\{d\}\)\\approx\\frac\{1\}\{p\(x\)m\_\{\\kappa\}\}I\_\{d\}andhϵ​\(O​\(ϵ2\)\)=O​\(ϵ2\)h\_\{\\epsilon\}\(O\(\\epsilon^\{2\}\)\)=O\(\\epsilon^\{2\}\)\. In other words,

hϵ​\(Cϵ​\(x\)\)=\(\[1p​\(x\)​mκ​Id000\]\)\+O​\(ϵ2\)\.h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)=\\left\(\\begin\{bmatrix\}\\frac\{1\}\{p\(x\)m\_\{\\kappa\}\}I\_\{d\}&0\\\\ 0&0\\end\{bmatrix\}\\right\)\+O\(\\epsilon^\{2\}\)\.\(52\)Putting the two bounds together, this implies

hϵ​\(Cϵ​\(x\)\)​J​\(∇ϵf​\(x\)\)=mκ−1​p−1​\(x\)​∇f​\(x\)\+O​\(ϵ\)\.h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)=m\_\{\\kappa\}^\{\-1\}p^\{\-1\}\(x\)\\nabla f\(x\)\+O\(\\epsilon\)\.\(53\)

### D\.3Smoothed Finsler Laplacian

We begin by bounding the difference betweenΔ​\[f\]​\(x\)\\Delta\[f\]\(x\)andΔϵ​\[f\]​\(x\)\\Delta\_\{\\epsilon\}\[f\]\(x\)\. For fixedxx, we apply a rigid transformation as before, so thatx=0x=0and the manifold is a local graph of the tangent planeyx​\(u\)=\(u,gx​\(u\)\)y\_\{x\}\(u\)=\(u,g\_\{x\}\(u\)\)\. For convenience, define the vector field on the tangent plane

Qx​\[f\]​\(u\)=J​\(∇f​\(yx​\(u\)\)\)p​\(x\)Q\_\{x\}\[f\]\(u\)=\\frac\{J\(\\nabla f\(y\_\{x\}\(u\)\)\)\}\{p\(x\)\}\(54\)so that by \([53](https://arxiv.org/html/2606.17185#A4.E53)\) we have

hϵ​\(Cϵ​\(yx​\(u\)\)\)​J​\(∇ϵf​\(yx​\(u\)\)\)=mκ−1​\(Qx​\[f\]​\(u\)\+O​\(ϵ\)\)\.h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(y\_\{x\}\(u\)\)\)J\(\\nabla\_\{\\epsilon\}f\(y\_\{x\}\(u\)\)\)=m\_\{\\kappa\}^\{\-1\}\\left\(Q\_\{x\}\[f\]\(u\)\+O\(\\epsilon\)\\right\)\.\(55\)Substituting \([55](https://arxiv.org/html/2606.17185#A4.E55)\) into \([29](https://arxiv.org/html/2606.17185#A4.E29)\) and rewriting the integral as one over the tangent plane yields, after a change of variables,

Δϵ​\[f\]​\(0\)=1mκ​ϵ2​∫κ​\(y​\(ϵ​v\)ϵ\)​\(Qx​\[f\]​\(ϵ​v\)\+Qx​\[f\]​\(0\)\+O​\(ϵ\)\)⋅y​\(ϵ​v\)​px​\(ϵ​v\)​𝑑v\.\\Delta\_\{\\epsilon\}\[f\]\(0\)=\\frac\{1\}\{m\_\{\\kappa\}\\epsilon^\{2\}\}\\int\\kappa\\left\(\\frac\{y\(\\epsilon v\)\}\{\\epsilon\}\\right\)\\left\(Q\_\{x\}\[f\]\(\\epsilon v\)\+Q\_\{x\}\[f\]\(0\)\+O\(\\epsilon\)\\right\)\\cdot y\(\\epsilon v\)p\_\{x\}\(\\epsilon v\)dv\.\(56\)We perform a Taylor expansion on the product\(Qx​\[f\]​\(ϵ​v\)\+Qx​\[f\]​\(0\)\+O​\(ϵ\)\)​px​\(ϵ​v\)\(Q\_\{x\}\[f\]\(\\epsilon v\)\+Q\_\{x\}\[f\]\(0\)\+O\(\\epsilon\)\)p\_\{x\}\(\\epsilon v\), and applying the same parity arguments as before, simplify this expression to

Δϵ​\[f\]​\(x\)\\displaystyle\\Delta\_\{\\epsilon\}\[f\]\(x\)=Tr⁡\(px​\(0\)​∇Qx​\[f\]​\(0\)\+2​Qx​\[f\]​\(0\)⊗∇px​\(0\)\)\+O​\(ϵ\)\\displaystyle=\\operatorname\{Tr\}\\left\(p\_\{x\}\(0\)\\nabla Q\_\{x\}\[f\]\(0\)\+2Q\_\{x\}\[f\]\(0\)\\otimes\\nabla p\_\{x\}\(0\)\\right\)\+O\(\\epsilon\)\(57\)=2​⟨J​\(∇f​\(x\)\),∇p​\(x\)⟩p​\(x\)\+p​\(x\)​div​\(J​\(∇f​\(x\)\)p​\(x\)\)\+O​\(ϵ\)\\displaystyle=2\\frac\{\\langle J\(\\nabla f\(x\)\),\\nabla p\(x\)\\rangle\}\{p\(x\)\}\+p\(x\)\\mathrm\{div\}\\left\(\\frac\{J\(\\nabla f\(x\)\)\}\{p\(x\)\}\\right\)\+O\(\\epsilon\)=1p​\(x\)​div​\(p​\(x\)​J​\(∇f​\(x\)\)\)\+O​\(ϵ\)\\displaystyle=\\frac\{1\}\{p\(x\)\}\\mathrm\{div\}\\left\(p\(x\)J\(\\nabla f\(x\)\)\\right\)\+O\(\\epsilon\)=Δ​\[f\]​\(x\)\+O​\(ϵ\)\.\\displaystyle=\\Delta\[f\]\(x\)\+O\(\\epsilon\)\.Thus, for fixedxxandff,Δϵ​\[f\]​\(x\)\\Delta\_\{\\epsilon\}\[f\]\(x\)is anO​\(ϵ\)O\(\\epsilon\)\-approximant ofΔ​\[f\]​\(x\)\\Delta\[f\]\(x\)\.

### D\.4Concentration Inequalities

We now bound the difference betweenΔϵ​\[f\]​\(x\)\\Delta\_\{\\epsilon\}\[f\]\(x\)andΔϵ,n​\[f\]​\(x\)\\Delta\_\{\\epsilon,n\}\[f\]\(x\)wherexxis assumed to be a sampled point\. The expression for the empirical Laplacian \([7](https://arxiv.org/html/2606.17185#S2.E7)\) closely resembles a Monte Carlo estimate of the smoothed Laplacian \([29](https://arxiv.org/html/2606.17185#A4.E29)\), with the exception of the smoothed gradients∇ϵ,nf\\nabla\_\{\\epsilon,n\}fcoupling the terms in the sum\. To account for this, we introduce an “oracle” empirical Laplacian that has access to the true \(smoothed\) gradients∇ϵf\\nabla\_\{\\epsilon\}f, and then bound the difference in two steps via the triangle inequality\. In particular, defining

Δ~ϵ,n​\[f\]​\(x\)=1n​ϵd\+2​∑i=1nκ​\(x−xiϵ\)\\displaystyle\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\)=\\frac\{1\}\{n\\epsilon^\{d\+2\}\}\\sum\_\{i=1\}^\{n\}\\kappa\\left\(\\frac\{x\-x\_\{i\}\}\{\\epsilon\}\\right\)\(hϵ\(Cϵ\(x\)\)J\(∇ϵf\(x\)\)\\displaystyle\\Big\(h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)\(58\)\+hϵ\(Cϵ\(xi\)\)J\(∇ϵf\(xi\)\)\)\(x−xi\)\\displaystyle\\quad\+h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\_\{i\}\)\)J\(\\nabla\_\{\\epsilon\}f\(x\_\{i\}\)\)\\Big\)\(x\-x\_\{i\}\)we consider the bound

\|Δϵ​\[f\]​\(x\)−Δϵ,n​\[f\]​\(x\)\|≤\|Δϵ​\[f\]​\(x\)−Δ~ϵ,n​\[f\]​\(x\)\|\+\|Δ~ϵ,n​\[f\]​\(x\)−Δϵ,n​\[f\]​\(x\)\|\.\\Big\|\\Delta\_\{\\epsilon\}\[f\]\(x\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\)\\Big\|\\leq\\Big\|\\Delta\_\{\\epsilon\}\[f\]\(x\)\-\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\)\\Big\|\+\\Big\|\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\)\\Big\|\.\(59\)
#### D\.4\.1Sampling Error

We consider the first term in the bound \([59](https://arxiv.org/html/2606.17185#A4.E59)\), which is the error due to discretization without the effects of gradient estimation\. Observe thatΔ~ϵ,n​\[f\]​\(xj\)\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)is an average ofi\.i\.d\.terms bounded byO​\(ϵ−\(d\+1\)\)O\(\\epsilon^\{\-\(d\+1\)\}\), with varianceO​\(ϵ−\(d\+2\)\)O\(\\epsilon^\{\-\(d\+2\)\}\), and whose expectation is equal toΔϵ​\[f\]​\(x\)\\Delta\_\{\\epsilon\}\[f\]\(x\)\. Recalling thatJJis assumed Lipschitz, Bernstein’s inequality for bounded random variables yields the concentration inequality for fixedxx:

ℙ​\{\|Δϵ​\[f\]​\(x\)−Δ~ϵ,n​\[f\]​\(x\)\|<tϵ\}\>1−2​exp⁡\(−t2​n​ϵd/3\)\.\\mathbb\{P\}\\Big\\\{\|\\Delta\_\{\\epsilon\}\[f\]\(x\)\-\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\)\|<\\frac\{t\}\{\\epsilon\}\\Big\\\}\>1\-2\\exp\\Big\(\-t^\{2\}n\\epsilon^\{d\}/3\\Big\)\.\(60\)

#### D\.4\.2Covariance Estimation

SinceCϵ,n​\(x\)C\_\{\\epsilon,n\}\(x\)is the average ofnni\.i\.d\.random matrices with operator norm uniformly bounded byO​\(ϵ−d\)O\(\\epsilon^\{\-d\}\)and expectationCϵ​\(x\)C\_\{\\epsilon\}\(x\), the matrix Hoeffding inequality implies that fort≥0t\\geq 0,

ℙ​\{‖Cϵ,n​\(x\)−Cϵ​\(x\)‖<t\}\>1−D⋅exp⁡\(−t2​n​ϵd/2\)\.\\mathbb\{P\}\\Big\\\{\\\|C\_\{\\epsilon,n\}\(x\)\-C\_\{\\epsilon\}\(x\)\\\|<t\\Big\\\}\>1\-D\\cdot\\exp\\left\(\-t^\{2\}n\\epsilon^\{d\}/2\\right\)\.\(61\)Fort=O​\(ϵ2\)t=O\(\\epsilon^\{2\}\), this implies

ℙ​\{‖hϵ​\(Cϵ,n​\(x\)\)−hϵ​\(Cϵ​\(x\)\)‖<t\}\>1−D⋅exp⁡\(−t2​n​ϵd/2\)\.\\mathbb\{P\}\\Big\\\{\\\|h\_\{\\epsilon\}\(C\_\{\\epsilon,n\}\(x\)\)\-h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)\\\|<t\\Big\\\}\>1\-D\\cdot\\exp\\left\(\-t^\{2\}n\\epsilon^\{d\}/2\\right\)\.\(62\)

#### D\.4\.3Cogradient Estimation

As a consequence of a previous approximation \([50](https://arxiv.org/html/2606.17185#A4.E50)\), we have

‖bϵ​\[f\]​\(x\)‖≤p​\(x\)​mκ​‖∇f​\(x\)‖\+O​\(ϵ2\)\.\\\|b\_\{\\epsilon\}\[f\]\(x\)\\\|\\leq p\(x\)m\_\{\\kappa\}\\\|\\nabla f\(x\)\\\|\+O\(\\epsilon^\{2\}\)\.\(63\)For fixedxx, the empirical cogradientbϵ,n​\[f\]​\(x\)b\_\{\\epsilon,n\}\[f\]\(x\)is the mean ofnni\.i\.d\.random vectors with norm uniformly bounded byO​\(ϵ−d\)O\(\\epsilon^\{\-d\}\), expected valuebϵ​\[f\]​\(x\)b\_\{\\epsilon\}\[f\]\(x\), and variance bounded byO​\(ϵ−d\)O\(\\epsilon^\{\-d\}\)\. The vector Bernstein inequality yields

ℙ​\{‖bϵ,n​\[f\]​\(x\)−bϵ​\[f\]​\(x\)‖<t\}\>1−2​exp⁡\(−t2​n​ϵd\)\.\\mathbb\{P\}\\Big\\\{\\\|b\_\{\\epsilon,n\}\[f\]\(x\)\-b\_\{\\epsilon\}\[f\]\(x\)\\\|<t\\Big\\\}\>1\-2\\exp\\left\(\-t^\{2\}n\\epsilon^\{d\}\\right\)\.\(64\)

#### D\.4\.4Gradient Approximation Error

We now consider the second term in the bound \([59](https://arxiv.org/html/2606.17185#A4.E59)\), which is the error in the empirical Laplacian due to estimation of the gradients at the sampled points\. By a union bound over \([62](https://arxiv.org/html/2606.17185#A4.E62)\) and \([64](https://arxiv.org/html/2606.17185#A4.E64)\), the empirical gradient satisfies

ℙ​\{‖∇ϵ,nf​\(x\)−∇ϵf​\(x\)‖<O​\(t\)\}\>1−\(2\+D\)​exp⁡\(−t2​n​ϵd/2\),\\mathbb\{P\}\\Big\\\{\\\|\\nabla\_\{\\epsilon,n\}f\(x\)\-\\nabla\_\{\\epsilon\}f\(x\)\\\|<O\(t\)\\Big\\\}\>1\-\(2\+D\)\\exp\\left\(\-t^\{2\}n\\epsilon^\{d\}/2\\right\),\(65\)where we assumet=O​\(ϵ2\)t=O\(\\epsilon^\{2\}\)\. SinceJJis assumed Lipschitz, these conditions imply our desired bound\. Namely,

ℙ​\{‖hϵ​\(Cϵ,n​\(x\)\)​J​\(∇ϵ,nf​\(x\)\)−hϵ​\(Cϵ​\(x\)\)​J​\(∇ϵf​\(x\)\)‖<O​\(t\)\}\>1−\(2\+D\)​exp⁡\(−t2​n​ϵd/2\)\.\\mathbb\{P\}\\Big\\\{\\\|h\_\{\\epsilon\}\(C\_\{\\epsilon,n\}\(x\)\)J\(\\nabla\_\{\\epsilon,n\}f\(x\)\)\-h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)\\\|<O\(t\)\\Big\\\}\>1\-\(2\+D\)\\exp\\left\(\-t^\{2\}n\\epsilon^\{d\}/2\\right\)\.\(66\)

### D\.5Proof of\\Crefthm:graph\-uniform\-convergence

For convenience, define

Φϵ​\(x\)\\displaystyle\\Phi\_\{\\epsilon\}\(x\)=hϵ​\(Cϵ​\(x\)\)​J​\(∇ϵf​\(x\)\)\\displaystyle=h\_\{\\epsilon\}\(C\_\{\\epsilon\}\(x\)\)J\(\\nabla\_\{\\epsilon\}f\(x\)\)\(67\)Φϵ,n​\(x\)\\displaystyle\\Phi\_\{\\epsilon,n\}\(x\)=hϵ​\(Cϵ,n​\(x\)\)​J​\(∇ϵ,nf​\(x\)\)\.\\displaystyle=h\_\{\\epsilon\}\(C\_\{\\epsilon,n\}\(x\)\)J\(\\nabla\_\{\\epsilon,n\}f\(x\)\)\.
For fixed indexii, \([66](https://arxiv.org/html/2606.17185#A4.E66)\) coupled with the law of total probability implies

ℙ​\{\|Φϵ​\(xi\)−Φϵ,n​\(xi\)\|<O​\(t\)\}\>1−\(2\+D\)​exp⁡\(−t2​\(n−1\)​ϵd/2\)\.\\mathbb\{P\}\\Big\\\{\|\\Phi\_\{\\epsilon\}\(x\_\{i\}\)\-\\Phi\_\{\\epsilon,n\}\(x\_\{i\}\)\|<O\(t\)\\Big\\\}\>1\-\(2\+D\)\\exp\\left\(\-t^\{2\}\(n\-1\)\\epsilon^\{d\}/2\\right\)\.\(68\)By a union bound over the indicesii, we have

ℙ​\{maxi⁡\|Φϵ​\(xi\)−Φϵ,n​\(xi\)\|<O​\(t\)\}\>1−\(2\+D\)​n​exp⁡\(−t2​\(n−1\)​ϵd/2\)\.\\mathbb\{P\}\\Big\\\{\\max\_\{i\}\|\\Phi\_\{\\epsilon\}\(x\_\{i\}\)\-\\Phi\_\{\\epsilon,n\}\(x\_\{i\}\)\|<O\(t\)\\Big\\\}\>1\-\(2\+D\)n\\exp\\left\(\-t^\{2\}\(n\-1\)\\epsilon^\{d\}/2\\right\)\.\(69\)These bounds taken together imply

\|Δ~ϵ,n​\[f\]​\(xj\)−Δϵ,n​\[f\]​\(xj\)\|=O​\(t​ϵ−\(d\+1\)\)\|\\widetilde\{\\Delta\}\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)\|=O\(t\\epsilon^\{\-\(d\+1\)\}\)\(70\)for all indices1≤j≤n1\\leq j\\leq nwith probability at least1−\(2\+D\)​n​exp⁡\(−t2​\(n−1\)​ϵd/2\)1\-\(2\+D\)n\\exp\(\-t^\{2\}\(n\-1\)\\epsilon^\{d\}/2\)\. Taken together with \([60](https://arxiv.org/html/2606.17185#A4.E60)\), we have

\|Δϵ​\[f\]​\(xj\)−Δϵ,n​\[f\]​\(xj\)\|=O​\(t​ϵ−\(d\+1\)\)\|\\Delta\_\{\\epsilon\}\[f\]\(x\_\{j\}\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)\|=O\(t\\epsilon^\{\-\(d\+1\)\}\)\(71\)with probability at least1−\(n​\(2\+D\)\+1\)​exp⁡\(−t2​\(n−1\)​ϵd/2\)1\-\(n\(2\+D\)\+1\)\\exp\(\-t^\{2\}\(n\-1\)\\epsilon^\{d\}/2\)\. We chooset,ϵt,\\epsilonvarying withnnaccording to

ϵn=\(\(4\+2​α\)​log⁡nn\)13​d\+4tn=ϵnd\+2\\begin\{gathered\}\\epsilon\_\{n\}=\\left\(\(4\+2\\alpha\)\\frac\{\\log n\}\{n\}\\right\)^\{\\frac\{1\}\{3d\+4\}\}\\\\ t\_\{n\}=\\epsilon\_\{n\}^\{d\+2\}\\end\{gathered\}\(72\)for someα\>0\\alpha\>0, so that

ℙ​\{maxj⁡\|Δϵ​\[f\]​\(xj\)−Δϵ,n​\[f\]​\(xj\)\|<O​\(ϵn\)\}\>1−O​\(Dn1\+α\)\.\\mathbb\{P\}\\left\\\{\\max\_\{j\}\|\\Delta\_\{\\epsilon\}\[f\]\(x\_\{j\}\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)\|<O\(\\epsilon\_\{n\}\)\\right\\\}\>1\-O\\left\(\\frac\{D\}\{n^\{1\+\\alpha\}\}\\right\)\.\(73\)Then, by the bound \([57](https://arxiv.org/html/2606.17185#A4.E57)\), we have

ℙ​\{maxj⁡\|Δ​\[f\]​\(xj\)−Δϵ,n​\[f\]​\(xj\)\|<O​\(ϵn\)\}\>1−O​\(Dn1\+α\)\.\\mathbb\{P\}\\left\\\{\\max\_\{j\}\|\\Delta\[f\]\(x\_\{j\}\)\-\\Delta\_\{\\epsilon,n\}\[f\]\(x\_\{j\}\)\|<O\(\\epsilon\_\{n\}\)\\right\\\}\>1\-O\\left\(\\frac\{D\}\{n^\{1\+\\alpha\}\}\\right\)\.\(74\)Noting that the empirical estimates are assumed independent, applying the Borel\-Cantelli Lemma establishes\\Crefthm:graph\-uniform\-convergence, as desired\.

## Appendix EImplementation Details

In this section, we detail the specifics of the experiments in\\Crefsec:experiments\. All experiments are implemented in Python using the JAX, Equinox, Optax, and Diffrax libraries\(jax2018;deepmind2020;kidger2021a;kidger2021b\)\.

### E\.1Convergence on the Finsler Graph Laplacian

For a torus with major radiusR=2\.0R=2\.0and minor radiusr=1\.0r=1\.0, points are sampled via rejection sampling\. Azimuthal and poloidal angles\(u,v\)\(u,v\)are uniformly proposed, and accepted according to the true area \(Hausdorff\) measured​A=r​\(R\+r​cos⁡\(v\)\)​d​u​d​vdA=r\(R\+r\\cos\(v\)\)dudv\. We computeCϵ,n†​\(x\)C\_\{\\epsilon,n\}^\{\\dagger\}\(x\)by taking the Moore\-Penrose pseudoinverse of the best rank\-22approximant of the local covariance matrixCϵ,n​\(x\)C\_\{\\epsilon,n\}\(x\)\.

### E\.2Learning Geometry using Finslerian Graph Neural Networks

The initial conditionsf​\(0\)f\(0\)are generated as a superposition of 5 random Fourier modes with frequencies sampled uniformly from\[1\.0,3\.0\]\[1\.0,3\.0\]and phases from\[0,2​π\]\[0,2\\pi\]\. The drift vector defining the Randers metric is set tov=\[0\.25,0\.25\]v=\[0\.25,0\.25\]\. The parameterizations ofJ^​\(ξ\)\\widehat\{J\}\(\\xi\)are constructed using a neural network with a single hidden layer of withE=64E=64\. The graph neural diffusion equations are integrated fromt=0t=0tot=1t=1using the Tsitouras 5/4 solver\(tsitouras2011\), with a time increment ofΔ​t=0\.05\\Delta t=0\.05\. Models are trained using the Adam optimizer with a learning rate of3×10−43\\times 10^\{\-4\}for10001000steps, with weight decay\. The loss function is the relative Mean Squared Error \(MSE\) computed exclusively over 20% of the nodes selected uniformly at random\.