Group Invariant Spectral Embedding

# Group Invariant Spectral Embedding
Source: [https://arxiv.org/html/2607.08987](https://arxiv.org/html/2607.08987)
Yeari VigderDepartment of Statistics and Operations Research, Tel Aviv University, Tel Aviv, Israel [mosco@tauex\.tau\.ac\.il](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(A\. Moscovich\),[vigderyeari@mail\.tau\.ac\.il](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(Y\. Vigder\)Equal contributionPaulina HoyosDepartment of Mathematics, University of Texas at Austin, Austin, TX, USA [paulinah@utexas\.edu](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(P\. Hoyos\),[jkileel@math\.utexas\.edu](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(J\. Kileel\)Equal contributionDavid ThongDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden [dthong@kth\.se](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(D\. Thong\),[janden@kth\.se](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(J\. Andén\)Joakim AndénDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden [dthong@kth\.se](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(D\. Thong\),[janden@kth\.se](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(J\. Andén\)Joe KileelDepartment of Mathematics, University of Texas at Austin, Austin, TX, USA [paulinah@utexas\.edu](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(P\. Hoyos\),[jkileel@math\.utexas\.edu](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(J\. Kileel\)Amit MoscovichDepartment of Statistics and Operations Research, Tel Aviv University, Tel Aviv, Israel [mosco@tauex\.tau\.ac\.il](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(A\. Moscovich\),[vigderyeari@mail\.tau\.ac\.il](https://arxiv.org/html/2607.08987v1/mailto:[email protected])\(Y\. Vigder\)

###### Abstract

Spectral embedding methods are widely used for dimensionality reduction and clustering of high\-dimensional datasets with intrinsic low\-dimensional structures\. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry\-related data points as unrelated\. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding\. We analyze the case of a Riemannian data manifoldℳ\\mathcal\{M\}with symmetries given by a compact Lie groupGGand prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second\-order differential operators on the quotient spaceℳ/G\\mathcal\{M\}/G\. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group\. We validate our approach on datasets withSO​\(2\)\\mathrm\{SO\}\(2\)orSO​\(3\)\\mathrm\{SO\}\(3\)symmetry, and show thatGG\-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data\.

> Keywords:dimensionality reduction, manifold learning, graph Laplacian, data symmetries, quotient manifold, sample complexity

MSC 2020:62R07, 62R30, 58J70, 58J50, 35R02

## 1Introduction

Spectral embedding methods are powerful tools for analyzing high\-dimensional data with intrinsic low\-dimensional structure\. Their basic operating principle is to construct a graph from pairwise affinities between data points and then use the eigenvectors of the graph Laplacian as low\-dimensional representations\. These methods are widely used for tasks such as dimensionality reduction, clustering, semi\-supervised learning, and data denoising\. In this work, we consider datasets that exhibit invariance under known symmetry transformations\. For example, in single\-particle cryo\-electron microscopy \(cryo\-EM\), molecular projection images are subject to random in\-plane rotations\. The 2D rotations are typically viewed as nuisance parameters, which computational methods need to account for\(Singer and Sigworth,[2020](https://arxiv.org/html/2607.08987#bib.bib51)\)\. As an example of a discrete symmetry, consider set\-structured data such as unlabeled graphs, stored using an adjacency matrix\. Despite the vertices having no natural ordering, they are nonetheless assigned arbitrary row/column indices\. However applying the same permutation on the rows and columns of the matrix results in an identical graph\(Zaheeret al\.,[2017](https://arxiv.org/html/2607.08987#bib.bib77); Maronet al\.,[2019](https://arxiv.org/html/2607.08987#bib.bib79)\)\.

Classical approaches for symmetry\-aware machine learning fall into two main categories: data augmentation and invariant features\. Data augmentation enlarges the dataset by including transformed copies of each data point; this is widely used in supervised learning and can provably reduce sample complexity\(Chenet al\.,[2020](https://arxiv.org/html/2607.08987#bib.bib50)\)\. A second approach relies on handcrafted invariant features such as rotation\-invariant descriptors for images\(Lowe,[2004](https://arxiv.org/html/2607.08987#bib.bib13)\)or permutation\-invariant statistics for sets\(Zaheeret al\.,[2017](https://arxiv.org/html/2607.08987#bib.bib77)\)\. In the unsupervised setting neither approach is entirely satisfactory\. Data augmentation treats different transformations of one data point as separate observations, failing to account for the fact that they represent the same instance, and can also incur significant memory overhead\. While invariant features are effective in some settings, they do not provide a general recipe that transfers across different domains and symmetry types easily\.

The central question we address in this paper is*how should one perform spectral embedding on datasets with known symmetries?*Rather than augmenting the data, we propose to “augment” the affinity kernel, so that observations are treated as representatives of an equivalence class\. Mathematically, we seek to pass to the quotient manifold by using group invariant kernel functions\. Figure[1](https://arxiv.org/html/2607.08987#S1.F1)illustrates this on a simple example\.

aaaabbbbℳ=𝕋2\\mathcal\{M\}=\\mathbb\{T\}^\{2\}xxx′=g⋅xx^\{\\prime\}=g\\cdot xyyπ\\pi𝒩=ℳ/G≅𝕊1\\mathcal\{N\}=\\mathcal\{M\}/G\\cong\\mathbb\{S\}^\{1\}endpoints identified\[x\]=\[x′\]\[x\]=\[x^\{\\prime\}\]\[y\]\[y\]d𝒩​\(\[x\],\[y\]\)d\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)

Figure 1:An illustration of group orbits and the quotient manifold\. The flat torusℳ=𝕋2\\mathcal\{M\}=\\mathbb\{T\}^\{2\}is drawn as a fundamental square with opposite edges identified\. The groupG=𝕊1G=\\mathbb\{S\}^\{1\}acts by translations in theaa\-direction, so each horizontal orbit is one equivalence class:xxandx′=g⋅xx^\{\\prime\}=g\\cdot xrepresent the same quotient point, whileyylies on a distinct orbit\. The quotient mapπ:ℳ→𝒩=ℳ/G\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}=\\mathcal\{M\}/Gcollapses each horizontal orbit to a point on thebb\-coordinate\. AGG\-invariant affinity kernelKG​\(x,y\)K\_\{G\}\(x,y\)is a real symmetric function that depends only on theGG\-orbits\[x\]\[x\]and\[y\]\[y\]\.### 1\.1Our Contributions

We study three broadly applicable classes of group invariant affinity kernels with respect to a known groupGG: \(i\) minimization overGG; \(ii\) integration overGG; and \(iii\)GG\-invariant features mapping\. We analyze the continuous case where the data lies on a Riemannian manifoldℳ\\mathcal\{M\}and the symmetries are given by a compact Lie groupGGthat acts smoothly and freely onℳ\\mathcal\{M\}via isometries\. Our main theoretical result \(Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)\) proves that graph Laplacians constructed fromGG\-invariant kernels converge pointwise to explicit second\-order differential operators on the quotient space𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/G\. Crucially, we establish an improvement in the convergence rate, from a rate of

O​\(ε\)\+OP​\(n−1/2​ε−1/2−dim​\(ℳ\)/4\)\(see Eq\. \([6](https://arxiv.org/html/2607.08987#S2.E6)\)\)\\displaystyle O\(\\varepsilon\)\+O\_\{P\}\\left\(n^\{\-1/2\}\\varepsilon^\{\-1/2\-\\mathrm\{dim\}\(\\mathcal\{M\}\)/4\}\\right\)\\hskip 18\.49988pt\\hskip 18\.49988pt\\hskip 18\.49988pt\\qquad\\ \\text\{\(see Eq\.~\\eqref\{eq:LRW\_convergence\}\)\}\(1\)for the standard graph Laplacian that converges to a differential operator onℳ\\mathcal\{M\}, to

O​\(ε\)\+OP​\(n−1/2​ε−1/2−\(dim​\(ℳ\)−dim\(G\)\)/4\)\(Corollary[3\.9](https://arxiv.org/html/2607.08987#S3.Thmtheorem9)\)\\displaystyle O\(\\varepsilon\)\+O\_\{P\}\\left\(n^\{\-1/2\}\\varepsilon^\{\-1/2\-\(\\mathrm\{dim\}\(\\mathcal\{M\}\)\-\\dim\(G\)\)/4\}\\right\)\\hskip 18\.49988pt\\hskip 18\.49988pt\\text\{\(Corollary~\\ref\{cor:improved\_convergence\_rate\}\)\}\(2\)usingGG\-invariant kernels that converge to operators onℳ/G\\mathcal\{M\}/G\. Hereε\\varepsilonis a bandwidth parameter andnnis the number of data points\. This reflects an effective dimension reduction by exploiting symmetries in the data, similar to results on augmentation in supervised learning\(Chenet al\.,[2020](https://arxiv.org/html/2607.08987#bib.bib50)\)\. Further, Corollary[3\.15](https://arxiv.org/html/2607.08987#S3.Thmtheorem15)shows that when all orbits of the action have the same volume there is a bijection between the eigenfunctions on𝒩\\mathcal\{N\}and theGG\-invariant eigenfunctions onℳ\\mathcal\{M\}\. In Section[5](https://arxiv.org/html/2607.08987#S5), we validate our framework on problems withSO​\(2\)\\mathrm\{SO\}\(2\)andSO​\(3\)\\mathrm\{SO\}\(3\)symmetry, demonstrating symmetry\-aware embeddings and their better sample efficiency and interpretability compared to standard spectral methods\.

### 1\.2Related Work

Manifold learning and spectral methods\.Manifold learning has a long history in unsupervised data analysis\. Early examples are Isomap and LLE\(Tenenbaumet al\.,[2000](https://arxiv.org/html/2607.08987#bib.bib34); Roweis and Saul,[2000](https://arxiv.org/html/2607.08987#bib.bib83)\), based on approximate geodesic distances and local linear approximations, respectively\. In this paper we consider spectral embeddings, two variants of which are Laplacian eigenmaps\(Belkin and Niyogi,[2003](https://arxiv.org/html/2607.08987#bib.bib33)\)and diffusion maps\(Coifman and Lafon,[2006](https://arxiv.org/html/2607.08987#bib.bib40)\)\. These methods use the eigenvectors of a graph Laplacian to obtain low\-dimensional coordinates that reflect geometry intrinsic to the data\. Convergence results for graph Laplacian methods were established inBelkin and Niyogi\([2008](https://arxiv.org/html/2607.08987#bib.bib1)\),Heinet al\.\([2007](https://arxiv.org/html/2607.08987#bib.bib99)\), andSinger\([2006](https://arxiv.org/html/2607.08987#bib.bib30)\), with improved rates obtained byCalder and Trillos\([2022](https://arxiv.org/html/2607.08987#bib.bib41)\)andCheng and Wu\([2022](https://arxiv.org/html/2607.08987#bib.bib17)\)\. The theory was extended to non\-Euclidean affinities byKileelet al\.\([2021](https://arxiv.org/html/2607.08987#bib.bib19)\)andXu and Singer\([2026](https://arxiv.org/html/2607.08987#bib.bib115)\)\. Spectral clustering leverages the same eigenvector embedding for unsupervised partitioning\(Nget al\.,[2001](https://arxiv.org/html/2607.08987#bib.bib24); von Luxburg,[2007](https://arxiv.org/html/2607.08987#bib.bib74)\)\. The consistency of spectral clustering was established byvon Luxburget al\.\([2008](https://arxiv.org/html/2607.08987#bib.bib75)\)\.

Symmetry\-aware neural networks\.Machine learning increasingly exploits symmetries to improve efficiency and generalization\. Data augmentation was analyzed byChenet al\.\([2020](https://arxiv.org/html/2607.08987#bib.bib50)\)where the authors proved that in the presence of group symmetries, adding transformed copies of data points reduces the sample complexity of training\. Theoretical analyses of the benefits of invariance for kernel methods and regression appeared inTahmasebi and Jegelka\([2023](https://arxiv.org/html/2607.08987#bib.bib82)\)\. Equivariant neural networks incorporate symmetry directly into neural network architectures using equivariant convolutional layers\(Cohen and Welling,[2016](https://arxiv.org/html/2607.08987#bib.bib76)\), steerable CNNs\(Cohen and Welling,[2017](https://arxiv.org/html/2607.08987#bib.bib78)\)and/or spherical CNNs\(Cohenet al\.,[2018](https://arxiv.org/html/2607.08987#bib.bib84)\)\. Tensor field networks\(Thomaset al\.,[2018](https://arxiv.org/html/2607.08987#bib.bib86)\)and SE\(3\)\-TransformersFuchset al\.\([2020](https://arxiv.org/html/2607.08987#bib.bib87)\)handle rotation and translation equivariance for 3D point clouds and E\(n\)\-equivariant graph neural networks\(Satorraset al\.,[2021](https://arxiv.org/html/2607.08987#bib.bib85)\)provide a flexible framework for Euclidean symmetries\. For works related to permutation\-symmetries of deep neural networks, seeZaheeret al\.\([2017](https://arxiv.org/html/2607.08987#bib.bib77)\)andMaronet al\.\([2019](https://arxiv.org/html/2607.08987#bib.bib79)\)\. A survey of geometric deep learning and equivariant architectures appears inBronsteinet al\.\([2021](https://arxiv.org/html/2607.08987#bib.bib80)\)\.

Symmetry\-aware spectral methods\.The steerable graph Laplacian ofLanda and Shkolnisky\([2018](https://arxiv.org/html/2607.08987#bib.bib31)\)is a symmetry\-aware extension of the graph Laplacian tailored to datasets of 2D images with in\-plane rotational symmetry\. That work constructs a rotation\-equivariant operator by explicitly incorporating rotated copies in the affinity construction\. Closely related to our setting,Rosenet al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib20)\)introduced theGG\-invariant graph Laplacian for data lying on a manifold that is closed under the action of a known Lie groupGG\. Its construction enforces invariance by incorporating distances across all pairs of points generated by the action ofGGon the dataset\. This amounts to implicit \(or closed\-form\) data augmentation, since the construction is conceptually equivalent to the standard graph Laplacian applied to the infinitely many data points generated by the action ofGG\. Similar to the approach of equivariant neural networks, we incorporate group invariance directly into the learning method rather than relying on data augmentation but do so in the unsupervised setting of spectral embedding\. In addition to integration overGG, we also consider minimization overGGand symmetry\-invariant feature maps\. Recent works that use rotation\-invariant metrics for 3D molecules includeDiepeveenet al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib112)\)andZhanget al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib113)\)\.

## 2Background on Spectral Embedding

Throughout this paper,ℳ⊆ℝD\\mathcal\{M\}\\subseteq\\mathbb\{R\}^\{D\}denotes add\-dimensional connected compact Riemannian submanifold ofℝD\\mathbb\{R\}^\{D\}without boundary\. The data points𝒳=\{x1,x2,…,xn\}\\mathcal\{X\}=\\\{x\_\{1\},x\_\{2\},\\dots,x\_\{n\}\\\}are sampled fromℳ\\mathcal\{M\}and given as vectors in the ambient spaceℝD\\mathbb\{R\}^\{D\}\. The geometric structure of the data manifoldℳ\\mathcal\{M\}is captured using a suitable affinity kernelK:ℝD×ℝD→ℝK\\colon\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}which measures the pairwise similarity of data points\. Common choices ofKKare the Gaussian kernelK​\(x,y\)=exp⁡\(−‖x−y‖2/ε\)K\(x,y\)=\\exp\(\-\\\|x\-y\\\|^\{2\}/\\varepsilon\)and the0/10/1kernelK​\(x,y\)=𝟏​\(‖x−y‖≤ε\)K\(x,y\)=\\mathbf\{1\}\(\\\|x\-y\\\|\\leq\\varepsilon\), where∥⋅∥\\\|\\cdot\\\|denotes the Euclidean norm inℝD\\mathbb\{R\}^\{D\}andε\>0\\varepsilon\>0is a bandwidth parameter\. See Appendix[D](https://arxiv.org/html/2607.08987#A4)for a table of the notation used in the paper\.

LetW∈ℝn×nW\\in\\mathbb\{R\}^\{n\\times n\}be a weight matrix defined by:

Wi​j:=K​\(xi,xj\)\.W\_\{ij\}:=K\(x\_\{i\},x\_\{j\}\)\.\(3\)The matrixWWdefines a weighted graph\(𝒳,E,W\)\(\\mathcal\{X\},E,W\)with edgesEEconsisting of the pairs\{xi,xj\}\\\{x\_\{i\},x\_\{j\}\\\}for whichWi​j\>0W\_\{ij\}\>0\. We assume throughout that this graph is connected\. LetDDbe the diagonal degree matrix with non\-zero entries given byDi​i:=∑jWi​j\.D\_\{ii\}:=\\sum\_\{j\}W\_\{ij\}\.The random\-walk normalized*graph Laplacian*is then×nn\\times nmatrix:

LR​W:=I−D−1​W,L\_\{RW\}\\mathrel\{\\mathchoice\{\\vbox\{\\hbox\{$\\displaystyle:$\}\}\}\{\\vbox\{\\hbox\{$\\textstyle:$\}\}\}\{\\vbox\{\\hbox\{$\\scriptstyle:$\}\}\}\{\\vbox\{\\hbox\{$\\scriptscriptstyle:$\}\}\}\{=\}\}I\-D^\{\-1\}W,\(4\)whereIIis the identity matrix\. SinceD−1​WD^\{\-1\}Wis a row\-stochastic matrix, it can be interpreted as the transition probability matrix of a random walk on the graph\(𝒳,E,W\)\(\\mathcal\{X\},E,W\)\. Since𝒳\\mathcal\{X\}is comprised ofnnpoints we can identify functionsf:𝒳→ℝf\\colon\\mathcal\{X\}\\to\\mathbb\{R\}with vectors inℝn\\mathbb\{R\}^\{n\}whose entries are indexed by the points in𝒳\\mathcal\{X\}\. The graph Laplacian defines a linear map on such functions, given by

\(LR​W​f\)​\(xi\)=1∑j=1nK​\(xi,xj\)​∑j=1nK​\(xi,xj\)​\(f​\(xi\)−f​\(xj\)\)\.\(L\_\{RW\}f\)\(x\_\{i\}\)=\\frac\{1\}\{\\sum\_\{j=1\}^\{n\}K\(x\_\{i\},x\_\{j\}\)\}\\sum\_\{j=1\}^\{n\}K\(x\_\{i\},x\_\{j\}\)\\left\(f\(x\_\{i\}\)\-f\(x\_\{j\}\)\\right\)\.\(5\)In words, at each pointxi∈𝒳x\_\{i\}\\in\\mathcal\{X\}, the graph Laplacian averages the differences between the valuef​\(xi\)f\(x\_\{i\}\)and the neighbors’ values, weighted by their affinities toxix\_\{i\}\.

The matrixLR​WL\_\{RW\}has real eigenvalues0=λ0<λ1≤⋯≤λn−10=\\lambda\_\{0\}<\\lambda\_\{1\}\\leq\\cdots\\leq\\lambda\_\{n\-1\}with corresponding eigenvectorsφ0,φ1,…,φn−1∈ℝn\\varphi\_\{0\},\\varphi\_\{1\},\\dots,\\varphi\_\{n\-1\}\\in\\mathbb\{R\}^\{n\}that form a basis forℝn\\mathbb\{R\}^\{n\}, whereφ0=n−1/2​𝟏\\varphi\_\{0\}=n^\{\-1/2\}\\mathbf\{1\}\(von Luxburg,[2007](https://arxiv.org/html/2607.08987#bib.bib74)\)\. The smoothness of each eigenvectorϕi\\phi\_\{i\}is measured by its Rayleigh quotient:λi=\(φiT​LR​W​φi\)/\(φiT​φi\)\\lambda\_\{i\}=\(\\varphi\_\{i\}^\{T\}L\_\{RW\}\\varphi\_\{i\}\)/\(\\varphi\_\{i\}^\{T\}\\varphi\_\{i\}\)\. The eigenvectors corresponding to themmsmallest nonzero eigenvalues vary most smoothly over the graph, and therefore capture the coarsest geometric structure of the data\. Mapping each data pointxix\_\{i\}to its coordinates in this low\-dimensional eigenvector basis, as detailed in Algorithm[1](https://arxiv.org/html/2607.08987#alg1), gives an embedding that reflects the intrinsic geometry of the data, bringing nearby points in the original space close together in the embedding\.111Several names were given to this method and its variants over the years\. Key works include Laplacian eigenmaps\(Belkin and Niyogi,[2003](https://arxiv.org/html/2607.08987#bib.bib33)\)and diffusion maps\(Coifman and Lafon,[2006](https://arxiv.org/html/2607.08987#bib.bib40)\)\.

Input:Data points

x1,…,xn∈ℝDx\_\{1\},\\dots,x\_\{n\}\\in\\mathbb\{R\}^\{D\}, positive integer

mm, affinity kernel

KK\.

2ptOutput:Embedded data points

x1,…,xn∈ℝmx\_\{1\},\\dots,x\_\{n\}\\in\\mathbb\{R\}^\{m\}\.

6pt

\(1\)Compute the weights matrix

WWvia Eq\. \([3](https://arxiv.org/html/2607.08987#S2.E3)\) and its corresponding degree matrix

DD\.

\(2\)Calculate the random\-walk normalized graph Laplacian

LR​W:=I−D−1​WL\_\{RW\}:=I\-D^\{\-1\}W\.

\(3\)Compute the eigenvectors

φ1,…,φm\\varphi\_\{1\},\\dots,\\varphi\_\{m\}of

LR​WL\_\{RW\}that correspond to the first

mmnonzero eigenvalues

λ1≤⋯≤λm\\lambda\_\{1\}\\leq\\cdots\\leq\\lambda\_\{m\}\.

\(4\)Map

xix\_\{i\}into

ℝm\\mathbb\{R\}^\{m\}via

xi↦xi=\(φ1​\(xi\),…,φm​\(xi\)\)x\_\{i\}\\mapsto x\_\{i\}=\\left\(\\varphi\_\{1\}\(x\_\{i\}\),\\dots,\\varphi\_\{m\}\(x\_\{i\}\)\\right\)\.

\(5\)return

x1,…,xnx\_\{1\},\\dots,x\_\{n\}\.

Algorithm 1Spectral EmbeddingIt is known that if one uses the Gaussian kernel to construct the weight matrixWWthen the random\-walk Laplacian converges in a pointwise sense to the Laplace–Beltrami operator on the manifoldℳ\\mathcal\{M\}\. A precise statement is as follows\.

###### Theorem 2\.1\(Singer \([2006](https://arxiv.org/html/2607.08987#bib.bib30)\)\)\.

Letℳ\\mathcal\{M\}be a compactdd\-dimensional Riemannian submanifold ofℝD\\mathbb\{R\}^\{D\}with Laplace–Beltrami operatorΔℳ​f:=−div​\(∇ℳf\)\\Delta\_\{\\mathcal\{M\}\}f:=\-\\mathrm\{div\}\(\\nabla\_\{\\mathcal\{M\}\}f\)\. For any smooth functionf:ℳ→ℝf:\\mathcal\{M\}\\to\\mathbb\{R\}and given pointsx1,…,xnx\_\{1\},\\dots,x\_\{n\}sampled independently and uniformly fromℳ\\mathcal\{M\}, we have that for all sufficiently smallε\\varepsilonand asn→∞n\\to\\infty

4ε​\(LR​W​f\)​\(xi\)=Δℳ​f​\(xi\)\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+d/4\),\\displaystyle\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=\\Delta\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)\+O\\left\(\\varepsilon\\right\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+d/4\}\}\\right\),\(6\)whereLR​WL\_\{RW\}is the random\-walk graph Laplacian based on the Gaussian kernelK​\(x,y\)=exp⁡\(−‖x−y‖2/ε\)K\(x,y\)=\\exp\(\-\\\|x\-y\\\|^\{2\}/\\varepsilon\)\.

HereO​\(ε\)O\(\\varepsilon\)denotes a quantity less thanC​εC\\varepsilonwhereCCis a constant that depends onℳ\\mathcal\{M\},ffandxix\_\{i\}but not onε\\varepsilonornn, whileOP​\(⋅\)O\_\{P\}\(\\cdot\)denotes the order in probability\. The theorem can be extended to non\-uniform sampling\. In that case, the random\-walk graph Laplacian converges to a weighted Laplacian \(or Fokker–Planck\) operator onℳ\\mathcal\{M\}, which has an additional drift term\.

###### Theorem 2\.2\(Coifman and Lafon \([2006](https://arxiv.org/html/2607.08987#bib.bib40)\)\)\.

Assume the setting of Theorem[2\.1](https://arxiv.org/html/2607.08987#S2.Thmtheorem1), except that the pointsx1,…,xn∈ℳx\_\{1\},\\dots,x\_\{n\}\\in\\mathcal\{M\}are now sampled from a probability densityρ​\(x\)\\rho\(x\)with respect to the uniform measure onℳ\\mathcal\{M\}\. Then for all sufficiently smallε\\varepsilonand asn→∞n\\to\\inftywe have that

4ε​\(LR​W​f\)​\(xi\)=Δℳ​f​\(xi\)−2​⟨∇ℳlog⁡ρ​\(xi\),∇ℳf​\(xi\)⟩\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+d/4\)\.\\displaystyle\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=\\Delta\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)\-2\\left\\langle\\nabla\_\{\\mathcal\{M\}\}\\log\\rho\(x\_\{i\}\),\\,\\nabla\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)\\right\\rangle\+O\(\\varepsilon\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+d/4\}\}\\right\)\.\(7\)

These theorems have been generalized to prove spectral consistency, where the eigenvalues and eigenvectors of the graph Laplacian converge to the eigenvalues and eigenfunctions of the Laplace–Beltrami operator\(von Luxburget al\.,[2008](https://arxiv.org/html/2607.08987#bib.bib75); Belkin and Niyogi,[2008](https://arxiv.org/html/2607.08987#bib.bib1); Trillos and Slepčev,[2016](https://arxiv.org/html/2607.08987#bib.bib4); Trilloset al\.,[2020](https://arxiv.org/html/2607.08987#bib.bib12); Calder and Trillos,[2022](https://arxiv.org/html/2607.08987#bib.bib41); Cheng and Wu,[2022](https://arxiv.org/html/2607.08987#bib.bib17)\)\. In this work, we focus on the pointwise consistency of our method\.

## 3Method and Main Results

Here we present our approach to incorporating symmetry invariance into spectral embedding\. We begin by definingGG\-invariant kernels and presenting three broadly applicable constructions \(Section[3\.1](https://arxiv.org/html/2607.08987#S3.SS1)\)\. We then describe the G\-invariant spectral embedding algorithm \(Section[3\.2](https://arxiv.org/html/2607.08987#S3.SS2)\), followed by our main theoretical result\. It is a pointwise convergence theorem showing that graph Laplacians built fromGG\-invariant kernels converge to explicit second\-order differential operators on the quotient manifold𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/G\(Section[3\.3](https://arxiv.org/html/2607.08987#S3.SS3)\)\. We conclude by analyzing the special case of constant orbit\-volume density \(Section[3\.4](https://arxiv.org/html/2607.08987#S3.SS4)\) and illustrate the improved rate with a numerical example \(Section[3\.5](https://arxiv.org/html/2607.08987#S3.SS5)\)\.

### 3\.1Group Invariant Kernels

The following definition formalizes the notion of a group invariant kernel\.

###### Definition 3\.1\.

LetGGbe a compact Lie group acting onℝD\\mathbb\{R\}^\{D\}, and letKG:ℝD×ℝD→ℝK\_\{G\}\\colon\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}be a symmetric kernel\. We say thatKGK\_\{G\}is aGG\-invariant kernelif

KG​\(α⋅x,β⋅y\)=KG​\(x,y\)K\_\{G\}\(\\alpha\\cdot x,\\beta\\cdot y\)=K\_\{G\}\(x,y\)for allx,y∈ℝDx,y\\in\\mathbb\{R\}^\{D\}andα,β∈G\\alpha,\\beta\\in G\.

In particular, we focus on the following three constructions ofGG\-invariant kernels\.

###### Proposition 3\.2\.

LetGGbe a compact Lie group acting by isometries onℝD\\mathbb\{R\}^\{D\}\. The following three kernels areGG\-invariant:

1. 1\.Theminimum kernel: Kmin​\(x,x\):=maxα∈G⁡K​\(x,α⋅x\)=exp⁡\(−minα∈G⁡‖x−α⋅x‖2/ε\)\.K\_\{\\mathrm\{min\}\}\(x,x\):=\\max\_\{\\alpha\\in G\}K\(x,\\alpha\\cdot x\)=\\exp\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot x\\\|^\{2\}/\\varepsilon\\right\)\.\(8\)
2. 2\.Theintegral kernel: Kint​\(x,x\):=∫GK​\(x,α⋅x\)​𝑑η​\(α\)=∫Gexp⁡\(−‖x−α⋅x‖2/ε\)​𝑑η​\(α\),K\_\{\\mathrm\{int\}\}\(x,x\):=\\int\_\{G\}K\(x,\\alpha\\cdot x\)d\\eta\(\\alpha\)=\\int\_\{G\}\\exp\\left\(\-\\\|x\-\\alpha\\cdot x\\\|^\{2\}/\\varepsilon\\right\)d\\eta\(\\alpha\),\(9\)whereη\\etadenotes the Haar measure onGGnormalized so thatVol​\(G\)=∫G1​𝑑η​\(α\)=1\\mathrm\{Vol\}\(G\)=\\int\_\{G\}1d\\eta\(\\alpha\)=1\.
3. 3\.Theinvariant features kernel: KIF​\(x,x\):=K​\(ϕ​\(x\),ϕ​\(x\)\)=exp⁡\(−‖ϕ​\(x\)−ϕ​\(x\)‖2/ε\),K\_\{\\mathrm\{IF\}\}\(x,x\):=K\(\\phi\(x\),\\phi\(x\)\)=\\exp\\left\(\-\\\|\\phi\(x\)\-\\phi\(x\)\\\|^\{2\}/\\varepsilon\\right\),\(10\)whereϕ:ℝD→ℝE\\phi:\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{E\}is aGG\-invariant map, i\.e\.ϕ​\(α⋅x\)=ϕ​\(x\)\\phi\(\\alpha\\cdot x\)=\\phi\(x\)for allα∈G,x∈ℝD\\alpha\\in G,x\\in\\mathbb\{R\}^\{D\}\.

###### Proof\.

Follows directly from the definitions\. ∎

The minimum and integral kernels are completely determined by the action ofGGonℝD\\mathbb\{R\}^\{D\}\. The invariant features kernel, on the other hand, further depends on the choice of aGG\-invariant mapϕ\\phi\. The following examples illustrate each of the threeGG\-invariant kernels in two different settings\.

###### Example 3\.3\(Point lists inℝd\\mathbb\{R\}^\{d\},G=O​\(d\)G=\\mathrm\{O\}\(d\)\)\.

Let the pointsx1,…,xn∈ℝdx\_\{1\},\\ldots,x\_\{n\}\\in\\mathbb\{R\}^\{d\}be given as rows of a matrixX∈ℝn×dX\\in\\mathbb\{R\}^\{n\\times d\}\. Any rotation \(and rotoreflection\) of the points can be written asX​R⊤XR^\{\\top\}whereR∈O​\(d\)R\\in\\mathrm\{O\}\(d\)is an orthogonal matrix\. The minimum kernel then corresponds to the orthogonal Procrustes problem forXXand another point listYY, which has a closed\-form solution\(Schönemann,[1966](https://arxiv.org/html/2607.08987#bib.bib11)\)\. Meanwhile for the integral kernel, one needs to approximate

∫O​\(d\)exp⁡\(−‖X−Y​RT‖2/ε\)​𝑑η​\(R\)\.\\displaystyle\\int\_\{\\mathrm\{O\}\(d\)\}\\exp\\left\(\-\\\|X\-YR^\{T\}\\\|^\{2\}/\\varepsilon\\right\)d\\eta\(R\)\.\(11\)Similar integrals without the exponential appear in steerable PCA works\(Zhaoet al\.,[2016](https://arxiv.org/html/2607.08987#bib.bib6); Fraimanet al\.,[2026](https://arxiv.org/html/2607.08987#bib.bib5)\)\.

For an instance of the invariant features kernel, the Gram matrix ofXXdefines a mapϕ:ℝn×d→ℝn×n\\phi\\colon\\mathbb\{R\}^\{n\\times d\}\\to\\mathbb\{R\}^\{n\\times n\}viaϕ​\(X\)=X​X⊤\\phi\(X\)=XX^\{\\top\}; this isO​\(d\)\\mathrm\{O\}\(d\)\-invariant since for allR∈O​\(d\),X∈ℝn×dR\\in\\mathrm\{O\}\(d\),X\\in\\mathbb\{R\}^\{n\\times d\}we haveϕ​\(X​R⊤\)=\(X​R⊤\)​\(X​R⊤\)⊤=X​X⊤=ϕ​\(X\)\\phi\(XR^\{\\top\}\)=\(XR^\{\\top\}\)\(XR^\{\\top\}\)^\{\\top\}=XX^\{\\top\}=\\phi\(X\)\. In fact, everyO​\(d\)\\mathrm\{O\}\(d\)\-invariant function factors through the Gram matrix, as the first fundamental theorem of invariant functions for the orthogonal group\(Weyl,[1946](https://arxiv.org/html/2607.08987#bib.bib57), Theorem 2\.9A\)shows: A functionf:ℝn×d→ℝf\\colon\\mathbb\{R\}^\{n\\times d\}\\to\\mathbb\{R\}is invariant under the action of the orthogonal groupO​\(d\)\\mathrm\{O\}\(d\)\(i\.e\.,f​\(X\)=f​\(X​R⊤\)f\(X\)=f\(XR^\{\\top\}\)for allR∈O​\(d\)R\\in\\mathrm\{O\}\(d\),X∈ℝn×dX\\in\\mathbb\{R\}^\{n\\times d\}\) if and only if there exists a functiong:ℝn×n→ℝg\\colon\\mathbb\{R\}^\{n\\times n\}\\to\\mathbb\{R\}such thatf​\(X\)=g​\(X​X⊤\)\.f\(X\)=g\(XX^\{\\top\}\)\.

###### Example 3\.4\(2D images,G=SO​\(2\)G=\\mathrm\{SO\}\(2\)\)\.

Consider 2D images as square integrable functionsX:ℝ2→ℝX\\colon\\mathbb\{R\}^\{2\}\\to\\mathbb\{R\}and letR∈SO​\(2\)R\\in\\mathrm\{SO\}\(2\)act onXXvia\(R⋅X\)​\(x\)=X​\(R⊤​x\)\(R\\cdot X\)\(x\)=X\(R^\{\\top\}x\)\. The minimum kernel corresponds to 2D alignment of two functions, while the integral kernel is given by

∫SO​\(2\)exp⁡\(−‖X−R⋅Y‖L22ε\)​𝑑η​\(R\)=12​π​exp⁡\(−‖X‖L22\+‖Y‖L22ε\)​∫02​πexp⁡\(2ε​∫ℝ2X​\(x\)​Y​\(Rθ⊤​x\)​𝑑x\)​𝑑θ\.\\displaystyle\\int\_\{\\mathrm\{SO\}\(2\)\}\\exp\\left\(\-\\tfrac\{\\\|X\-R\\cdot Y\\\|\_\{L^\{2\}\}^\{2\}\}\{\\varepsilon\}\\right\)d\\eta\(R\)=\\frac\{1\}\{2\\pi\}\\exp\\left\(\-\\tfrac\{\\\|X\\\|\_\{L^\{2\}\}^\{2\}\+\\\|Y\\\|\_\{L^\{2\}\}^\{2\}\}\{\\varepsilon\}\\right\)\\int\_\{0\}^\{2\\pi\}\\exp\\\!\\left\(\\frac\{2\}\{\\varepsilon\}\\int\_\{\\mathbb\{R\}^\{2\}\}X\(x\)\\,Y\(R\_\{\\theta\}^\{\\top\}x\)\\,dx\\right\)d\\theta\.\(12\)
For the invariant features kernel, one can define a set of rotational invariants based on a Fourier–Bessel decomposition:

X​\(r,θ\)=∑m=0\+∞∑k=1\+∞ak,m​ψk,m​\(r,θ\),\\displaystyle X\(r,\\theta\)=\\sum\_\{m=0\}^\{\+\\infty\}\\sum\_\{k=1\}^\{\+\\infty\}a\_\{k,m\}\\psi\_\{k,m\}\(r,\\theta\),\(13\)whereψk,m​\(r,θ\)\\psi\_\{k,m\}\(r,\\theta\)are the basis functions\(Zhao and Singer,[2013](https://arxiv.org/html/2607.08987#bib.bib26)\)\. The*\(rotational\) bispectrum*ofXXis given byϕ​\(X\)\\phi\(X\)with elementsak1,m1​ak2,m2​ak3,m1\+m2∗a\_\{k\_\{1\},m\_\{1\}\}a\_\{k\_\{2\},m\_\{2\}\}a\_\{k\_\{3\},m\_\{1\}\+m\_\{2\}\}^\{\*\}wherek1=1,…,kmax​\(m1\)k\_\{1\}=1,\\ldots,k\_\{\\max\}\(m\_\{1\}\),k2=1,…,kmax​\(m2\)k\_\{2\}=1,\\ldots,k\_\{\\max\}\(m\_\{2\}\),k3=1,…,kmax​\(m1\+m2\)k\_\{3\}=1,\\ldots,k\_\{\\max\}\(m\_\{1\}\+m\_\{2\}\),m1=0,…,⌊mmax/2⌋m\_\{1\}=0,\\ldots,\\lfloor m\_\{\\max\}/2\\rfloorandm2=0,…,mmax−m1m\_\{2\}=0,\\ldots,m\_\{\\max\}\-m\_\{1\}\. Heremmaxm\_\{\\max\}is an angular\-frequency cutoff, andkmax​\(m\)k\_\{\\max\}\(m\)is the number of radial Fourier–Bessel coefficients retained for angular frequencymm\. The bispectrum is anSO​\(2\)\\mathrm\{SO\}\(2\)\-invariant map\(Zhao and Singer,[2014](https://arxiv.org/html/2607.08987#bib.bib27)\)\.

#### Computation of Invariant Kernels

Computation of the minimum kernel requires evaluatingminα∈G⁡‖x−α⋅y‖2\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}, which is a non\-convex global optimization problem\. ForG=SO​\(n\)G=\\mathrm\{SO\}\(n\)andG=O​\(n\)G=\\mathrm\{O\}\(n\)acting onℝn×m\\mathbb\{R\}^\{n\\times m\}by left multiplication, there are closed\-form solutions based on the SVD\(Kabsch,[1976](https://arxiv.org/html/2607.08987#bib.bib7); Umeyama,[1991](https://arxiv.org/html/2607.08987#bib.bib9); Schönemann,[1966](https://arxiv.org/html/2607.08987#bib.bib11)\)\. When different actions ofG=SO​\(n\)G=\\mathrm\{SO\}\(n\)orG=O​\(n\)G=\\mathrm\{O\}\(n\)are considered, one obtains generally non\-trivial alignment problems\. ForG=SO​\(2\)G=\\mathrm\{SO\}\(2\), one may compute the distance as a trigonometric polynomial and solve the minimization using univariate search or Fourier methods\. ForG=SO​\(3\)G=\\mathrm\{SO\}\(3\), deterministic low\-discrepancy designs, such as Fibonacci and super\-Fibonacci samplings\(Alexa,[2022](https://arxiv.org/html/2607.08987#bib.bib16)\), provide efficient alternatives to random sampling\. For general compact Lie groups, there exist branch\-and\-bound algorithms\(Campbell and Petersson,[2016](https://arxiv.org/html/2607.08987#bib.bib107); Hartleyet al\.,[2013](https://arxiv.org/html/2607.08987#bib.bib102)\)\. Manifold optimization methods\(Boumal,[2023](https://arxiv.org/html/2607.08987#bib.bib22)\)include gradient descent and Newton’s method over general Lie groups; however, these methods are not guaranteed to find a global minimum in general\.

For the integral kernel, a naive quadrature rule requiresO​\(Lp\)O\(L^\{p\}\)operations per pair of data points, whereLLrepresents a resolution parameter andppis the dimension ofGG\. ForG=SO​\(2\)G=\\mathrm\{SO\}\(2\), uniform trapezoidal rules on the circle yield exponentially fast convergence for smooth integrands\(Trefethen and Weideman,[2014](https://arxiv.org/html/2607.08987#bib.bib10)\)\. ForG=SO​\(3\)G=\\mathrm\{SO\}\(3\), one can apply Monte Carlo sampling with exact Haar draws from quaternion parameterizations\(Shoemake,[1992](https://arxiv.org/html/2607.08987#bib.bib101)\)or quasi–Monte Carlo rules using low\-discrepancy point sets\(Alexa,[2022](https://arxiv.org/html/2607.08987#bib.bib16)\)\. For a general compact Lie group, there exist quadrature schemes based on FFT\-type algorithms\(Maslen,[2004](https://arxiv.org/html/2607.08987#bib.bib8); Kostelec and Rockmore,[2008](https://arxiv.org/html/2607.08987#bib.bib103); Pottset al\.,[2009](https://arxiv.org/html/2607.08987#bib.bib104)\)\.

Finally, computation of the invariant features kernel depends on the choice ofGG\-invariant mapϕ\\phi\. A generic cost includes two evaluations ofϕ\\phiplusO​\(E\)O\(E\)operations to evaluate‖ϕ​\(x\)−ϕ​\(x\)‖2\\\|\\phi\(x\)\-\\phi\(x\)\\\|^\{2\}, whereEEis the output dimension ofϕ\\phi\. Returning to the examples above, the computation of the Gram matrix in Example[3\.3](https://arxiv.org/html/2607.08987#S3.Thmtheorem3)requiresO​\(n2​d\)O\(n^\{2\}d\)operations\. Moreover, the bispectrum of a 2D image in Example[3\.4](https://arxiv.org/html/2607.08987#S3.Thmtheorem4)can be calculated inO​\(L3\)O\(L^\{3\}\)time\(Zhao and Singer,[2013](https://arxiv.org/html/2607.08987#bib.bib26)\), where each image is represented on a discreteL×LL\\times Lgrid\. In general, we also note it may sometimes be possible to evaluate‖ϕ​\(x\)−ϕ​\(y\)‖2\\\|\\phi\(x\)\-\\phi\(y\)\\\|^\{2\}without formingϕ​\(x\)\\phi\(x\)andϕ​\(y\)\\phi\(y\)explicitly, i\.e\. for someGG\-invariant maps there may be a kernel trick available\.

### 3\.2Spectral Embedding with Group Invariant Kernels

Given a dataset with a known symmetry groupGG, our proposal is to replace the generic affinity kernelKKwith aGG\-invariant kernelKGK\_\{G\}in the construction of the graph LaplacianLR​WL\_\{RW\}\(Eqs\. \([3](https://arxiv.org/html/2607.08987#S2.E3)\)\-\([5](https://arxiv.org/html/2607.08987#S2.E5)\)\)\. In principle, anyGG\-invariant kernel may be used in this construction\. The resulting graph LaplacianLR​WL\_\{RW\}is then used directly in Algorithm[1](https://arxiv.org/html/2607.08987#alg1), with no other modification to the spectral embedding procedure\. Theoretical guarantees for the case of the minimum, integral, and invariant features kernels introduced in Section[3\.1](https://arxiv.org/html/2607.08987#S3.SS1)are established in Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)below\.

### 3\.3Convergence of the Graph Laplacian with Group Invariant Kernels

LetGGbe a compact Lie group acting by isometries onℝD\\mathbb\{R\}^\{D\}, andℳ\\mathcal\{M\}a compactdd\-dimensional Riemannian submanifold ofℝD\\mathbb\{R\}^\{D\}that isGG\-invariant, i\.e\.,α⋅ℳ=ℳ\\alpha\\cdot\\mathcal\{M\}=\\mathcal\{M\}for allα∈G\\alpha\\in G\. It follows that the restricted action ofGGonℳ\\mathcal\{M\}is by isometries\. We further assume that this action is smooth and free\. Denote the corresponding quotient manifold by𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/G\. It inherits a Riemannian metric naturally\. See Appendix[A](https://arxiv.org/html/2607.08987#A1)for details\. Our analysis focuses on group invariant functions, defined as follows\.

###### Definition 3\.5\.

We say thatf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}is a𝐆\\mathbf\{G\}\-invariant functioniff​\(α⋅x\)=f​\(x\)f\(\\alpha\\cdot x\)=f\(x\)for allα∈G\\alpha\\in G,x∈ℳx\\in\\mathcal\{M\}\. In this case, the unique functionf¯:𝒩→ℝ\\overline\{f\}\\colon\\mathcal\{N\}\\to\\mathbb\{R\}such thatf¯​\(\[x\]\)=f​\(x\)\\overline\{f\}\(\[x\]\)=f\(x\)for allx∈ℳx\\in\\mathcal\{M\}is called theinduced functionon𝒩\\mathcal\{N\}\.

Restricting toGG\-invariant functions is a natural choice for datasets with known symmetries, since we aim to treat symmetry\-related observations as being the same\. For example, the underlying molecular structure of projection images in cryo\-EM is unaffected by in\-plane rotations, so meaningful quantities should be rotation invariant\. Next, we follow\(Helgason,[2000](https://arxiv.org/html/2607.08987#bib.bib21), Section II\.3\.3\)and introduce a functionδ\\deltaonℳ\\mathcal\{M\}which measures the relative volume of the orbits ofGG\.

###### Definition 3\.6\.

Givenx∈ℳx\\in\\mathcal\{M\}, letd​VG⋅xdV\_\{G\\cdot x\}denote the Riemannian measure on the orbitG⋅xG\\cdot xinduced by the ambient metric onℳ\\mathcal\{M\}\. Fix a left\-invariant Haar measured​ηd\\etaonGGsuch thatVol​\(G\)=∫G1​𝑑η​\(α\)=1\\mathrm\{Vol\}\(G\)=\\int\_\{G\}1d\\eta\(\\alpha\)=1\. Thedensity functionδ:ℳ→ℝ\\delta\\colon\\mathcal\{M\}\\to\\mathbb\{R\}is defined by the formula

d​VG⋅x=δ​\(x\)​d​η,x∈ℳ,\\displaystyle dV\_\{G\\cdot x\}=\\delta\(x\)d\\eta,\\qquad x\\in\\mathcal\{M\},\(14\)

where we identifyG⋅xG\\cdot xwithGGvia the orbit mapθ\(x\):G→G⋅x\\theta^\{\(x\)\}\\colon G\\to G\\cdot xgiven byθ\(x\)​\(α\)=α⋅x\\theta^\{\(x\)\}\(\\alpha\)=\\alpha\\cdot x\. To see thatδ\\deltais well\-defined, note that the orbit map is a diffeomorphism sinceGGacts freely onℳ\\mathcal\{M\}, which provides an identification ofGGwithG⋅xG\\cdot xfor eachx∈ℳx\\in\\mathcal\{M\}, and it follows thatd​VG⋅xdV\_\{G\\cdot x\}andd​ηd\\etamust be proportional by uniqueness of the Haar measure onGG\. It is clear from the definition thatδ\\deltais aGG\-invariant function, so it descends to a functionδ¯\\overline\{\\delta\}on the quotient manifold𝒩\\mathcal\{N\}\.

We are now ready to state the main theoretical result regarding the convergence of the graph Laplacian based on a group invariant kernel\.

###### Theorem 3\.7\(Main theoretical result\)\.

LetGGbe a compact Lie group of dimensionppacting isometrically onℝD\\mathbb\{R\}^\{D\}\. Letℳ\\mathcal\{M\}be a connected compactdd\-dimensional Riemannian submanifold ofℝD\\mathbb\{R\}^\{D\}without boundary\. Assume thatℳ\\mathcal\{M\}isGG\-invariant and thatGGacts smoothly and freely onℳ\\mathcal\{M\}\. Denote the Riemannian quotient manifoldℳ/G\\mathcal\{M\}/Gby𝒩\\mathcal\{N\}\. Letδ¯\\overline\{\\delta\}denote the function on𝒩\\mathcal\{N\}induced by the density functionδ\\deltadefined in Eq\. \([14](https://arxiv.org/html/2607.08987#S3.E14)\), which we assume to be smooth\. Let data pointsx1,…,xnx\_\{1\},\\dots,x\_\{n\}be i\.i\.d\. samples from the uniform measured​VℳdV\_\{\\mathcal\{M\}\}onℳ\\mathcal\{M\}\. Letf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}be a smoothGG\-invariant function andf¯:𝒩→ℝ\\overline\{f\}\\colon\\mathcal\{N\}\\to\\mathbb\{R\}the corresponding induced function\. LetLR​WL\_\{RW\}be the normalized graph Laplacian defined in Eq\. \([4](https://arxiv.org/html/2607.08987#S2.E4)\) using aGG\-invariant kernel\. Then for all sufficiently smallε\\varepsilonand asn→∞n\\to\\inftywe have

4ε​\(LR​W​f\)​\(xi\)=D​f¯​\(\[xi\]\)\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+\(d−p\)/4\),\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=D\\overline\{f\}\(\[x\_\{i\}\]\)\+O\(\\varepsilon\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+\(d\-p\)/4\}\}\\right\),\(15\)whereDDis a second\-order differential operator on𝒩\\mathcal\{N\}which depends on the choice ofGG\-invariant kernel as follows:

1. 1\.For the minimum kernel in Eq\. \([8](https://arxiv.org/html/2607.08987#S3.E8)\), D=Δ𝒩−2​⟨∇𝒩log⁡δ¯,∇𝒩\(⋅\)⟩\.\\displaystyle D=\\Delta\_\{\\mathcal\{N\}\}\-2\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\rangle\.\(16\)
2. 2\.For the integral kernel in Eq\. \([9](https://arxiv.org/html/2607.08987#S3.E9)\), D=Δ𝒩−⟨∇𝒩log⁡δ¯,∇𝒩\(⋅\)⟩\.\\displaystyle D=\\Delta\_\{\\mathcal\{N\}\}\-\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\rangle\.\(17\)
3. 3\.For the invariant features kernel in Eq\. \([10](https://arxiv.org/html/2607.08987#S3.E10)\), D=ϕ¯∗​Dim​ϕ\.\\displaystyle D=\\bar\{\\phi\}^\{\\ast\}D\_\{\\mathrm\{im\}\\phi\}\.\(18\)Hereϕ:ℳ→ℝE\\phi\\colon\\mathcal\{M\}\\to\\mathbb\{R\}^\{E\}is a smoothGG\-invariant map such that the induced mapϕ¯:𝒩→im​ϕ\\bar\{\\phi\}\\colon\\mathcal\{N\}\\to\\mathrm\{im\}\\phiis a diffeomorphism\. The operatorDim​ϕD\_\{\\mathrm\{im\}\\phi\}is defined onim​ϕ\\mathrm\{im\}\\phiby Dim​ϕ=Δim​ϕ−2​⟨∇im​ϕlog⁡pϕ,∇im​ϕ\(⋅\)⟩,\\displaystyle D\_\{\\mathrm\{im\}\\phi\}=\\Delta\_\{\\mathrm\{im\}\\phi\}\-2\\left\\langle\\nabla\_\{\\mathrm\{im\}\\phi\}\\log p\_\{\\phi\},\\nabla\_\{\\mathrm\{im\}\\phi\}\(\\cdot\)\\right\\rangle,\(19\)wherepϕp\_\{\\phi\}is the density of the pushforward measureϕ∗​\(d​Vℳ\)\\phi\_\{\\ast\}\(dV\_\{\\mathcal\{M\}\}\)with respect to the Riemannian volume measured​Vim​ϕdV\_\{\\mathrm\{im\}\\phi\}induced onim​ϕ\\mathrm\{im\}\\phiby the ambient Euclidean metric, that is,ϕ∗​\(d​Vℳ\)=pϕ​d​Vim​ϕ\.\\phi\_\{\\ast\}\(dV\_\{\\mathcal\{M\}\}\)=p\_\{\\phi\}\\,dV\_\{\\mathrm\{im\}\\phi\}\.

Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)is proved in Section[4](https://arxiv.org/html/2607.08987#S4)\.

The theorem can be extended to non\-uniform sampling of data points, where the operatorDDacquires an additional first\-order term depending on the average overGGof the sampling density\.

###### Corollary 3\.9\.

Assume the conditions of Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7), but with data pointsx1,…,xnx\_\{1\},\\dots,x\_\{n\}sampled from a smooth probability densityρ\\rhowith respect to the uniform measure onℳ\\mathcal\{M\}\. Letρ~\\tilde\{\\rho\}be the function on𝒩\\mathcal\{N\}defined by

ρ~​\(\[x\]\)=∫Gρ​\(α⋅x\)​𝑑η​\(α\)\.\\tilde\{\\rho\}\(\[x\]\)=\\int\_\{G\}\\rho\(\\alpha\\cdot x\)d\\eta\(\\alpha\)\.\(20\)Then for all sufficiently smallε\\varepsilonand asn→∞n\\to\\inftywe have

4ε​\(LR​W​f\)​\(xi\)=Dρ​f¯​\(\[xi\]\)\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+\(d−p\)/4\),\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=D\_\{\\rho\}\\overline\{f\}\(\[x\_\{i\}\]\)\+O\(\\varepsilon\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+\(d\-p\)/4\}\}\\right\),\(21\)whereDρD\_\{\\rho\}adds a first\-order term to the operatorDDin Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)as follows:

1. 1\.For the minimum kernel in Eq\. \([8](https://arxiv.org/html/2607.08987#S3.E8)\) andDDin Eq\. \([16](https://arxiv.org/html/2607.08987#S3.E16)\), Dρ=D−2​⟨∇𝒩log⁡ρ~,∇𝒩\(⋅\)⟩\.D\_\{\\rho\}=D\-2\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\tilde\{\\rho\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\rangle\.\(22\)
2. 2\.For the integral kernel in Eq\. \([9](https://arxiv.org/html/2607.08987#S3.E9)\) andDDin Eq\. \([17](https://arxiv.org/html/2607.08987#S3.E17)\), Dρ=D−2​⟨∇𝒩log⁡ρ~,∇𝒩\(⋅\)⟩\.D\_\{\\rho\}=D\-2\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\tilde\{\\rho\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\rangle\.\(23\)
3. 3\.For the invariant features kernel in Eq\. \([10](https://arxiv.org/html/2607.08987#S3.E10)\) andDDin Eq\. \([18](https://arxiv.org/html/2607.08987#S3.E18)\), Dρ=D−2​ϕ¯∗​⟨∇im​ϕlog⁡\(ρ~∘ϕ¯−1\),∇im​ϕ\(⋅\)⟩\.D\_\{\\rho\}=D\-2\\bar\{\\phi\}^\{\\ast\}\\left\\langle\\nabla\_\{\\mathrm\{im\}\\phi\}\\log\(\\tilde\{\\rho\}\\circ\\bar\{\\phi\}^\{\-1\}\),\\nabla\_\{\\mathrm\{im\}\\phi\}\(\\cdot\)\\right\\rangle\.\(24\)

###### Proof\.

For the minimum kernel, Eq\. \([35](https://arxiv.org/html/2607.08987#S4.E35)\) below shows thatδ¯↦δ¯​ρ~\\bar\{\\delta\}\\mapsto\\bar\{\\delta\}\\tilde\{\\rho\}\. For the invariant features kernel, given any measurable setA⊆im​ϕA\\subseteq\\mathrm\{im\}\\phiwe have

ϕ∗​\(ρ​d​Vℳ\)​\(A\)=∫π−1​\(ϕ¯−1​\(A\)\)ρ​𝑑Vℳ=∫ϕ¯−1​\(A\)ρ~​\(\[x\]\)​δ¯​\(\[x\]\)​𝑑V𝒩​\(\[x\]\)=∫A\(ρ~∘ϕ¯−1\)​\(y\)​𝑑μ​\(y\),\\displaystyle\\phi\_\{\\ast\}\(\\rho\\,dV\_\{\\mathcal\{M\}\}\)\(A\)=\\int\_\{\\pi^\{\-1\}\(\\bar\{\\phi\}^\{\-1\}\(A\)\)\}\\rho\\,dV\_\{\\mathcal\{M\}\}=\\int\_\{\\bar\{\\phi\}^\{\-1\}\(A\)\}\\tilde\{\\rho\}\(\[x\]\)\\,\\bar\{\\delta\}\(\[x\]\)\\,dV\_\{\\mathcal\{N\}\}\(\[x\]\)=\\int\_\{A\}\(\\tilde\{\\rho\}\\circ\\bar\{\\phi\}^\{\-1\}\)\(y\)d\\mu\(y\),\(25\)whereμ=ϕ¯∗​\(δ¯​d​V𝒩\)\\mu=\\bar\{\\phi\}\_\{\\ast\}\(\\bar\{\\delta\}dV\_\{\\mathcal\{N\}\}\)is the pushforward underϕ¯\\bar\{\\phi\}of the measureδ¯​d​V𝒩\\bar\{\\delta\}dV\_\{\\mathcal\{N\}\}\. Takingρ=1\\rho=1givesϕ∗​\(ρ​d​Vℳ\)​\(A\)=∫A𝑑μ\\phi\_\{\\ast\}\(\\rho\\,dV\_\{\\mathcal\{M\}\}\)\(A\)=\\int\_\{A\}d\\mu, which implies thatμ=ϕ∗​\(ρ​d​Vℳ\)=pϕ​d​Vim​ϕ\\mu=\\phi\_\{\\ast\}\(\\rho\\,dV\_\{\\mathcal\{M\}\}\)=p\_\{\\phi\}dV\_\{\\mathrm\{im\}\\phi\}, so Eq\. \([25](https://arxiv.org/html/2607.08987#S3.E25)\) becomes

ϕ∗​\(ρ​d​Vℳ\)​\(A\)=∫A\(ρ~∘ϕ¯−1\)​\(y\)​pϕ​\(y\)​𝑑Vim​ϕ​\(y\)\.\\displaystyle\\phi\_\{\\ast\}\(\\rho\\,dV\_\{\\mathcal\{M\}\}\)\(A\)=\\int\_\{A\}\(\\tilde\{\\rho\}\\circ\\bar\{\\phi\}^\{\-1\}\)\(y\)\\,p\_\{\\phi\}\(y\)\\,dV\_\{\\mathrm\{im\}\\,\\phi\}\(y\)\.\(26\)Hence, the pushforward measure satisfiesϕ∗​\(ρ​d​Vℳ\)=pϕ⋅\(ρ~∘ϕ¯−1\)​d​Vim​ϕ\\phi\_\{\\ast\}\(\\rho dV\_\{\\mathcal\{M\}\}\)=p\_\{\\phi\}\\cdot\(\\tilde\{\\rho\}\\circ\\bar\{\\phi\}^\{\-1\}\)dV\_\{\\mathrm\{im\}\\phi\}, meaning thatpϕ↦pϕ⋅\(ρ~∘ϕ¯−1\)p\_\{\\phi\}\\mapsto p\_\{\\phi\}\\cdot\(\\tilde\{\\rho\}\\circ\\bar\{\\phi\}^\{\-1\}\)\. The result then follows for these two kernels by the additivity oflog\\logand the linearity of the inner product\. For the integral kernel,Rosenet al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib20)\)show thatΔℳ\\Delta\_\{\\mathcal\{M\}\}picks up the extra term−2​⟨∇ℳlog⁡π∗​ρ~,∇ℳ\(⋅\)⟩\-2\\langle\\nabla\_\{\\mathcal\{M\}\}\\log\\pi^\{\\ast\}\\tilde\{\\rho\},\\nabla\_\{\\mathcal\{M\}\}\(\\cdot\)\\rangle, which equals−2​⟨∇𝒩log⁡ρ~,∇𝒩\(⋅\)⟩\-2\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\tilde\{\\rho\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\ranglesince the quotient mapπ:ℳ→𝒩\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}is a Riemannian submersion\. ∎

### 3\.4Special Case of Constant Densityδ\\delta

In the special case that the density functionδ\\deltadefined by Eq\. \([14](https://arxiv.org/html/2607.08987#S3.E14)\) is constant, our main result simplifies considerably\. Eqs\. \([15](https://arxiv.org/html/2607.08987#S3.E15)\), \([16](https://arxiv.org/html/2607.08987#S3.E16)\) and \([17](https://arxiv.org/html/2607.08987#S3.E17)\) show that in this case the graph LaplacianLR​WL\_\{RW\}based on the minimum kernel or the integral kernel converges to the Laplace–Beltrami operatorΔ𝒩\\Delta\_\{\\mathcal\{N\}\}on the quotient\. Moreover, in this setting we obtain a further relation between the Laplace–Beltrami operatorsΔℳ\\Delta\_\{\\mathcal\{M\}\}andΔ𝒩\\Delta\_\{\\mathcal\{N\}\}, and theGG\-invariant eigenfunctions ofΔℳ\\Delta\_\{\\mathcal\{M\}\}are in bijection with the eigenfunctions ofΔ𝒩\\Delta\_\{\\mathcal\{N\}\}\. To make the discussion precise, define the*projection*ofΔℳ\\Delta\_\{\\mathcal\{M\}\}as the differential operatorP​\(Δℳ\)P\(\\Delta\_\{\\mathcal\{M\}\}\)on𝒩\\mathcal\{N\}such that

π∗​\(P​\(Δℳ\)​h\)=Δℳ​\(π∗​h\)\\pi^\{\\ast\}\\left\(P\(\\Delta\_\{\\mathcal\{M\}\}\)h\\right\)=\\Delta\_\{\\mathcal\{M\}\}\(\\pi^\{\\ast\}h\)\(29\)for any smooth functionh:𝒩→ℝh\\colon\\mathcal\{N\}\\to\\mathbb\{R\}, whereπ:ℳ→𝒩\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}is the quotient map\. The following lemma gives an explicit formula forP​\(Δℳ\)P\(\\Delta\_\{\\mathcal\{M\}\}\)\. See Proposition 3\.1 and Remark \(c\)\(ii\) in\(Le,[2001](https://arxiv.org/html/2607.08987#bib.bib2)\)for a proof\.

###### Lemma 3\.13\.

The projection ofΔℳ\\Delta\_\{\\mathcal\{M\}\}onto𝒩\\mathcal\{N\}is given by the formula

P​\(Δℳ\)=Δ𝒩−⟨∇𝒩log⁡δ¯,∇𝒩\(⋅\)⟩\.P\(\\Delta\_\{\\mathcal\{M\}\}\)=\\Delta\_\{\\mathcal\{N\}\}\-\\left\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\},\\nabla\_\{\\mathcal\{N\}\}\(\\cdot\)\\right\\rangle\.\(30\)

In the special case thatδ\\deltais constant, the first\-order term in Eq\. \([30](https://arxiv.org/html/2607.08987#S3.E30)\) vanishes and the projection ofΔℳ\\Delta\_\{\\mathcal\{M\}\}is precisely the Laplace–Beltrami operator on the quotient\.

###### Corollary 3\.14\.

Assume the density functionδ\\deltais constant\. ThenP​\(Δℳ\)=Δ𝒩\.P\(\\Delta\_\{\\mathcal\{M\}\}\)=\\Delta\_\{\\mathcal\{N\}\}\.

In this regime the spectral analysis ofGG\-invariant functions onℳ\\mathcal\{M\}agrees*exactly*with the spectral analysis of functions on the quotient𝒩\\mathcal\{N\}, simplifying both theory and computation\. Concretely, eigenpairs ofΔℳ\\Delta\_\{\\mathcal\{M\}\}restricted toGG\-invariant functions are in bijection with eigenpairs ofΔ𝒩\\Delta\_\{\\mathcal\{N\}\}\.

###### Corollary 3\.15\(Eigenfunction correspondence\)\.

Assume that the density functionδ\\deltais constant, letf∈C∞​\(ℳ\)f\\in C^\{\\infty\}\(\\mathcal\{M\}\)beGG–invariant and writef=π∗​f¯f=\\pi^\{\\ast\}\\bar\{f\}forf¯∈C∞​\(𝒩\)\\bar\{f\}\\in C^\{\\infty\}\(\\mathcal\{N\}\)\. ThenΔℳ​f=λ​f\\Delta\_\{\\mathcal\{M\}\}f=\\lambda fif and only ifΔ𝒩​f¯=λ​f¯\\Delta\_\{\\mathcal\{N\}\}\\bar\{f\}=\\lambda\\bar\{f\}\.

###### Proof\.

By Eq\. \([29](https://arxiv.org/html/2607.08987#S3.E29)\) and Corollary[3\.14](https://arxiv.org/html/2607.08987#S3.Thmtheorem14),Δℳ​f=λ​f\\Delta\_\{\\mathcal\{M\}\}f=\\lambda fif and only ifπ∗​\(Δ𝒩​f¯\)=λ​π∗​f¯\\pi^\{\\ast\}\(\\Delta\_\{\\mathcal\{N\}\}\\bar\{f\}\)=\\lambda~\\pi^\{\\ast\}\\bar\{f\}\. The result follows from the surjectivity ofπ\\pi\. ∎

We illustrate these results with the following example\.

###### Example 3\.16\.

Considerℳ=SO​\(3\)⊂ℝ3×3\\mathcal\{M\}=\\mathrm\{SO\}\(3\)\\subset\\mathbb\{R\}^\{3\\times 3\}with the induced Frobenius metric222This metric differs from the more standard bi\-invariant metric inChirikjian and Kyatkin \([2016](https://arxiv.org/html/2607.08987#bib.bib48)\)bygFrob=2​gbi−invg\_\{\\mathrm\{Frob\}\}=2g\_\{\\mathrm\{bi\-inv\}\}, which impliesΔSO​\(3\)Frob=ΔSO​\(3\)bi−inv\\Delta\_\{\\mathrm\{SO\}\(3\)\}^\{\\mathrm\{Frob\}\}=\\Delta\_\{\\mathrm\{SO\}\(3\)\}^\{\\mathrm\{bi\-inv\}\}/2\.fromℝ3×3\\mathbb\{R\}^\{3\\times 3\}\. LetG=SO​\(2\)G=\\mathrm\{SO\}\(2\)act onS​O​\(3\)SO\(3\)byR⋅A=A​RTR\\cdot A=AR^\{T\}forR∈SO​\(2\)R\\in\\mathrm\{SO\}\(2\)andA∈SO​\(3\)A\\in\\mathrm\{SO\}\(3\), whereSO​\(2\)\\mathrm\{SO\}\(2\)is embedded inSO​\(3\)\\mathrm\{SO\}\(3\)as rotations about thezz\-axis\. The quotient manifold𝒩=SO​\(3\)/SO​\(2\)\\mathcal\{N\}=\\mathrm\{SO\}\(3\)/\\mathrm\{SO\}\(2\)is diffeomorphic to the 2\-sphere𝕊2\\mathbb\{S\}^\{2\}\(Lee,[2003](https://arxiv.org/html/2607.08987#bib.bib14), Example 21\.19\(a\)\)\. Moreover, all orbits of this action have volume2​π​22\\pi\\sqrt\{2\}since they are all isometric toSO​\(2\)\\mathrm\{SO\}\(2\), so the density functionδ\\deltais constant\. If we parametrizeR∈SO​\(2\)R\\in\\mathrm\{SO\}\(2\)by the rotation angleθ∈\[0,2​π\)\\theta\\in\[0,2\\pi\)andA∈SO​\(3\)A\\in\\mathrm\{SO\}\(3\)withZ​Y​ZZYZEuler anglesα∈\[0,2​π\),β∈\[0,π\],γ∈\[0,2​π\)\\alpha\\in\[0,2\\pi\),\\beta\\in\[0,\\pi\],\\gamma\\in\[0,2\\pi\), thenRRacts onAAbyR​\(θ\)⋅A​\(α,β,γ\)=A​\(α,β,γ−θ\)\.R\(\\theta\)\\cdot A\(\\alpha,\\beta,\\gamma\)=A\(\\alpha,\\beta,\\gamma\-\\theta\)\.It follows that theSO​\(2\)\\mathrm\{SO\}\(2\)\-invariant functionsf:SO​\(3\)→ℝf:\\mathrm\{SO\}\(3\)\\to\\mathbb\{R\}are precisely those which do not depend on the Euler angleγ\\gamma\. The Laplace–Beltrami operator onSO​\(3\)\\mathrm\{SO\}\(3\)is\(Chirikjian and Kyatkin,[2016](https://arxiv.org/html/2607.08987#bib.bib48), Eq\. \(9\.22\)\)

ΔSO​\(3\)=12​\(−∂2∂β2−cot⁡β​∂∂β−1sin2⁡β​\(∂2∂α2−2​cos⁡β​∂2∂α​∂γ\+∂2∂γ2\)\)\.\\displaystyle\\Delta\_\{\\mathrm\{SO\}\(3\)\}=\\frac\{1\}\{2\}\\left\(\-\\frac\{\\partial^\{2\}\}\{\\partial\\beta^\{2\}\}\-\\cot\\beta\\frac\{\\partial\}\{\\partial\\beta\}\-\\frac\{1\}\{\\sin^\{2\}\\beta\}\\left\(\\frac\{\\partial^\{2\}\}\{\\partial\\alpha^\{2\}\}\-2\\cos\\beta\\frac\{\\partial^\{2\}\}\{\\partial\\alpha\\partial\\gamma\}\+\\frac\{\\partial^\{2\}\}\{\\partial\\gamma^\{2\}\}\\right\)\\right\)\.\(31\)Since aSO​\(2\)\\mathrm\{SO\}\(2\)\-invariant functionffdoes not depend onγ\\gamma, theγ\\gamma\-derivatives vanish, and Corollary[3\.14](https://arxiv.org/html/2607.08987#S3.Thmtheorem14)gives

ΔSO​\(3\)​f=12​\(−∂2f∂β2−cot⁡β​∂f∂β−1sin2⁡β​∂2f∂α2\)=π∗​\(Δ𝕊2​f¯\)\\displaystyle\\Delta\_\{\\mathrm\{SO\}\(3\)\}f=\\frac\{1\}\{2\}\\left\(\-\\frac\{\\partial^\{2\}f\}\{\\partial\\beta^\{2\}\}\-\\cot\\beta\\frac\{\\partial f\}\{\\partial\\beta\}\-\\frac\{1\}\{\\sin^\{2\}\\beta\}\\frac\{\\partial^\{2\}f\}\{\\partial\\alpha^\{2\}\}\\right\)=\\pi^\{\\ast\}\(\\Delta\_\{\\mathbb\{S\}^\{2\}\}\\overline\{f\}\)\(32\)where𝕊2\\mathbb\{S\}^\{2\}is parameterized by the polar angleβ\\betaand the azimuthal angleα\\alpha\. The eigenfunctions ofΔSO​\(3\)\\Delta\_\{\\mathrm\{SO\}\(3\)\}are linear combinations of the unitary irreducible representations

Um​nℓ​\(A​\(α,β,γ\)\)=e−i​m​α​Pm​nℓ​\(cos⁡β\)​e−i​n​γ,\\displaystyle U^\{\\ell\}\_\{mn\}\(A\(\\alpha,\\beta,\\gamma\)\)=e^\{\-im\\alpha\}P^\{\\ell\}\_\{mn\}\(\\cos\\beta\)e^\{\-in\\gamma\},\(33\)with eigenvalueℓ​\(ℓ\+1\)/2\\ell\(\\ell\+1\)/2, wherePm​nℓP^\{\\ell\}\_\{mn\}denotes the generalized associated Legendre polynomials\(Chirikjian and Kyatkin,[2016](https://arxiv.org/html/2607.08987#bib.bib48), Eq\. \(9\.41\)\)\. TheSO​\(2\)\\mathrm\{SO\}\(2\)\-invariant eigenfunctions, those withn=0n=0, are therefore linear combinations of

Um​0ℓ​\(A​\(α,β,γ\)\)=e−i​m​α​Pm​0ℓ​\(cos⁡β\)=\(−1\)m​4​π2​ℓ\+1​Yℓm¯​\(β,α\),\\displaystyle U^\{\\ell\}\_\{m0\}\\left\(A\(\\alpha,\\beta,\\gamma\)\\right\)=e^\{\-im\\alpha\}P^\{\\ell\}\_\{m0\}\(\\cos\\beta\)=\(\-1\)^\{m\}\\sqrt\{\\frac\{4\\pi\}\{2\\ell\+1\}\}\\overline\{Y^\{m\}\_\{\\ell\}\}\(\\beta,\\alpha\),\(34\)whereYℓmY^\{m\}\_\{\\ell\}denotes the spherical harmonic on the unit sphere\(Chirikjian and Kyatkin,[2016](https://arxiv.org/html/2607.08987#bib.bib48), Eq\. \(9\.42\)\), in agreement with Corollary[3\.15](https://arxiv.org/html/2607.08987#S3.Thmtheorem15)\. In particular, forℓ=1\\ell=1andm=0m=0we haveU001​\(A​\(α,β,γ\)\)=P001​\(cos⁡β\)=cos⁡βU^\{1\}\_\{00\}\\left\(A\(\\alpha,\\beta,\\gamma\)\\right\)=P^\{1\}\_\{00\}\(\\cos\\beta\)=\\cos\\beta, an eigenfunction ofΔSO​\(3\)\\Delta\_\{\\mathrm\{SO\}\(3\)\}with eigenvalue1​\(1\+1\)/2=11\(1\+1\)/2=1\.

### 3\.5A Numerical Example

We illustrate the improved convergence rate in Eq\. \([15](https://arxiv.org/html/2607.08987#S3.E15)\) compared to spectral embeddings that are not symmetry\-aware with a numerical example in the special case of a constant densityδ\\delta\. Examples with non\-constantδ\\deltathat illustrate Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)are in Section[5](https://arxiv.org/html/2607.08987#S5)\.

Consider the action ofG=SO​\(2\)G=\\mathrm\{SO\}\(2\)onℳ=SO​\(3\)⊆ℝ3×3\\mathcal\{M\}=\\mathrm\{SO\}\(3\)\\subseteq\\mathbb\{R\}^\{3\\times 3\}from Example[3\.16](https://arxiv.org/html/2607.08987#S3.Thmtheorem16)and theGG\-invariant functionf​\(R\)=cos⁡β​\(R\),f\(R\)=\\cos\\beta\(R\),where0≤β​\(R\)≤π0\\leq\\beta\(R\)\\leq\\piis the second Euler angle in theZ​Y​ZZYZparametrization ofR∈SO​\(3\)R\\in\\mathrm\{SO\}\(3\)\. The functionffsatisfiesΔℳ​f=f\\Delta\_\{\\mathcal\{M\}\}f=f\(Example[3\.16](https://arxiv.org/html/2607.08987#S3.Thmtheorem16)\), and sinceδ\\deltais constant, Corollary[3\.14](https://arxiv.org/html/2607.08987#S3.Thmtheorem14)implies thatΔ𝒩​f¯=f¯\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}=\\overline\{f\}\. Therefore, atR0=I3R\_\{0\}=I\_\{3\}, we haveΔℳ​f​\(I3\)=1\\Delta\_\{\\mathcal\{M\}\}f\(\{I\_\{3\}\}\)=1andΔ𝒩​f¯​\(\[I3\]\)=1\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\{\[I\_\{3\}\]\}\)=1\. By writing the Riemannian metric elements in terms of Euler angles and using the formula for the Riemannian gradient, a straightforward calculation shows that∇ℳf​\(I3\)=0\\nabla\_\{\\mathcal\{M\}\}f\(\{I\_\{3\}\}\)=0\.

We uniformly samplen=10000n=10000data points from the Haar distribution onSO​\(3\)\\mathrm\{SO\}\(3\)andm=200m=200group elements from the Haar distribution onSO​\(2\)\\mathrm\{SO\}\(2\)\. We then use the Euclidean, minimum, and integral kernels to compute\(LR​W​f\)​\(I3\)\(L\_\{RW\}f\)\(I\_\{3\}\)for each kernel\. By Theorem[2\.1](https://arxiv.org/html/2607.08987#S2.Thmtheorem1), for the classical Euclidean kernel,4ε​\(LR​W​f\)​\(I3\)\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(I\_\{3\}\)approximatesΔℳ​f​\(I3\)=1\\Delta\_\{\\mathcal\{M\}\}f\(\{I\_\{3\}\}\)=1\. Meanwhile, by Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7), for the minimum and integral kernels,4ε​\(LR​W​f\)​\(I3\)\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(I\_\{3\}\)approximatesD​f¯​\(\[I3\]\)=Δ𝒩​f¯​\(\[I3\]\)=1D\\overline\{f\}\(\[I\_\{3\}\]\)=\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\{\[I\_\{3\}\]\}\)=1sinceδ\\deltais constant\. Figure[2](https://arxiv.org/html/2607.08987#S3.F2)shows a logarithmic plot of the mean absolute error overT=1000T=1000trials between4ε​\(LR​W​f\)​\(I3\)\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(I\_\{3\}\)and11versusε\\varepsilonfor each kernel\. As explained in Remark[3\.12](https://arxiv.org/html/2607.08987#S3.Thmtheorem12), the condition∇ℳf​\(I3\)=0\\nabla\_\{\\mathcal\{M\}\}f\(I\_\{3\}\)=0implies that the predicted slopes of the error in the variance\-dominated region \(small values ofε\\varepsilon\) are−0\.75\-0\.75for the Euclidean kernel and−0\.5\-0\.5for the minimum and integral kernels\. Linear fits give slopes of−0\.853\-0\.853,−0\.503\-0\.503and−0\.505\-0\.505for the Euclidean, minimum and integral kernels, respectively\. For the minimum and integral kernels, the fitted slopes are consistent with the corresponding predictions\. By contrast, for the Euclidean kernel, approximately 20 times more data points are required for a linear fit to be correct to one decimal place\.

![Refer to caption](https://arxiv.org/html/2607.08987v1/x1.png)Figure 2:Error between4ε​\(LR​W​f\)​\(I3\)\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(I\_\{3\}\)and11versusε\\varepsilonfor theGG\-invariant functionf​\(R\)=cos⁡β​\(R\)f\(R\)=\\cos\\beta\(R\)using the Euclidean, minimum, and integral kernels\. Shaded regions show the 95% confidence intervals across 1000 trials\.

## 4Proof of Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)

We split the proof of Theorem[3\.7](https://arxiv.org/html/2607.08987#S3.Thmtheorem7)into three parts, corresponding to the three G\-invariant kernels considered in the statement of the theorem: the minimum kernel \(Section[4\.1](https://arxiv.org/html/2607.08987#S4.SS1)\), the integral kernel \(Section[4\.2](https://arxiv.org/html/2607.08987#S4.SS2)\), and the invariant features kernel \(Section[4\.3](https://arxiv.org/html/2607.08987#S4.SS3)\)\. We note that the implicit constants in theOOnotation below depend onℳ,G,𝒩\\mathcal\{M\},G,\\mathcal\{N\},hhandxix\_\{i\}but not onnnorε\\varepsilon\.

### 4\.1Minimum Kernel

Fix a sufficiently smallε\>0\\varepsilon\>0and a data pointxi∈ℳx\_\{i\}\\in\\mathcal\{M\}\. LetF​\(y\):=Kmin​\(xi,y\)​f​\(y\)F\(y\):=K\_\{\\min\}\(x\_\{i\},y\)f\(y\)andG​\(y\):=Kmin​\(xi,y\)G\(y\):=K\_\{\\min\}\(x\_\{i\},y\)for ally∈ℳy\\in\\mathcal\{M\}\. We prove the desired result in three steps\. First, we derive an asymptotic approximation for𝔼​\[F\]/𝔼​\[G\]\\mathbb\{E\}\[F\]/\\mathbb\{E\}\[G\]up to orderO​\(ε\)O\(\\varepsilon\)\. Second, we use Bernstein’s inequality to show that∑j≠iF​\(xj\)/∑j≠iG​\(xj\)\\sum\_\{j\\neq i\}F\(x\_\{j\}\)/\\sum\_\{j\\neq i\}G\(x\_\{j\}\)converges to𝔼​\[F\]/𝔼​\[G\]\\mathbb\{E\}\[F\]/\\mathbb\{E\}\[G\]at rateOP​\(n−1/2​ε1/2−\(d−p\)/4\)O\_\{P\}\\left\(\{n^\{\-1/2\}\\varepsilon^\{1/2\-\(d\-p\)/4\}\}\\right\)\. Third, we combine steps 1 and 2 to obtain the stated error bound onLR​WL\_\{RW\}\.

Step 1\. To compute the expectations ofFFandGG, we convert the expectations overℳ\\mathcal\{M\}into integrals over the quotient𝒩\\mathcal\{N\}by using the following Fubini\-type integration formula\. This is a special case of a classical result for Riemannian submersions\(Sakai,[1996](https://arxiv.org/html/2607.08987#bib.bib70), Thm\. 5\.6\), which we adapt to our setting in Appendix[B](https://arxiv.org/html/2607.08987#A2)\.

###### Lemma 4\.1\(Fubini\-like quotient integration formula\)\.

Let\(ℳ,gℳ\)\(\\mathcal\{M\},g\_\{\\mathcal\{M\}\}\)be a Riemannian manifold and letGGbe a compact Lie group acting smoothly, freely, and isometrically onℳ\\mathcal\{M\}\. Let\(𝒩,g𝒩\)\(\\mathcal\{N\},g\_\{\\mathcal\{N\}\}\)denote the Riemannian quotient manifold arising from the action ofGGonℳ\\mathcal\{M\}\. Letδ¯\\overline\{\\delta\}denote the function on𝒩\\mathcal\{N\}induced by the density functionδ\\deltagiven in Definition[3\.6](https://arxiv.org/html/2607.08987#S3.Thmtheorem6)\. Then given any integrable functionf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}, we have that

∫ℳf​\(x\)​𝑑Vℳ​\(x\)=∫𝒩\(∫Gf​\(α⋅x\)​𝑑η​\(α\)\)​δ¯​\(\[x\]\)​𝑑V𝒩​\(\[x\]\),\\displaystyle\\int\_\{\\mathcal\{M\}\}f\(x\)dV\_\{\\mathcal\{M\}\}\(x\)=\\int\_\{\\mathcal\{N\}\}\\left\(\\int\_\{G\}f\(\\alpha\\cdot x\)d\\eta\(\\alpha\)\\right\)\\overline\{\\delta\}\(\[x\]\)dV\_\{\\mathcal\{N\}\}\(\[x\]\),\(35\)whered​VℳdV\_\{\\mathcal\{M\}\}andd​V𝒩dV\_\{\\mathcal\{N\}\}denote the Riemannian measures onℳ\\mathcal\{M\}and𝒩\\mathcal\{N\}determined by the metricsgℳg\_\{\\mathcal\{M\}\}andg𝒩g\_\{\\mathcal\{N\}\}, respectively, andd​ηd\\etais the Haar measure onGGsuch thatη​\(G\)=1\\eta\(G\)=1\. If, in addition, the functionf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}is G\-invariant, this identity simplifies to

∫ℳf​\(x\)​𝑑Vℳ​\(x\)=∫𝒩f¯​\(\[x\]\)​δ¯​\(\[x\]\)​𝑑V𝒩​\(\[x\]\)\.\\displaystyle\\int\_\{\\mathcal\{M\}\}f\(x\)dV\_\{\\mathcal\{M\}\}\(x\)=\\int\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\]\)\\overline\{\\delta\}\(\[x\]\)dV\_\{\\mathcal\{N\}\}\(\[x\]\)\.\(36\)

We now consider local smoothing with the minimum kernel\. Due to symmetry,KminK\_\{\\mathrm\{min\}\}induces a kernelK¯min\\overline\{K\}\_\{\\mathrm\{min\}\}on the quotient manifold𝒩\\mathcal\{N\}given by

K¯min​\(\[x\],\[y\]\)=exp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\),\[x\],\[y\]∈𝒩\.\\displaystyle\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\],\[y\]\)=\\exp\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\),\\qquad\\ \[x\],\[y\]\\in\\mathcal\{N\}\.\(37\)The minimum value overGGis continuous sinceGGis a compact Lie group, so for a fixedxix\_\{i\}the minimum kernelKmin​\(xi,y\)K\_\{\\mathrm\{min\}\}\(x\_\{i\},y\)is a continuous function onℳ\\mathcal\{M\}and thus integrable by the compactness ofℳ\\mathcal\{M\}\. By Eq\. \([36](https://arxiv.org/html/2607.08987#S4.E36)\) we have that

𝔼​\[F\]=1Vol​\(ℳ\)​∫ℳKmin​\(xi,y\)​f​\(y\)​𝑑Vℳ​\(y\)=1Vol​\(ℳ\)​∫𝒩K¯min​\(\[xi\],\[y\]\)​f¯​\(\[y\]\)​δ¯​\(\[y\]\)​𝑑V𝒩​\(\[y\]\)\.\\displaystyle\\begin\{split\}\\mathbb\{E\}\\left\[F\\right\]&=\\frac\{1\}\{\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\int\_\{\\mathcal\{M\}\}K\_\{\\min\}\(x\_\{i\},y\)f\(y\)dV\_\{\\mathcal\{M\}\}\(y\)=\\frac\{1\}\{\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\int\_\{\\mathcal\{N\}\}\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\_\{i\}\],\[y\]\)\\overline\{f\}\(\[y\]\)\\overline\{\\delta\}\(\[y\]\)dV\_\{\\mathcal\{N\}\}\(\[y\]\)\.\\end\{split\}\(38\)The following lemma, proved in Appendix[C](https://arxiv.org/html/2607.08987#A3), is needed to evaluate the integral in Equation \([38](https://arxiv.org/html/2607.08987#S4.E38)\)\.

###### Lemma 4\.2\(Asymptotics of local smoothing with the minimum kernel\)\.

Letq=dim\(𝒩\)q=\\dim\(\\mathcal\{N\}\)\. Given a smooth functionh:𝒩→ℝh\\colon\\mathcal\{N\}\\to\\mathbb\{R\}and a fixed\[x\]∈𝒩\[x\]\\in\\mathcal\{N\}, for all sufficiently smallε\\varepsilonwe have that

1\(π​ε\)q/2​∫𝒩K¯min​\(\[x\],\[y\]\)​h​\(\[y\]\)​𝑑V𝒩​\(\[y\]\)=h​\(\[x\]\)\+ε4​\(E​\(\[x\]\)​h​\(\[x\]\)−Δ𝒩​h​\(\[x\]\)\)\+O​\(ε2\),\\displaystyle\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathcal\{N\}\}\\overline\{K\}\_\{\\min\}\(\[x\],\[y\]\)h\(\[y\]\)dV\_\{\\mathcal\{N\}\}\(\[y\]\)=h\(\[x\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\(E\(\[x\]\)h\(\[x\]\)\-\\Delta\_\{\\mathcal\{N\}\}h\(\[x\]\)\\right\)\+O\(\\varepsilon^\{2\}\),\(39\)whered​V𝒩dV\_\{\\mathcal\{N\}\}denotes the Riemannian measure on𝒩\\mathcal\{N\}determined by the metricg𝒩g\_\{\\mathcal\{N\}\}, andE​\(\[x\]\)E\(\[x\]\)is some function on𝒩\\mathcal\{N\}that depends on the curvature of𝒩\\mathcal\{N\}at\[x\]\[x\]and on the embedding ofℳ\\mathcal\{M\}intoℝD\\mathbb\{R\}^\{D\}\.

The explicit form ofEEis not needed later; only the fact that it is a scalar function multiplyinghh\. Under the assumption that bothδ:ℳ→ℝ\\delta:\\mathcal\{M\}\\to\\mathbb\{R\}andf:ℳ→ℝf:\\mathcal\{M\}\\to\\mathbb\{R\}are smooth it follows that the projectionsδ¯,f¯:𝒩→ℝ\\overline\{\\delta\},\\overline\{f\}:\\mathcal\{N\}\\to\\mathbb\{R\}are both smooth by the characteristic property of surjective smooth submersions \[Theorem 4\.29,Lee\([2003](https://arxiv.org/html/2607.08987#bib.bib14)\)\]\. Using Lemma[4\.2](https://arxiv.org/html/2607.08987#S4.Thmtheorem2)withh=f¯​δ¯h=\\overline\{f\}\\overline\{\\delta\}gives

𝔼​\[F\]=\(π​ε\)q/2Vol​\(ℳ\)​\[f¯​\(\[xi\]\)​δ¯​\(\[xi\]\)\+ε4​\(E​\(\[xi\]\)​f¯​\(\[xi\]\)​δ¯​\(\[xi\]\)−Δ𝒩​\(f¯​δ¯\)​\(\[xi\]\)\)\+O​\(ε2\)\]\.\\displaystyle\\mathbb\{E\}\\left\[F\\right\]=\\frac\{\(\\pi\\varepsilon\)^\{q/2\}\}\{\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\left\[\\overline\{f\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\(E\(\[x\_\{i\}\]\)\\overline\{f\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\-\\Delta\_\{\\mathcal\{N\}\}\(\\overline\{f\}~\\overline\{\\delta\}\)\(\[x\_\{i\}\]\)\\right\)\+O\(\\varepsilon^\{2\}\)\\right\]\.\(40\)By takingf=1f=1, we obtain

𝔼​\[G\]=\(π​ε\)q/2Vol​\(ℳ\)​\[δ¯​\(\[xi\]\)\+ε4​\(E​\(\[xi\]\)​δ¯​\(\[xi\]\)−Δ𝒩​δ¯​\(\[xi\]\)\)\+O​\(ε2\)\]\.\\displaystyle\\mathbb\{E\}\\left\[G\\right\]=\\frac\{\(\\pi\\varepsilon\)^\{q/2\}\}\{\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\left\[\\overline\{\\delta\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\(E\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\-\\Delta\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\(\[x\_\{i\}\]\)\\right\)\+O\(\\varepsilon^\{2\}\)\\right\]\.\(41\)Using the Taylor expansion1/\(1\+a​ε\)=1−a​ε\+O​\(ε2\)1/\(1\+a\\varepsilon\)=1\-a\\varepsilon\+O\(\\varepsilon^\{2\}\), a straightforward calculation shows that the ratio is

𝔼​\[F\]𝔼​\[G\]=f¯​\(\[xi\]\)\+ε4​\[Δ𝒩​δ¯​\(\[xi\]\)δ¯​\(\[xi\]\)​f¯​\(\[xi\]\)−Δ𝒩​\(f¯​δ¯\)​\(\[xi\]\)δ¯​\(\[xi\]\)\]\+O​\(ε2\)=f¯​\(\[xi\]\)\+ε4​\[−Δ𝒩​f¯​\(\[xi\]\)\+2δ¯​\(\[xi\]\)​⟨∇𝒩δ¯​\(\[xi\]\),∇𝒩f¯​\(\[xi\]\)⟩\]\+O​\(ε2\),\\displaystyle\\begin\{split\}\\frac\{\\mathbb\{E\}\\left\[F\\right\]\}\{\\mathbb\{E\}\\left\[G\\right\]\}&=\\overline\{f\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\[\\frac\{\\Delta\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\(\[x\_\{i\}\]\)\}\{\\overline\{\\delta\}\(\[x\_\{i\}\]\)\}\\overline\{f\}\(\[x\_\{i\}\]\)\-\\frac\{\\Delta\_\{\\mathcal\{N\}\}\(\\overline\{f\}~\\overline\{\\delta\}\)\(\[x\_\{i\}\]\)\}\{\\overline\{\\delta\}\(\[x\_\{i\}\]\)\}\\right\]\+O\(\\varepsilon^\{2\}\)\\\\ &=\\overline\{f\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\[\-\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\+\\frac\{2\}\{\\overline\{\\delta\}\(\[x\_\{i\}\]\)\}\\langle\\nabla\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\(\[x\_\{i\}\]\),\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\rangle\\right\]\+O\(\\varepsilon^\{2\}\),\\end\{split\}\(42\)where the second line follows from applying the product rule for the Laplace–Beltrami operator:Δ𝒩​\(f¯​δ¯\)=f¯​Δ𝒩​δ¯−2​⟨∇𝒩f¯,∇𝒩δ¯⟩\+δ¯​Δ𝒩​f¯\\Delta\_\{\\mathcal\{N\}\}\(\\overline\{f\}\\,\\overline\{\\delta\}\)=\\overline\{f\}\\Delta\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\-2\\langle\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\},\\nabla\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\\rangle\+\\overline\{\\delta\}\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\. Note also thatδ¯​\(\[x\]\)\>0\\overline\{\\delta\}\(\[x\]\)\>0for all\[x\]∈𝒩\[x\]\\in\\mathcal\{N\}since the action ofGGonℳ\\mathcal\{M\}is free, so all orbits have a positive volume\.

Step 2\. Letp\+​\(n,α\)p\_\{\+\}\(n,\\alpha\)andp−​\(n,α\)p\_\{\-\}\(n,\\alpha\)denote the probability of having an error greater thanα\\alphaor less than−α\-\\alpha, respectively, i\.e\.,

p\+​\(n,α\)=ℙ​\[∑j≠iF​\(xj\)∑j≠iG​\(xj\)−𝔼​\[F\]𝔼​\[G\]\>α\],p−​\(n,α\)=ℙ​\[∑j≠iF​\(xj\)∑j≠iG​\(xj\)−𝔼​\[F\]𝔼​\[G\]<−α\]\.\\displaystyle p\_\{\+\}\(n,\\alpha\)=\\mathbb\{P\}\\left\[\\frac\{\\sum\_\{j\\neq i\}F\(x\_\{j\}\)\}\{\\sum\_\{j\\neq i\}G\(x\_\{j\}\)\}\-\\frac\{\\mathbb\{E\}\[F\]\}\{\\mathbb\{E\}\[G\]\}\>\\alpha\\right\],\\qquad p\_\{\-\}\(n,\\alpha\)=\\mathbb\{P\}\\left\[\\frac\{\\sum\_\{j\\neq i\}F\(x\_\{j\}\)\}\{\\sum\_\{j\\neq i\}G\(x\_\{j\}\)\}\-\\frac\{\\mathbb\{E\}\[F\]\}\{\\mathbb\{E\}\[G\]\}<\-\\alpha\\right\]\.\(43\)DefineYj=𝔼​\[G\]​F​\(xj\)−𝔼​\[F\]​G​\(xj\)\+α​𝔼​\[G\]​\(𝔼​\[G\]−G​\(xj\)\)Y\_\{j\}=\\mathbb\{E\}\[G\]F\(x\_\{j\}\)\-\\mathbb\{E\}\[F\]G\(x\_\{j\}\)\+\\alpha\\mathbb\{E\}\[G\]\(\\mathbb\{E\}\[G\]\-G\(x\_\{j\}\)\)for allj≠ij\\neq i\. Multiplying both sides of Eq\. \([43](https://arxiv.org/html/2607.08987#S4.E43)\) by𝔼​\[G\]​∑j≠iG​\(xj\)\\mathbb\{E\}\[G\]\\sum\_\{j\\neq i\}G\(x\_\{j\}\)and rearranging gives

p\+​\(n,α\)=ℙ​\[∑j≠iYj\>\(n−1\)​α​\(𝔼​\[G\]\)2\]\.\\displaystyle p\_\{\+\}\(n,\\alpha\)=\\mathbb\{P\}\\left\[\\sum\_\{j\\neq i\}Y\_\{j\}\>\(n\-1\)\\alpha\(\\mathbb\{E\}\[G\]\)^\{2\}\\right\]\.\(44\)Note thatYjY\_\{j\}are i\.i\.d\. random variables with zero mean and variance given by

𝔼\[Yj2\]=𝔼\[G\]2𝔼\[F2\]−2𝔼\[F\]𝔼\[G\]𝔼\[\\displaystyle\\mathbb\{E\}\[Y\_\{j\}^\{2\}\]=\\mathbb\{E\}\[G\]^\{2\}\\mathbb\{E\}\[F^\{2\}\]\-2\\mathbb\{E\}\[F\]\\mathbb\{E\}\[G\]\\mathbb\{E\}\[FG\]\+𝔼\[F\]2𝔼\[G2\]\+O\(α\)\.\\displaystyle FG\]\+\\mathbb\{E\}\[F\]^\{2\}\\mathbb\{E\}\[G^\{2\}\]\+O\(\\alpha\)\.\(45\)Next we use Lemma[4\.1](https://arxiv.org/html/2607.08987#S4.Thmtheorem1)and Lemma[4\.2](https://arxiv.org/html/2607.08987#S4.Thmtheorem2)to compute the second moments ofFFandGG\. We use the fact thatK¯min2​\(\[xi\],\[y\];ε\)=K¯min​\(\[xi\],\[y\];ε/2\)\\overline\{K\}\_\{\\mathrm\{min\}\}^\{2\}\(\[x\_\{i\}\],\[y\];\\varepsilon\)=\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\_\{i\}\],\[y\];\\varepsilon/2\)for the integral in Lemma[4\.2](https://arxiv.org/html/2607.08987#S4.Thmtheorem2)\. We obtain

𝔼​\[F2\]=\(π​ε\)q/22q/2​Vol​\(ℳ\)​\[f¯2​\(\[xi\]\)​δ¯​\(\[xi\]\)\+ε8​\(E​\(\[xi\]\)​f¯2​\(\[xi\]\)​δ¯​\(\[xi\]\)−Δ𝒩​\(f¯2​δ¯\)​\(\[xi\]\)\)\+O​\(ε2\)\],\\displaystyle\\mathbb\{E\}\[F^\{2\}\]=\\frac\{\(\\pi\\varepsilon\)^\{q/2\}\}\{2^\{q/2\}\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\left\[\\overline\{f\}^\{2\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{8\}\\left\(E\(\[x\_\{i\}\]\)\\overline\{f\}^\{2\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\-\\Delta\_\{\\mathcal\{N\}\}\(\\overline\{f\}^\{2\}~\\overline\{\\delta\}\)\(\[x\_\{i\}\]\)\\right\)\+O\(\\varepsilon^\{2\}\)\\right\],\(46\)
𝔼​\[G2\]=\(π​ε\)q/22q/2​Vol​\(ℳ\)​\[δ¯​\(\[xi\]\)\+ε8​\(E​\(\[xi\]\)​δ¯​\(\[xi\]\)−Δ𝒩​δ¯​\(\[xi\]\)\)\+O​\(ε2\)\],\\displaystyle\\mathbb\{E\}\\left\[G^\{2\}\\right\]=\\frac\{\(\\pi\\varepsilon\)^\{q/2\}\}\{2^\{q/2\}\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\left\[\\overline\{\\delta\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{8\}\\left\(E\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\-\\Delta\_\{\\mathcal\{N\}\}\\overline\{\\delta\}\(\[x\_\{i\}\]\)\\right\)\+O\(\\varepsilon^\{2\}\)\\right\],\(47\)and

𝔼​\[F​G\]=\(π​ε\)q/22q/2​Vol​\(ℳ\)​\[f¯​\(\[xi\]\)​δ¯​\(\[xi\]\)\+ε8​\(E​\(\[xi\]\)​f¯​\(\[xi\]\)​δ¯​\(\[xi\]\)−Δ𝒩​\(f¯​δ¯\)​\(\[xi\]\)\)\+O​\(ε2\)\]\.\\displaystyle\\mathbb\{E\}\[FG\]=\\frac\{\(\\pi\\varepsilon\)^\{q/2\}\}\{2^\{q/2\}\\mathrm\{Vol\}\(\\mathcal\{M\}\)\}\\left\[\\overline\{f\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\+\\frac\{\\varepsilon\}\{8\}\\left\(E\(\[x\_\{i\}\]\)\\overline\{f\}\(\[x\_\{i\}\]\)\\overline\{\\delta\}\(\[x\_\{i\}\]\)\-\\Delta\_\{\\mathcal\{N\}\}\(\\overline\{f\}~\\overline\{\\delta\}\)\(\[x\_\{i\}\]\)\\right\)\+O\(\\varepsilon^\{2\}\)\\right\]\.\(48\)Substituting Eqs\. \([40](https://arxiv.org/html/2607.08987#S4.E40)\), \([41](https://arxiv.org/html/2607.08987#S4.E41)\), \([46](https://arxiv.org/html/2607.08987#S4.E46)\), \([47](https://arxiv.org/html/2607.08987#S4.E47)\), and \([48](https://arxiv.org/html/2607.08987#S4.E48)\) into Eq\. \([45](https://arxiv.org/html/2607.08987#S4.Ex2)\) gives

𝔼​\[Yj2\]=\(π​ε\)3​q/2​δ¯3​\(\[xi\]\)2q/2​Vol​\(ℳ\)3​ε8​\[−Δ𝒩​f¯2​\(\[xi\]\)\+2​f¯​\(\[xi\]\)​Δ𝒩​f¯​\(\[xi\]\)\+O​\(ε\)\]\+O​\(α\),\\displaystyle\\mathbb\{E\}\[Y\_\{j\}^\{2\}\]=\\frac\{\(\\pi\\varepsilon\)^\{3q/2\}\\overline\{\\delta\}^\{3\}\(\[x\_\{i\}\]\)\}\{2^\{q/2\}\\mathrm\{Vol\}\(\\mathcal\{M\}\)^\{3\}\}\\frac\{\\varepsilon\}\{8\}\\left\[\-\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}^\{2\}\(\[x\_\{i\}\]\)\+2\\overline\{f\}\(\[x\_\{i\}\]\)\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\+O\(\\varepsilon\)\\right\]\+O\(\\alpha\),\(49\)and after applying the identity−Δ𝒩​f¯2​\(\[xi\]\)\+2​f¯​\(\[xi\]\)​Δ𝒩​f¯​\(\[xi\]\)=2​‖∇𝒩f¯​\(\[xi\]\)‖2\-\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}^\{2\}\(\[x\_\{i\}\]\)\+2\\overline\{f\}\(\[x\_\{i\}\]\)\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)=2\\\|\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\\|^\{2\}, this simplifies to

𝔼​\[Yj2\]=\(π​ε\)3​q/2​δ¯3​\(\[xi\]\)2q/2​Vol​\(ℳ\)3​ε4​\[‖∇𝒩f¯​\(\[xi\]\)‖2\+O​\(ε\)\]\+O​\(α\)\.\\displaystyle\\mathbb\{E\}\[Y\_\{j\}^\{2\}\]=\\frac\{\(\\pi\\varepsilon\)^\{3q/2\}\\overline\{\\delta\}^\{3\}\(\[x\_\{i\}\]\)\}\{2^\{q/2\}\\mathrm\{Vol\}\(\\mathcal\{M\}\)^\{3\}\}\\frac\{\\varepsilon\}\{4\}\\left\[\\\|\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\\|^\{2\}\+O\(\\varepsilon\)\\right\]\+O\(\\alpha\)\.\(50\)We have thatFFandGGare bounded random variables sinceℳ\\mathcal\{M\}is compact, so by construction the random variablesYjY\_\{j\}are also bounded, say\|Yj\|<3​C\|Y\_\{j\}\|<3Cfor allj≠ij\\neq ifor someC\>0C\>0\. It follows from Bernstein’s inequality that

p\+​\(n,α\)≤exp⁡\(−\(n−1\)2​α2​𝔼​\[G\]42​\(n−1\)​𝔼​\[Yj2\]\+2​C​\(n−1\)​α​𝔼​\[G\]2\)=exp⁡\(−\(n−1\)​α2​δ¯​\(\[xi\]\)​2q/2\+1​\(π​ε\)q/2Vol​\(ℳ\)​ε​\(‖∇𝒩f¯​\(\[xi\]\)‖2\+O​\(ε\)\)\+O​\(α\)\)\.\\displaystyle\\begin\{split\}p\_\{\+\}\(n,\\alpha\)&\\leq\\exp\\left\(\-\\frac\{\(n\-1\)^\{2\}\\alpha^\{2\}\\mathbb\{E\}\[G\]^\{4\}\}\{2\(n\-1\)\\mathbb\{E\}\[Y\_\{j\}^\{2\}\]\+2C\(n\-1\)\\alpha\\mathbb\{E\}\[G\]^\{2\}\}\\right\)\\\\ &=\\exp\\left\(\-\\frac\{\(n\-1\)\\alpha^\{2\}\\bar\{\\delta\}\(\[x\_\{i\}\]\)2^\{q/2\+1\}\(\\pi\\varepsilon\)^\{q/2\}\}\{\\mathrm\{Vol\}\(\\mathcal\{M\}\)\\varepsilon\(\\\|\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\\|^\{2\}\+O\(\\varepsilon\)\)\+O\(\\alpha\)\}\\right\)\.\\end\{split\}\(51\)Assume∇𝒩f¯​\(\[xi\]\)≠0\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\neq 0; the case∇𝒩f¯​\(\[xi\]\)=0\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)=0is covered in Remark[3\.12](https://arxiv.org/html/2607.08987#S3.Thmtheorem12)\. Fix a constantM\>0M\>0and setα=M​n−1/2​ε1/2−q/4\\alpha=Mn^\{\-1/2\}\\varepsilon^\{1/2\-q/4\}\. Then Eq\. \([51](https://arxiv.org/html/2607.08987#S4.E51)\) becomes

p\+​\(n,α\)≤exp⁡\(−C0​M21\+C1​M​n−1/2​ε−1/2−q/4\)\\displaystyle p\_\{\+\}\(n,\\alpha\)\\leq\\exp\\left\(\-\\frac\{C\_\{0\}M^\{2\}\}\{1\+C\_\{1\}Mn^\{\-1/2\}\\varepsilon^\{\-1/2\-q/4\}\}\\right\)\(52\)for appropriate constantsC0C\_\{0\}andC1C\_\{1\}\. We obtain the same bound forp−​\(n,α\)p\_\{\-\}\(n,\\alpha\)by changing the sign ofα\\alphain the definition ofYjY\_\{j\}, which does not affect the approximation of𝔼​\[Yj2\]\\mathbb\{E\}\[Y\_\{j\}^\{2\}\]in Eq\. \([45](https://arxiv.org/html/2607.08987#S4.Ex2)\)\. Note that there exists a positive integerNNsuch thatC1​n−1/2​ε−1/2−q/4<1C\_\{1\}n^\{\-1/2\}\\varepsilon^\{\-1/2\-q/4\}<1for alln\>Nn\>N, and in this casep±​\(n,α\)≤exp⁡\(−C0​M2/\(1\+M\)\)p\_\{\\pm\}\(n,\\alpha\)\\leq\\exp\(\-C\_\{0\}M^\{2\}/\(1\+M\)\)\. Thus for anyδ\>0\\delta\>0we can choose sufficiently large constantsMMandNNsuch thatp±​\(n,α\)<δp\_\{\\pm\}\(n,\\alpha\)<\\deltafor alln\>Nn\>N\. Recalling thatα=M​n−1/2​ε1/2−q/4\\alpha=Mn^\{\-1/2\}\\varepsilon^\{1/2\-q/4\}and the definition ofp±​\(n,a\)p\_\{\\pm\}\(n,a\)in Eq\. \([43](https://arxiv.org/html/2607.08987#S4.E43)\) gives the error

\|∑j≠iF​\(xj\)∑j≠iG​\(xj\)−𝔼​\[F\]𝔼​\[G\]\|=OP​\(1n1/2​ε−1/2\+q/4\)\.\\displaystyle\\Bigg\|\\frac\{\\sum\_\{j\\neq i\}F\(x\_\{j\}\)\}\{\\sum\_\{j\\neq i\}G\(x\_\{j\}\)\}\-\\frac\{\\mathbb\{E\}\[F\]\}\{\\mathbb\{E\}\[G\]\}\\Bigg\|=O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{\-1/2\+q/4\}\}\\right\)\.\(53\)
Step 3\. With the notationF​\(y\)=Kmin​\(xi,y\)​f​\(y\)F\(y\)=K\_\{\\min\}\(x\_\{i\},y\)f\(y\)andG​\(y\)=Kmin​\(xi,y\)G\(y\)=K\_\{\\min\}\(x\_\{i\},y\),y∈ℳy\\in\\mathcal\{M\}, we can rewrite Eq\. \([5](https://arxiv.org/html/2607.08987#S2.E5)\) as

4ε​LR​W​f​\(xi\)=4ε​\[f​\(xi\)−∑j=1nF​\(xj\)∑j=1nG​\(xj\)\]\.\\displaystyle\\frac\{4\}\{\\varepsilon\}L\_\{RW\}f\(x\_\{i\}\)=\\frac\{4\}\{\\varepsilon\}\\left\[f\(x\_\{i\}\)\-\\frac\{\\sum\_\{j=1\}^\{n\}F\(x\_\{j\}\)\}\{\\sum\_\{j=1\}^\{n\}G\(x\_\{j\}\)\}\\right\]\.\(54\)As shown byLanda and Shkolnisky\([2018](https://arxiv.org/html/2607.08987#bib.bib31)\), removing the diagonal terms from the sums introduces anO​\(1/\(n​ε\(d−p\)/2\)\)O\\left\(\{1\}/\{\(n\\varepsilon^\{\(d\-p\)/2\}\}\)\\right\)error, which is negligible compared to the variance error term in Eq\. \([53](https://arxiv.org/html/2607.08987#S4.E53)\)\. Then from Eqs\. \([42](https://arxiv.org/html/2607.08987#S4.E42)\), \([53](https://arxiv.org/html/2607.08987#S4.E53)\) and \([54](https://arxiv.org/html/2607.08987#S4.E54)\) we see that

\|4ε​LR​W​f​\(xi\)−\(Δ𝒩​f¯​\(\[xi\]\)−2​⟨∇𝒩log⁡δ¯​\(\[xi\]\),∇𝒩f¯​\(\[xi\]\)⟩\+O​\(ε\)\)\|=\|4ε​\(f¯​\(\[xi\]\)−∑j≠iF​\(xj\)∑j≠iG​\(xj\)\)\+4ε​\(𝔼​\[F\]𝔼​\[G\]−f¯​\(\[xi\]\)\)\|=OP​\(1n1/2​ε1/2\+q/4\)\.\\displaystyle\\begin\{split\}&\\Bigg\|\\frac\{4\}\{\\varepsilon\}L\_\{RW\}f\(x\_\{i\}\)\-\\Big\(\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\-2\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\}\(\[x\_\{i\}\]\),\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\rangle\+O\(\\varepsilon\)\\Big\)\\Bigg\|\\\\ &=\\Bigg\|\\frac\{4\}\{\\varepsilon\}\\left\(\\bar\{f\}\(\[x\_\{i\}\]\)\-\\frac\{\\sum\_\{j\\neq i\}F\(x\_\{j\}\)\}\{\\sum\_\{j\\neq i\}G\(x\_\{j\}\)\}\\right\)\+\\frac\{4\}\{\\varepsilon\}\\left\(\\frac\{\\mathbb\{E\}\[F\]\}\{\\mathbb\{E\}\[G\]\}\-\\bar\{f\}\(\[x\_\{i\}\]\)\\right\)\\Bigg\|=O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+q/4\}\}\\right\)\.\\end\{split\}\(55\)Recalling thatq=dim​\(𝒩\)=d−pq=\\mathrm\{dim\}\(\\mathcal\{N\}\)=d\-pgives the desired result\.

### 4\.2Integral Kernel

Our proof for the integral kernel is based on the results ofRosenet al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib20)\), which constructed aGG\-invariant graph Laplacian \(G\-GL\) by considering the distances between all the pairs of points generated by the action ofGGon the dataset, followed by integration overGG\. The key result is their Theorem 11, which shows that the normalizedGG\-GL converges to the Laplace–Beltrami operatorΔℳ\\Delta\_\{\\mathcal\{M\}\}onℳ\\mathcal\{M\}with an improved variance error ofOP​\(1/\(n1/2​ε1/2\+\(d−p\)/4\)\)O\_\{P\}\(1/\(n^\{1/2\}\\varepsilon^\{1/2\+\(d\-p\)/4\}\)\)\. We then use the projection ofΔℳ\\Delta\_\{\\mathcal\{M\}\}onto the quotient manifold𝒩\\mathcal\{N\}to obtain a differential operator on𝒩\\mathcal\{N\}, as given in Lemma[3\.13](https://arxiv.org/html/2607.08987#S3.Thmtheorem13)\.

Given data pointsx1,…,xn∈ℳx\_\{1\},\\dots,x\_\{n\}\\in\\mathcal\{M\}, letL~\\tilde\{L\}denote the normalizedGG\-GL Laplacian introduced byRosenet al\.\([2024](https://arxiv.org/html/2607.08987#bib.bib20)\), which acts on functionsg:\{1,…,n\}×G→ℝg\\colon\\\{1,\\dots,n\\\}\\times G\\to\\mathbb\{R\}by

L~​g​\(i,α\)=g​\(i,α\)−∑j=1n∫Gexp⁡\(−‖α⋅xi−β⋅xj‖2ε\)​g​\(j,β\)​𝑑η​\(β\)∑j=1n∫Gexp⁡\(−‖xi−β⋅xj‖2ε\)​𝑑η​\(β\)\.\\displaystyle\\tilde\{L\}g\(i,\\alpha\)=g\(i,\\alpha\)\-\\frac\{\\sum\_\{j=1\}^\{n\}\\int\_\{G\}\\exp\\left\(\-\\frac\{\\\|\\alpha\\cdot x\_\{i\}\-\\beta\\cdot x\_\{j\}\\\|^\{2\}\}\{\\varepsilon\}\\right\)g\(j,\\beta\)d\\eta\(\\beta\)\}\{\\sum\_\{j=1\}^\{n\}\\int\_\{G\}\\exp\\left\(\-\\frac\{\\\|x\_\{i\}\-\\beta\\cdot x\_\{j\}\\\|^\{2\}\}\{\\varepsilon\}\\right\)d\\eta\(\\beta\)\}\.\(56\)Now, given aGG\-invariant smooth functionf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}, we obtain a functiong:\{1,…,n\}×G→ℝg\\colon\\\{1,\\dots,n\\\}\\times G\\to\\mathbb\{R\}by definingg​\(i,α\)=f​\(α⋅xi\)g\(i,\\alpha\)=f\(\\alpha\\cdot x\_\{i\}\)\. Note thatggis independent ofGG, i\.e\.g​\(i,α\)=g​\(i,e\)g\(i,\\alpha\)=g\(i,e\)for allα∈G\\alpha\\in G, due to theGG\-invariance offf\. It follows immediately from the above definitions that\(LR​W​f\)​\(xi\)\(L\_\{RW\}f\)\(x\_\{i\}\)coincides with\(L~​g\)​\(i,e\)\(\\tilde\{L\}g\)\(i,e\)for allx1,…,xnx\_\{1\},\\dots,x\_\{n\}\. Therefore, by the convergence of theGG\-GL operatorL~\\tilde\{L\}\(Rosenet al\.,[2024](https://arxiv.org/html/2607.08987#bib.bib20), Thm 11\), we obtain

4ε​\(LR​W​f\)​\(xi\)=4ε​\(L~​g\)​\(i,e\)=Δℳ​f​\(xi\)\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+\(d−p\)/4\)\.\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=\\frac\{4\}\{\\varepsilon\}\(\\tilde\{L\}g\)\(i,e\)=\\Delta\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)\+O\(\\varepsilon\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+\(d\-p\)/4\}\}\\right\)\.\(57\)SinceffisGG\-invariant, we have thatf=π∗​f¯f=\\pi^\{\\ast\}\\bar\{f\}withf¯\\bar\{f\}the induced function on𝒩\\mathcal\{N\}, soΔℳ​f​\(xi\)=P​\(Δℳ​f¯\)​\(\[xi\]\)\\Delta\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)=P\(\\Delta\_\{\\mathcal\{M\}\}\\bar\{f\}\)\(\[x\_\{i\}\]\)by the definition ofP​\(Δℳ\)P\(\\Delta\_\{\\mathcal\{M\}\}\)in Eq\. \([29](https://arxiv.org/html/2607.08987#S3.E29)\)\. Applying Lemma[3\.13](https://arxiv.org/html/2607.08987#S3.Thmtheorem13)then gives

Δℳ​f​\(xi\)=Δ𝒩​f¯​\(\[xi\]\)−⟨∇𝒩log⁡δ¯​\(\[xi\]\),∇𝒩f¯​\(\[xi\]\)⟩\.\\displaystyle\\Delta\_\{\\mathcal\{M\}\}f\(x\_\{i\}\)=\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\-\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\}\(\[x\_\{i\}\]\),\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\rangle\.\(58\)Hence,

4ε​\(LR​W​f\)​\(xi\)=Δ𝒩​f¯​\(\[xi\]\)−⟨∇𝒩log⁡δ¯​\(\[xi\]\),∇𝒩f¯​\(\[xi\]\)⟩\+O​\(ε\)\+OP​\(1n1/2​ε1/2\+\(d−p\)/4\),\\displaystyle\\frac\{4\}\{\\varepsilon\}\(L\_\{RW\}f\)\(x\_\{i\}\)=\\Delta\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\-\\langle\\nabla\_\{\\mathcal\{N\}\}\\log\\overline\{\\delta\}\(\[x\_\{i\}\]\),\\nabla\_\{\\mathcal\{N\}\}\\overline\{f\}\(\[x\_\{i\}\]\)\\rangle\+O\(\\varepsilon\)\+O\_\{P\}\\left\(\\frac\{1\}\{n^\{1/2\}\\varepsilon^\{1/2\+\(d\-p\)/4\}\}\\right\),\(59\)as we wished to prove\.

### 4\.3Invariant Features Kernel

Letϕ\\phibe theGG\-invariant map as defined in Eq\. \([10](https://arxiv.org/html/2607.08987#S3.E10)\)\. By assumption,im​ϕ\\mathrm\{im\}\\phiis a submanifold ofℝE\\mathbb\{R\}^\{E\}diffeomorphic to𝒩\\mathcal\{N\}and therefore it has dimensionq=d−pq=d\-p\. Given data pointsx1,…,xnx\_\{1\},\\dots,x\_\{n\}sampled from the uniform measured​VℳdV\_\{\\mathcal\{M\}\}onℳ\\mathcal\{M\}, thenϕ​\(x1\),…,ϕ​\(xn\)\\phi\(x\_\{1\}\),\\dots,\\phi\(x\_\{n\}\)are sampled from the pushforward measureϕ∗​\(d​Vℳ\)\\phi\_\{\\ast\}\(dV\_\{\\mathcal\{M\}\}\)onim​ϕ\\mathrm\{im\}\\phi\. Letpϕp\_\{\\phi\}denote the PDF ofϕ∗​\(d​Vℳ\)\\phi\_\{\\ast\}\(dV\_\{\\mathcal\{M\}\}\)\. LetL~R​W\\tilde\{L\}\_\{RW\}denote the normalized graph Laplacian constructed from the standard Gaussian kernel onim​\(ϕ\)\\mathrm\{im\}\(\\phi\)and letϕ¯\\bar\{\\phi\}be the projection ofϕ\\phionto𝒩\\mathcal\{N\}\. Consider the smooth functionf¯∘ϕ¯−1:im​ϕ→ℝ\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\\colon\\mathrm\{im\}\\phi\\to\\mathbb\{R\}and note thatf​\(x\)=f¯∘ϕ¯−1​\(ϕ​\(x\)\)f\(x\)=\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\(\\phi\(x\)\)for allx∈ℳx\\in\\mathcal\{M\}, so it follows thatLR​W​f​\(xi\)=L~R​W​f¯∘ϕ¯−1​\(ϕ​\(xi\)\)L\_\{RW\}f\(x\_\{i\}\)=\\tilde\{L\}\_\{RW\}\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\(\\phi\(x\_\{i\}\)\)for all data pointsxix\_\{i\}\. Denotingxi:=ϕ​\(xi\)x\_\{i\}:=\\phi\(x\_\{i\}\), Theorem[2\.2](https://arxiv.org/html/2607.08987#S2.Thmtheorem2)implies

4ε​LR​W​f​\(xi\)=4ε​L~R​W​f¯∘ϕ¯−1​\(xi\)=Δim​ϕ​f¯∘ϕ¯−1​\(xi\)−2​⟨∇im​ϕlog⁡pϕ​\(xi\),∇im​ϕf¯∘ϕ¯−1​\(xi\)⟩\+OP​\(ε\+1n1/2​ε1/2\+\(d−p\)/4\)\.\\displaystyle\\begin\{split\}&\\frac\{4\}\{\\varepsilon\}L\_\{RW\}f\(x\_\{i\}\)=\\frac\{4\}\{\\varepsilon\}\\widetilde\{L\}\_\{RW\}\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\(x\_\{i\}\)\\\\ &=\\Delta\_\{\\mathrm\{im\}\\phi\}\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\(x\_\{i\}\)\-2\\,\\bigl\\langle\\nabla\_\{\\mathrm\{im\}\\phi\}\\log p\_\{\\phi\}\\left\(x\_\{i\}\\right\),\\nabla\_\{\\mathrm\{im\}\\phi\}\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\(x\_\{i\}\)\\bigr\\rangle\+O\_\{P\}\\\!\\left\(\\varepsilon\+\\frac\{1\}\{n^\{1/2\}\\,\\varepsilon^\{1/2\+\(d\-p\)/4\}\}\\right\)\.\\end\{split\}\(60\)To conclude the proof, let us write

Dim​ϕ​\(f¯∘ϕ¯−1\)​\(xi\)=Δim​ϕ​\(f¯∘ϕ¯−1\)​\(xi\)−2​⟨∇im​ϕlog⁡pϕ​\(xi\),∇im​ϕ\(f¯∘ϕ¯−1\)⁡\(xi\)⟩,\\displaystyle D\_\{\\mathrm\{im\}\\phi\}\(\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\)\\left\(x\_\{i\}\\right\)=\\Delta\_\{\\mathrm\{im\}\\phi\}\(\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\)\(x\_\{i\}\)\-2\\langle\\nabla\_\{\\mathrm\{im\}\\phi\}\\log p\_\{\\phi\}\(x\_\{i\}\),\\nabla\_\{\\mathrm\{im\}\\phi\}\(\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\)\(x\_\{i\}\)\\rangle,\(61\)and note that

Dim​ϕ​\(f¯∘ϕ¯−1\)​\(ϕ​\(xi\)\)=Dim​ϕ​\(f¯∘ϕ¯−1\)​\(ϕ¯​\(\[xi\]\)\)=\(ϕ¯∗​Dim​ϕ\)​f¯​\(\[xi\]\),\\displaystyle D\_\{\\mathrm\{im\}\\phi\}\\left\(\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\\right\)\\left\(\\phi\(x\_\{i\}\)\\right\)=D\_\{\\mathrm\{im\}\\phi\}\(\\overline\{f\}\\circ\\bar\{\\phi\}^\{\-1\}\)\\left\(\\bar\{\\phi\}\(\[x\_\{i\}\]\)\\right\)=\(\\bar\{\\phi\}^\{\\ast\}D\_\{\\mathrm\{im\}\\phi\}\)\\overline\{f\}\(\[x\_\{i\}\]\),\(62\)whereϕ¯∗​Dim​ϕ\\bar\{\\phi\}^\{\\ast\}D\_\{\\mathrm\{im\}\\phi\}denotes the pullback of the operatorDim​ϕD\_\{\\mathrm\{im\}\\phi\}to𝒩\\mathcal\{N\}\.

## 5Experiments

In this section, we demonstrate our framework through several numerical experiments, performing spectral embedding with each invariant kernel and comparing to the standard Euclidean kernelexp⁡\(−‖x−y‖2/ε\)\\exp\(\-\\\|x\-y\\\|^\{2\}/\\varepsilon\)\. The code for reproducing these results is available at:[https://github\.com/yearivig/G\_invariant\_spectral\_embedding](https://github.com/yearivig/G_invariant_spectral_embedding)

### 5\.13D Point Clouds

We obtained synthetic point clouds from the molecular structure of the Glucagon polypeptide, entry 1GCN\(Sasakiet al\.,[1975](https://arxiv.org/html/2607.08987#bib.bib71)\)in the Protein Data Bank \(PDB\)\(Roseet al\.,[2017](https://arxiv.org/html/2607.08987#bib.bib3)\), each consisting of208020803D points\. We generatedn=800n=800point clouds by rotating theψ\\psi\-torsion angle of the 19th amino acid residue within the Glucagon over a full circle, see Figure[3](https://arxiv.org/html/2607.08987#S5.F3)\. The motion of the residue comprises the intrinsic geometry in this experiment\.

![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/exp1/frame_0001a.png)\(a\)A single conformation\.
![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/exp1/overlay3.png)\(b\)Overlay of all frames\.

Figure 3:The Glucagon molecule as a point cloud \(the displayed sticks are for visualization purposes\): a single conformation \(left\) and the overlay of all frames showing the full rotation movement of theψ\\psi\-torsion angle \(right\), shown at the same viewing angle\. In our experiments each conformation is treated as a point cloud of the20802080atomic coordinates\.We then centered each point cloud and applied a random global rotation sampled uniformly fromSO​\(3\)\\mathrm\{SO\}\(3\)\. The global rotations represent nuisance parameters and extrinsic motion\. The generated point clouds constituted the clean dataset\. The noisy dataset was created by adding i\.i\.d\. Gaussian noise with mean zero and variance given by 1/10 the Frobenius norm of each point cloud\. The point clouds lie \(approximately\) on a manifoldℳ\\mathcal\{M\}of dimensiond=4d=4, and their intrinsic geometry is a circle, i\.e\.,𝒩≅𝕊1\\mathcal\{N\}\\cong\\mathbb\{S\}^\{1\}\.

We performed two\-dimensional spectral embeddings of the data using Algorithm[1](https://arxiv.org/html/2607.08987#alg1)with each of the following kernels: Euclidean, minimum kernel, integral kernel, and invariant features kernel\. For the three invariant kernels, the symmetry group isG=SO​\(3\)G=\\mathrm\{SO\}\(3\), acting on a point cloudX∈ℝ2080×3X\\in\\mathbb\{R\}^\{2080\\times 3\}viaX​R⊤XR^\{\\top\}for a3×33\\times 3rotation matrixRR\. For invariant features, we use the map given by the Gram matrix333SinceSO​\(3\)⊂O​\(3\)\\mathrm\{SO\}\(3\)\\subset\\mathrm\{O\}\(3\), the Gram matrixX​X⊤XX^\{\\top\}is in particularSO​\(3\)\\mathrm\{SO\}\(3\)\-invariant\.\(Example[3\.3](https://arxiv.org/html/2607.08987#S3.Thmtheorem3)in Section[3\.1](https://arxiv.org/html/2607.08987#S3.SS1)\)\. The bandwidth parameterε\\varepsilonwas chosen from a fixed default grid:ε=47\\varepsilon=47for the minimum and integral kernels, andε=3000\\varepsilon=3000for the Euclidean and invariant features kernels\. Figure[4](https://arxiv.org/html/2607.08987#S5.F4)shows the resulting embeddings for both the clean and noisy datasets\. The key finding is that the embeddings based on each of the threeSO​\(3\)\\mathrm\{SO\}\(3\)\-invariant kernels recover the correct intrinsic geometry of the data: a circle\. In contrast, the embedding based on the Euclidean kernel fails to capture this circular geometry even for a large number of samples\. The Euclidean embedding is also visibly noisier, consistent with its slower convergence rate\.

NoisennMinimumIntegralInvariantFeaturesEuclidean200![Refer to caption](https://arxiv.org/html/2607.08987v1/x2.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x3.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x4.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x5.png)Clean400![Refer to caption](https://arxiv.org/html/2607.08987v1/x6.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x7.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x8.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x9.png)800![Refer to caption](https://arxiv.org/html/2607.08987v1/x10.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x11.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x12.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x13.png)200![Refer to caption](https://arxiv.org/html/2607.08987v1/x14.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x15.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x16.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x17.png)Noisy400![Refer to caption](https://arxiv.org/html/2607.08987v1/x18.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x19.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x20.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x21.png)800![Refer to caption](https://arxiv.org/html/2607.08987v1/x22.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x23.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x24.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x25.png)Figure 4:Spectral embedding of clean and noisy 3D point clouds of the Glucagon molecule with rotational torsion based on the Euclidean vs\.SO​\(3\)\\mathrm\{SO\}\(3\)\-invariant kernels forn∈\{200,400,800\}n\\in\\\{200,400,800\\\}data points\.
### 5\.2Tomographic Images

In this test, we generated 2D projection images from the 3D molecular structure of the Glucagon polypeptide \(PDB ID 1GCN\(Sasakiet al\.,[1975](https://arxiv.org/html/2607.08987#bib.bib71)\)\)\. Similarly to the previous experiment, we createdn=1000n=1000volumes by rotating theψ\\psi\-torsion angle of the 19th amino acid residue in the Glucagon molecule, but in this case the motion followed a half\-circle trajectory only\. This constitutes the intrinsic motion in our data\.

Using the ASPIRE package\(Wrightet al\.,[2025](https://arxiv.org/html/2607.08987#bib.bib73)\), each volume was projected to a 2D image using tomographic projection from a fixed point of view\. Finally, we rotated each image by a random element inSO​\(2\)\\mathrm\{SO\}\(2\), which gives nuisance parameters and extrinsic motion in the data\. This constituted the clean dataset\. The noisy dataset was created by adding i\.i\.d\. Gaussian noise with mean zero and variance given by 0\.79 times the Frobenius norm of each image\. These tomographic images lie \(approximately\) on a manifoldℳ\\mathcal\{M\}of dimensiond=2d=2, and their intrinsic geometry is a half\-circle\. The groupG=SO​\(2\)G=\\mathrm\{SO\}\(2\)acts by in\-plane rotation of the 2D images\.

As in the previous experiment, we performed different two\-dimensional spectral embeddings of the data by applying Algorithm[1](https://arxiv.org/html/2607.08987#alg1)with each of the following kernels: Euclidean, minimum, integral, and invariant features\. In this case, the symmetry group isG=SO​\(2\)G=\\mathrm\{SO\}\(2\)and the invariant features map is given by the rotational bispectrum of the 2D images \(see Example[3\.4](https://arxiv.org/html/2607.08987#S3.Thmtheorem4)in Section[3\.1](https://arxiv.org/html/2607.08987#S3.SS1)\)\. The bandwidth parameter was set toε=0\.005\\varepsilon=0\.005for the minimum, integral, and Euclidean kernels, and toε=3×10−8\\varepsilon=3\\times 10^\{\-8\}for the bispectrum invariant features kernel\. Figure[5](https://arxiv.org/html/2607.08987#S5.F5)shows the resulting embeddings for both the clean and noisy datasets\. We observe that the embeddings based on each of the threeSO​\(2\)\\mathrm\{SO\}\(2\)\-invariant kernels recover the correct intrinsic geometry of the data, a half\-circle, while the embedding based on the Euclidean kernel fails to capture this geometry\.

NoiseMinimumIntegralInvariantFeaturesEuclidean \(intrinsicrotation color\)Euclidean \(extrinsicrotation color\)Clean![Refer to caption](https://arxiv.org/html/2607.08987v1/x26.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x27.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x28.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x29.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x30.png)Noisy![Refer to caption](https://arxiv.org/html/2607.08987v1/x31.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x32.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x33.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x34.png)![Refer to caption](https://arxiv.org/html/2607.08987v1/x35.png)Figure 5:Spectral embedding of clean and noisy tomographic images based on the Euclidean vs\.SO​\(2\)\\mathrm\{SO\}\(2\)\-invariant kernels forn=1000n=1000data points\. The last two columns show the Euclidean kernel embedding colored according to the intrinsic and extrinsic rotation angles, respectively\.![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/two_motions/example_combined_image_no_title_white_bg.png)\(a\)Original image
![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/two_motions/example_combined_rotated_image_no_title_white_bg.png)\(b\)After a random in\-plane rotation
![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/two_motions/3d_scatter_126_color_left.png)\(c\)Embedding colored byθ1\\theta\_\{1\}
![Refer to caption](https://arxiv.org/html/2607.08987v1/figures/two_motions/3d_scatter_126_color_right.png)\(d\)Embedding colored byθ2\\theta\_\{2\}

Figure 6:Spinning toys experiment\. \(a\) A 2D image of two independently rotating 3D toy figurines\. \(b\) The same image after a random in\-planeSO​\(2\)\\mathrm\{SO\}\(2\)rotation\. \(c\)–\(d\) Three\-dimensional spectral embeddings based on theSO​\(2\)\\mathrm\{SO\}\(2\)\-minimum kernel, showing the same set of embedded points from the same viewpoint, colored by the first intrinsic rotation angleθ1\\theta\_\{1\}in \(c\) and by the second intrinsic rotation angleθ2\\theta\_\{2\}in \(d\)\. Both intrinsic degrees of freedom are smoothly parameterized, and the torus\-like geometry of𝕊1×𝕊1\\mathbb\{S\}^\{1\}\\times\\mathbb\{S\}^\{1\}is recovered\.
### 5\.3Spinning Toys

We use a dataset ofLederman and Talmon\([2018](https://arxiv.org/html/2607.08987#bib.bib90)\)which consists of 2D images obtained by taking snapshots of two spinning 3D toy figurines, where each toy was spinning independently with a non\-constant rotational velocity\. We then rotated each 2D image by a uniformly\-drawn angle\. Figure[6](https://arxiv.org/html/2607.08987#S5.F6)\(a\)–\(b\) shows two example images from this dataset\. Given the two independent rotations of the 3D objects, the intrinsic geometry of the data is topologically equivalent to a torus𝕊1×𝕊1\\mathbb\{S\}^\{1\}\\times\\mathbb\{S\}^\{1\}\. The groupG=SO​\(2\)G=\\mathrm\{SO\}\(2\)acts on each image by in\-plane rotations\.

We embedded the data in three\-dimensional space using Algorithm[1](https://arxiv.org/html/2607.08987#alg1)with the minimum kernel\. The bandwidth parameter was set toε=0\.005\\varepsilon=0\.005\. Figure[6](https://arxiv.org/html/2607.08987#S5.F6)presents example images and the resulting embedding\. The embedded data, colored according to each of the two intrinsic rotation angles, shows that both degrees of freedom are adequately captured and that the embedding recovers the correct torus\-like intrinsic geometry\. This experiment shows the success of the method in a case where the intrinsic geometry is two\-dimensional rather than one\-dimensional\.

## 6Conclusion

We have proposed a framework for the spectral embedding of datasets with known symmetries, by incorporating group invariance directly into the affinity kernels\. Our theoretical analysis focused on the setting of a Riemannian data manifoldℳ\\mathcal\{M\}, with symmetries given by a compact Lie groupGG\. We proved that graph Laplacians constructed from three broadly applicable classes ofGG\-invariant kernels converge pointwise to explicit second\-order differential operators on the quotient manifold𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/G\. In all three cases, the differential operators include a first\-order term related to the action ofGGonℳ\\mathcal\{M\}\. The utility of the framework was demonstrated through experiments on datasets withSO​\(2\)\\mathrm\{SO\}\(2\)andSO​\(3\)\\mathrm\{SO\}\(3\)symmetries\. Our experiments showed thatGG\-invariant affinity kernels recover the correct intrinsic geometry of the data, while the standard Euclidean kernel fails to do so even at large sample sizes\. The resulting symmetry\-aware embeddings faithfully parameterize the degrees of freedom of the quotient manifold\. A key feature of our results is the improved convergence rate: the variance error drops fromn−1/2​ε−1/2−d/4n^\{\-1/2\}\\varepsilon^\{\-1/2\-d/4\}for the standard graph Laplacian ton−1/2​ε−1/2−\(d−p\)/4n^\{\-1/2\}\\varepsilon^\{\-1/2\-\(d\-p\)/4\}for the graph Laplacian constructed from a group invariant kernel, effectively reducing the dimension of the problem byp=dim\(G\)p=\\dim\(G\)\. This is similar to the known improved convergence rates when using symmetry augmentations in supervised learning\(Chenet al\.,[2020](https://arxiv.org/html/2607.08987#bib.bib50); Tahmasebi and Jegelka,[2023](https://arxiv.org/html/2607.08987#bib.bib82)\)\.

Several directions remain open for future work\. First, while we establish pointwise convergence of the graph Laplacian, convergence of the eigenvalues and eigenvectors to those of the limiting operator remains to be proven\. Second, our analysis assumes a free action of a compact Lie group, so that𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/Gis itself a smooth manifold\. Extending the framework to non\-free actions, would broaden its applicability to cases such asSO​\(3\)\\mathrm\{SO\}\(3\)acting on symmetric molecules\. For non\-free actions the orbit space is an orbifold with singular strata of varying isotropy\. Third, it would be useful to consider situations where the data manifold has only approximate symmetriesTahmasebi and Weber\([2026](https://arxiv.org/html/2607.08987#bib.bib111)\)\.

### Acknowledgments

We thank Nicolas Boumal, Yu\-Fang Hsieh, Ofir Karin, Yael Karshon, Roy Lederman, Ofir Lindenbaum, Haggai Maron, Keren Mor Waknin, Amitai Netser Zenik, Eitan Rosen, Yoel Shkolnisky, Amit Singer, and Nir Sochen for helpful discussions\. J\.K\. is partially supported by the United States National Science Foundation \(Grant Nos\. 2309782, 2436499 and 2312746\), the Department of Energy \(Grant No\. SC0025312\), and the Sloan Foundation\. A\.M\. is partially supported by the United States\-Israel Binational Science Foundation \(Grant No\. 2022778\) and by the Israel Science Foundation \(Grant No\. 1662/22\)\. D\.T\. is partially supported by the Knut and Alice Wallenberg Foundation through a joint WASP–DDLS grant\. J\.A\. is partially supported by the Swedish Research Council \(Grant No\. 2023\-04143\)\.

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## Appendix ARiemannian Quotient Manifold

This appendix recalls some basic notation and results related to group actions and quotient manifolds\. For more on these topics, refer to the textbooks byLee\([2003](https://arxiv.org/html/2607.08987#bib.bib14),[2018](https://arxiv.org/html/2607.08987#bib.bib65)\)\.

LetGGbe a compact Lie group with identity element denoted byee\. A*left action*ofGGon the Riemannian manifoldℳ\\mathcal\{M\}is a mapθ:G×ℳ→ℳ\\theta:G\\times\\mathcal\{M\}\\to\\mathcal\{M\}, denoted asα⋅x=θ​\(α,x\)\\alpha\\cdot x=\\theta\(\\alpha,x\), such thatα⋅\(β⋅x\)=\(α​β\)⋅x\\alpha\\cdot\(\\beta\\cdot x\)=\(\\alpha\\beta\)\\cdot xande⋅x=xe\\cdot x=xfor allα,β∈G\\alpha,\\beta\\in Gandx∈ℳx\\in\\mathcal\{M\}\. The group action ofGGonℳ\\mathcal\{M\}is said to be*free*if

∃x∈ℳ:θ​\(α,x\)=x⟹α=e\.\\displaystyle\\exists x\\in\\mathcal\{M\}:\\theta\(\\alpha,x\)=x\\qquad\\Longrightarrow\\qquad\\alpha=e\.\(63\)i\.e\., non\-identity actions have no fixed points\. The action is called*smooth*ifθ​\(α,x\)\\theta\(\\alpha,x\)is a smooth map with respect to bothα\\alphaandxx\. In that case, it induces a mapping of the tangent bundled​θα,x:Tx​ℳ→Tθ​\(α,x\)​ℳd\\theta\_\{\\alpha,x\}:T\_\{x\}\\mathcal\{M\}\\to T\_\{\\theta\(\\alpha,x\)\}\\mathcal\{M\}known as the*differential*ofθα\\theta\_\{\\alpha\}atxx\. We sayGG*acts by isometries*if for everyα∈G\\alpha\\in Gthe diffeomorphism given byθα\\theta\_\{\\alpha\}preserves the Riemannian metric,

⟨u,v⟩x=⟨d​θα,x​\(u\),d​θα,x​\(v\)⟩θ​\(α,x\)​for all​x∈ℳ​and all​u,v∈Tx\.\\displaystyle\\langle u,v\\rangle\_\{x\}=\\langle d\\theta\_\{\\alpha,x\}\(u\),d\\theta\_\{\\alpha,x\}\(v\)\\rangle\_\{\\theta\(\\alpha,x\)\}\\\!\\qquad\\text\{for all \}x\\in\\mathcal\{M\}\\text\{ and all \}u,v\\in T\_\{x\}\.\(64\)
The*orbit*of a pointx∈ℳx\\in\\mathcal\{M\}under the action ofGGis the set

\[x\]=G⋅x=\{α⋅x∣α∈G\}⊆ℳ\.\[x\]=G\\cdot x=\\\{\\alpha\\cdot x\\mid\\alpha\\in G\\\}\\subseteq\\mathcal\{M\}\.\(65\)The orbits are the equivalence classes of theGG\-action, i\.e\.x∼yx\\sim yif and only if there existsα∈G\\alpha\\in Gsuch thaty=α⋅xy=\\alpha\\cdot x\. As a set, the quotientℳ/G\\mathcal\{M\}/Gis the set of orbits ofGG, also known as the*orbit space*of the action\. The mapπ:ℳ→ℳ/G\\pi:\\mathcal\{M\}\\to\\mathcal\{M\}/Gdefined byπ​\(x\)=\[x\]\\pi\(x\)=\[x\]is called the*quotient map*\. We say thatπ\\piis a Riemannian submersion if it is a smooth map, its differentiald​πd\\piis surjective at each point, and

gℳ​\(u,v\)=g𝒩​\(d​πx​\(u\),d​πx​\(v\)\)g\_\{\\mathcal\{M\}\}\(u,v\)=g\_\{\\mathcal\{N\}\}\(d\\pi\_\{x\}\(u\),d\\pi\_\{x\}\(v\)\)\(66\)for allu,vu,vin the orthogonal complement toker⁡\(d​πx\)\\ker\(d\\pi\_\{x\}\)\. The following theorem gives sufficient conditions forπ\\pito be a Riemannian submersion\.

###### Theorem A\.1\(Riemannian Quotient Manifold\)\.

If a compact Lie groupGGacts smoothly, freely and isometrically onℳ\\mathcal\{M\}, then the orbit space𝒩:=ℳ/G\\mathcal\{N\}\\mathrel\{\\mathchoice\{\\vbox\{\\hbox\{$\\displaystyle:$\}\}\}\{\\vbox\{\\hbox\{$\\textstyle:$\}\}\}\{\\vbox\{\\hbox\{$\\scriptstyle:$\}\}\}\{\\vbox\{\\hbox\{$\\scriptscriptstyle:$\}\}\}\{=\}\}\\mathcal\{M\}/Gis a smooth manifold of dimensiondim\(ℳ\)−dim\(G\)\\dim\(\\mathcal\{M\}\)\-\\dim\(G\)with a unique Riemannian metricg𝒩g\_\{\\mathcal\{N\}\}such that the quotient mapπ:ℳ→𝒩\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}is a Riemannian submersion\.

A proof can be found inLee\([2018](https://arxiv.org/html/2607.08987#bib.bib65), Cor 2\.29\), noting that the action of a compact group is always proper\. We call\(𝒩,g𝒩\)\(\\mathcal\{N\},g\_\{\\mathcal\{N\}\}\)the*Riemannian quotient manifold*coming from the action ofGGonℳ\\mathcal\{M\}\.

## Appendix BProof of Lemma[4\.1](https://arxiv.org/html/2607.08987#S4.Thmtheorem1)

By Theorem[A\.1](https://arxiv.org/html/2607.08987#A1.Thmtheorem1), the quotient mapπ:ℳ→𝒩\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}is a Riemannian submersion, and its fibers are given byπ−1​\(\[x\]\)=G⋅x\\pi^\{\-1\}\(\[x\]\)=G\\cdot x\. Applying\(Sakai,[1996](https://arxiv.org/html/2607.08987#bib.bib70), Thm\. 5\.6\)then gives

∫ℳf​\(x\)​𝑑Vℳ​\(x\)=∫𝒩\(∫G⋅xf​\(α⋅x\)​𝑑VG⋅x​\(α⋅x\)\)​𝑑V𝒩​\(\[x\]\),\\displaystyle\\int\_\{\\mathcal\{M\}\}f\(x\)\\,dV\_\{\\mathcal\{M\}\}\(x\)=\\int\_\{\\mathcal\{N\}\}\\left\(\\int\_\{G\\cdot x\}f\(\\alpha\\cdot x\)dV\_\{G\\cdot x\}\(\\alpha\\cdot x\)\\right\)dV\_\{\\mathcal\{N\}\}\(\[x\]\),\(67\)whered​VG⋅xdV\_\{G\\cdot x\}is the Riemannian measure onG⋅xG\\cdot xinduced by the ambient metric onℳ\\mathcal\{M\}, which is related to the Haar measured​ηd\\etaonGGbyd​VG⋅x​\(α⋅x\)=δ​\(x\)​d​η​\(α\)dV\_\{G\\cdot x\}\(\\alpha\\cdot x\)=\\delta\(x\)d\\eta\(\\alpha\)\(Definition[3\.6](https://arxiv.org/html/2607.08987#S3.Thmtheorem6)\)\. Thus Eq\. \([67](https://arxiv.org/html/2607.08987#A2.E67)\) becomes

∫ℳf​\(x\)​𝑑Vℳ​\(x\)=∫𝒩\(∫Gf​\(α⋅x\)​δ​\(x\)​𝑑η​\(α\)\)​𝑑V𝒩​\(\[x\]\)\.\\displaystyle\\int\_\{\\mathcal\{M\}\}f\(x\)\\,dV\_\{\\mathcal\{M\}\}\(x\)=\\int\_\{\\mathcal\{N\}\}\\left\(\\int\_\{G\}f\(\\alpha\\cdot x\)\\delta\(x\)d\\eta\(\\alpha\)\\right\)dV\_\{\\mathcal\{N\}\}\(\[x\]\)\.\(68\)Sinceδ​\(x\)=δ¯​\(\[x\]\)\\delta\(x\)=\\bar\{\\delta\}\(\[x\]\)depends only on the orbitG⋅xG\\cdot x, this gives Eq\. \([35](https://arxiv.org/html/2607.08987#S4.E35)\)\. To conclude the proof, let us further assume thatf:ℳ→ℝf\\colon\\mathcal\{M\}\\to\\mathbb\{R\}is aGG\-invariant function\. Letf¯\\bar\{f\}denote the induced function on𝒩\\mathcal\{N\}\. By substitutingf​\(α⋅x\)=f¯​\(\[x\]\)f\(\\alpha\\cdot x\)=\\overline\{f\}\(\[x\]\)into Eq\. \([35](https://arxiv.org/html/2607.08987#S4.E35)\), and since the Haar measured​ηd\\etais chosen so thatη​\(G\)=1\\eta\(G\)=1, we obtain Eq\. \([36](https://arxiv.org/html/2607.08987#S4.E36)\) as desired\.

## Appendix CProof of Lemma[4\.2](https://arxiv.org/html/2607.08987#S4.Thmtheorem2)

We derive an asymptotic expansion of the integral

I​\(\[x\]\)=1\(π​ε\)q/2​∫𝒩K¯min​\(\[x\],\[y\]\)​h​\(\[y\]\)​𝑑V𝒩​\(\[y\]\),I\(\[x\]\)=\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathcal\{N\}\}\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\],\[y\]\)h\(\[y\]\)dV\_\{\\mathcal\{N\}\}\(\[y\]\),\(69\)with

K¯min​\(\[x\],\[y\]\)=exp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\),\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\],\[y\]\)=\\exp\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\),\(70\)The exponential map then lets us rewrite the approximation ofI​\(\[x\]\)I\(\[x\]\)as an integral overqq\-dimensional Euclidean space\. Finally, standard properties of the Gaussian distribution in Euclidean space are used to conclude the proof\.

### Step 1: Comparison of Riemannian and Euclidean Distances onℳ\\mathcal\{M\}

In this step, we prove three inequalities that relate the Riemannian distancedℳ​\(x,y\)d\_\{\\mathcal\{M\}\}\(x,y\)and the Euclidean distance‖x−y‖\\\|x\-y\\\|for any two pointsx,y∈ℳx,y\\in\\mathcal\{M\}\. These inequalities imply that the Riemannian and Euclidean distances are uniformly equivalent metrics onℳ\\mathcal\{M\}\.

###### Lemma C\.1\.

Let\(ℳ,gℳ\)\(\\mathcal\{M\},g\_\{\\mathcal\{M\}\}\)be a compact and connected Riemannian submanifold ofℝD\\mathbb\{R\}^\{D\}\. Then there exist positive constantsCℳC\_\{\\mathcal\{M\}\}andKℳK\_\{\\mathcal\{M\}\}such that

0≤dℳ2​\(x,y\)−‖x−y‖2≤Cℳ​dℳ4​\(x,y\)≤Kℳ​‖x−y‖4\\displaystyle 0\\leq d^\{2\}\_\{\\mathcal\{M\}\}\(x,y\)\-\\\|x\-y\\\|^\{2\}\\leq C\_\{\\mathcal\{M\}\}d^\{4\}\_\{\\mathcal\{M\}\}\(x,y\)\\leq K\_\{\\mathcal\{M\}\}\\\|x\-y\\\|^\{4\}\(71\)for allx,y∈ℳx,y\\in\\mathcal\{M\}\.

###### Proof\.

The first inequality follows directly from the definition of the Riemannian and Euclidean distances onℳ\\mathcal\{M\}\. To prove the second inequality, note that, sinceℳ\\mathcal\{M\}is compact and connected, the global injectivity radiusrℳr\_\{\\mathcal\{M\}\}is a positive constant such that for allx,y∈ℳx,y\\in\\mathcal\{M\}withdℳ​\(x,y\)<rℳd\_\{\\mathcal\{M\}\}\(x,y\)<r\_\{\\mathcal\{M\}\}there is a unique minimizing geodesicγx​yℳ\\gamma\_\{xy\}^\{\\mathcal\{M\}\}inℳ\\mathcal\{M\}starting atxxand ending atyy\. Assume thatγx​yℳ\\gamma\_\{xy\}^\{\\mathcal\{M\}\}is parametrized by arc\-length\. As proved in\(Smolyanovet al\.,[2007](https://arxiv.org/html/2607.08987#bib.bib62), Proposition 6\), the following limit holds uniformly with respect tox,y∈ℳx,y\\in\\mathcal\{M\}:

limdℳ​\(x,y\)→0dℳ2​\(x,y\)−‖x−y‖2dℳ4​\(x,y\)=112​‖II​\(γ˙x​yℳ​\(0\),γ˙x​yℳ​\(0\)\)‖2,\\displaystyle\\lim\_\{d\_\{\\mathcal\{M\}\}\(x,y\)\\to 0\}\\frac\{d\_\{\\mathcal\{M\}\}^\{2\}\(x,y\)\-\\\|x\-y\\\|^\{2\}\}\{d\_\{\\mathcal\{M\}\}^\{4\}\(x,y\)\}=\\frac\{1\}\{12\}\\left\\\|\\mathrm\{II\}\(\\dot\{\\gamma\}^\{\\mathcal\{M\}\}\_\{xy\}\(0\),\\dot\{\\gamma\}^\{\\mathcal\{M\}\}\_\{xy\}\(0\)\)\\right\\\|^\{2\},\(72\)whereII\\mathrm\{II\}denotes the second fundamental form ofℳ\\mathcal\{M\}\. Let

C=112​supv∈U​ℳ‖II​\(v,v\)‖2\+1,\\displaystyle C=\\frac\{1\}\{12\}\\sup\_\{v\\in U\\mathcal\{M\}\}\\left\\\|\\mathrm\{II\}\(v,v\)\\right\\\|^\{2\}\+1,\(73\)whereU​ℳU\\mathcal\{M\}denotes the unit tangent bundle ofℳ\\mathcal\{M\}; the supremum is finite sinceII\\mathrm\{II\}is continuous andU​ℳU\\mathcal\{M\}is compact\. By the uniform convergence of the limit above, there exists a constantη∈\(0,rℳ\)\\eta\\in\(0,r\_\{\\mathcal\{M\}\}\)such that if0<dℳ​\(x,y\)<η0<d\_\{\\mathcal\{M\}\}\(x,y\)<\\etathen

dℳ2​\(x,y\)−‖x−y‖2dℳ4​\(x,y\)≤112​‖II​\(γ˙x​yℳ​\(0\),γ˙x​yℳ​\(0\)\)‖2\+1≤C\.\\displaystyle\\frac\{d\_\{\\mathcal\{M\}\}^\{2\}\(x,y\)\-\\\|x\-y\\\|^\{2\}\}\{d\_\{\\mathcal\{M\}\}^\{4\}\(x,y\)\}\\leq\\frac\{1\}\{12\}\\left\\\|\\mathrm\{II\}\(\\dot\{\\gamma\}^\{\\mathcal\{M\}\}\_\{xy\}\(0\),\\dot\{\\gamma\}^\{\\mathcal\{M\}\}\_\{xy\}\(0\)\)\\right\\\|^\{2\}\+1\\leq C\.\(74\)On the other hand, ifη≤dℳ​\(x,y\)\\eta\\leq d\_\{\\mathcal\{M\}\}\(x,y\), then

dℳ​\(x,y\)2−‖x−y‖2≤dℳ​\(x,y\)2≤1η2​dℳ​\(x,y\)4\.\\displaystyle d\_\{\\mathcal\{M\}\}\(x,y\)^\{2\}\-\\\|x\-y\\\|^\{2\}\\leq d\_\{\\mathcal\{M\}\}\(x,y\)^\{2\}\\leq\\frac\{1\}\{\\eta^\{2\}\}d\_\{\\mathcal\{M\}\}\(x,y\)^\{4\}\.\(75\)In any case, for all pairsx,y∈ℳx,y\\in\\mathcal\{M\}we have

dℳ​\(x,y\)2−‖x−y‖2≤Cℳ​dℳ​\(x,y\)4,\\displaystyle d\_\{\\mathcal\{M\}\}\(x,y\)^\{2\}\-\\\|x\-y\\\|^\{2\}\\leq C\_\{\\mathcal\{M\}\}d\_\{\\mathcal\{M\}\}\(x,y\)^\{4\},\(76\)whereCℳ=max⁡\{1/η2,C\}C\_\{\\mathcal\{M\}\}=\\max\\\{1/\\eta^\{2\},C\\\}\. Finally, we use inequality \([76](https://arxiv.org/html/2607.08987#A3.E76)\) to prove that there exists a nonzero constantKKsuch that

dℳ​\(x,y\)≤K​‖x−y‖\\displaystyle d\_\{\\mathcal\{M\}\}\(x,y\)\\leq K\\\|x\-y\\\|\(77\)for allx,y∈ℳx,y\\in\\mathcal\{M\}\. The last inequality then follows by takingKℳ=Cℳ​K4K\_\{\\mathcal\{M\}\}=C\_\{\\mathcal\{M\}\}K^\{4\}\.

Assume, for the sake of contradiction, that for allN∈ℕN\\in\\mathbb\{N\}there existxN,xN∈ℳx\_\{N\},x\_\{N\}\\in\\mathcal\{M\}such that

dℳ2​\(xN,xN\)\>N​‖xN−xN‖2\.\\displaystyle d\_\{\\mathcal\{M\}\}^\{2\}\(x\_\{N\},x\_\{N\}\)\>N\\\|x\_\{N\}\-x\_\{N\}\\\|^\{2\}\.\(78\)Due to the compactness ofℳ\\mathcal\{M\}, and by passing to a convergent subsequence if necessary, we may assume that\(xN,xN\)→\(x∗,y∗\)∈ℳ×ℳ\(x\_\{N\},x\_\{N\}\)\\to\(x^\{\\ast\},y^\{\\ast\}\)\\in\\mathcal\{M\}\\times\\mathcal\{M\}asN→∞N\\to\\infty\. Eq\. \([78](https://arxiv.org/html/2607.08987#A3.E78)\) implies‖xN−xN‖<dℳ​\(xN,xN\)/N<D/N→0\\\|x\_\{N\}\-x\_\{N\}\\\|<d\_\{\\mathcal\{M\}\}\(x\_\{N\},x\_\{N\}\)/\\sqrt\{N\}<D/\\sqrt\{N\}\\to 0, whereD=diam​\(ℳ\)<∞D=\\mathrm\{diam\}\(\\mathcal\{M\}\)<\\inftyby compactness ofℳ\\mathcal\{M\}, so it follows thatx∗=y∗x^\{\\ast\}=y^\{\\ast\}\. Combining Eqs\. \([76](https://arxiv.org/html/2607.08987#A3.E76)\) and \([78](https://arxiv.org/html/2607.08987#A3.E78)\) gives

dℳ2​\(xN,xN\)\>N​‖xN−xN‖2≥N​dℳ2​\(xN,xN\)−N​Cℳ​dℳ4​\(xN,xN\)\.\\displaystyle d\_\{\\mathcal\{M\}\}^\{2\}\(x\_\{N\},x\_\{N\}\)\>N\\\|x\_\{N\}\-x\_\{N\}\\\|^\{2\}\\geq Nd\_\{\\mathcal\{M\}\}^\{2\}\(x\_\{N\},x\_\{N\}\)\-NC\_\{\\mathcal\{M\}\}d\_\{\\mathcal\{M\}\}^\{4\}\(x\_\{N\},x\_\{N\}\)\.\(79\)For allN\>1N\>1, this implies

1dℳ​\(xN,xN\)2<N​CℳN−1\.\\displaystyle\\frac\{1\}\{d\_\{\\mathcal\{M\}\}\(x\_\{N\},x\_\{N\}\)^\{2\}\}<\\frac\{NC\_\{\\mathcal\{M\}\}\}\{N\-1\}\.\(80\)This gives a contradiction asN→∞N\\to\\infty, sincex∗=y∗x^\{\\ast\}=y^\{\\ast\}implies1/dℳ2​\(xN,xN\)→∞1/d\_\{\\mathcal\{M\}\}^\{2\}\(x\_\{N\},x\_\{N\}\)\\to\\inftybutN​Cℳ/\(N−1\)→CℳNC\_\{\\mathcal\{M\}\}/\(N\-1\)\\to C\_\{\\mathcal\{M\}\}\. ∎

### Step 2: Approximation of the Minimum Kernel

We now use Lemma[C\.1](https://arxiv.org/html/2607.08987#A3.Thmtheorem1)to show thatminα∈G⁡‖x−α⋅y‖2\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}approximates the Riemannian distanced𝒩​\(\[x\],\[y\]\)d\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)up to order three\. We then use this approximation to obtain an asymptotic expansion of the minimum kernel asε→0\\varepsilon\\to 0\. Let us start by noting that Lemma[C\.1](https://arxiv.org/html/2607.08987#A3.Thmtheorem1)gives

‖x−α⋅y‖2≤dℳ2​\(x,α⋅y\)\\displaystyle\\\|x\-\\alpha\\cdot y\\\|^\{2\}\\leq d\_\{\\mathcal\{M\}\}^\{2\}\(x,\\alpha\\cdot y\)\(81\)and

dℳ2​\(x,α⋅y\)≤‖x−α⋅y‖2\+Kℳ​‖x−α⋅y‖4\\displaystyle d\_\{\\mathcal\{M\}\}^\{2\}\(x,\\alpha\\cdot y\)\\leq\\\|x\-\\alpha\\cdot y\\\|^\{2\}\+K\_\{\\mathcal\{M\}\}\\\|x\-\\alpha\\cdot y\\\|^\{4\}\(82\)for allx,y∈ℳx,y\\in\\mathcal\{M\}and allα∈G\\alpha\\in G\. On the other hand, the Riemannian distances onℳ\\mathcal\{M\}and𝒩\\mathcal\{N\}are related by the formula

d𝒩​\(\[x\],\[y\]\)=minα∈G⁡dℳ​\(x,α⋅y\)d\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)=\\min\_\{\\alpha\\in G\}d\_\{\\mathcal\{M\}\}\(x,\\alpha\\cdot y\)\(83\)for all\[x\],\[y\]∈𝒩\[x\],\[y\]\\in\\mathcal\{N\}\(see\(Boumal,[2023](https://arxiv.org/html/2607.08987#bib.bib22), Exercise 10\.15\)\)\. Although the minimizing group elements for the Euclidean and Riemannian distances need not coincide, the inequalities in Lemma[C\.1](https://arxiv.org/html/2607.08987#A3.Thmtheorem1)hold uniformly for everyα∈G\\alpha\\in G\. Consequently, taking the minimum overα∈G\\alpha\\in Gin inequalities \([81](https://arxiv.org/html/2607.08987#A3.E81)\) and \([82](https://arxiv.org/html/2607.08987#A3.E82)\), using relation \([83](https://arxiv.org/html/2607.08987#A3.E83)\), and combining the results yields

−d𝒩2​\(\[x\],\[y\]\)≤−minα∈G⁡‖x−α⋅y‖2≤−d𝒩2​\(\[x\],\[y\]\)\+Kℳ​d𝒩4​\(\[x\],\[y\]\)\\displaystyle\-d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)\\leq\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}\\leq\-d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)\+K\_\{\\mathcal\{M\}\}d^\{4\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)\(84\)for all\[x\],\[y\]∈𝒩\[x\],\[y\]\\in\\mathcal\{N\}\. It follows that, whend𝒩​\(\[x\],\[y\]\)<εγd\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)<\\varepsilon^\{\\gamma\}with1/4<γ1/4<\\gamma, the minimum kernel satisfies

K¯min​\(\[x\],\[y\]\)=exp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\)\\displaystyle\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\],\[y\]\)=\\exp\\\!\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\)=exp⁡\(−d𝒩2​\(\[x\],\[y\]\)/ε\)​\(1\+O​\(d𝒩4​\(\[x\],\[y\]\)/ε\)\)\\displaystyle=\\exp\\\!\\left\(\-d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)/\\varepsilon\\right\)\\left\(1\+O\\\!\\left\(d\_\{\\mathcal\{N\}\}^\{4\}\(\[x\],\[y\]\)/\\varepsilon\\right\)\\right\)\(85\)for all sufficiently smallε\\varepsilon, where the implied constant isKℳ′=exp⁡\(Kℳ\)−1K^\{\\prime\}\_\{\\mathcal\{M\}\}=\\exp\(K\_\{\\mathcal\{M\}\}\)\-1\.

### Step 3: Reduction of the Domain of Integration

As proved in Lemma[C\.1](https://arxiv.org/html/2607.08987#A3.Thmtheorem1), there exists a positive constantK\>0K\>0such thatdℳ​\(x,α⋅y\)≤K​‖x−α⋅y‖d\_\{\\mathcal\{M\}\}\(x,\\alpha\\cdot y\)\\leq K\\\|x\-\\alpha\\cdot y\\\|for allx,y∈ℳ,α∈Gx,y\\in\\mathcal\{M\},\\alpha\\in G\. Since this inequality holds uniformly for everyα∈G\\alpha\\in G, taking the minimum overα∈G\\alpha\\in Ggives

d𝒩​\(\[x\],\[y\]\)=minα∈G⁡dℳ​\(x,α⋅y\)≤K​minα∈G⁡‖x−α⋅y‖\.d\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)=\\min\_\{\\alpha\\in G\}d\_\{\\mathcal\{M\}\}\(x,\\alpha\\cdot y\)\\leq K\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|\.\(86\)LetBBdenote the geodesic ball in𝒩\\mathcal\{N\}around\[x\]\[x\]of radiusεγ\\varepsilon^\{\\gamma\}with1/4<γ<1/21/4<\\gamma<1/2\. Then Eq\. \([86](https://arxiv.org/html/2607.08987#A3.E86)\) implies that for any\[y\]∈𝒩∖B\[y\]\\in\\mathcal\{N\}\\setminus Bwe have

minα∈G⁡‖x−α⋅y‖2≥1K2​d𝒩2​\(\[x\],\[y\]\)≥1K2​ε2​γ\.\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}\\geq\\frac\{1\}\{K^\{2\}\}d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)\\geq\\frac\{1\}\{K^\{2\}\}\\varepsilon^\{2\\gamma\}\.\(87\)It follows that restricting the domain of integration ofI​\(\[x\]\)I\(\[x\]\)in Eq\. \([69](https://arxiv.org/html/2607.08987#A3.E69)\) toBBgenerates an error of order

\|1\(π​ε\)q/2​∫𝒩∖Bexp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\)​h​\(\[y\]\)​𝑑V𝒩​\(\[y\]\)\|≤Vol​\(𝒩∖B\)\(π​ε\)q/2​‖h‖∞​exp⁡\(−ε2​γ−1/K2\),\\displaystyle\\left\\lvert\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathcal\{N\}\\setminus B\}\\exp\\\!\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\)h\(\[y\]\)\\,dV\_\{\\mathcal\{N\}\}\(\[y\]\)\\right\\rvert\\leq\\frac\{\\mathrm\{Vol\}\(\\mathcal\{N\}\\setminus B\)\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\,\\\|h\\\|\_\{\\infty\}\\,\\exp\\left\(\-\\varepsilon^\{2\\gamma\-1\}/K^\{2\}\\right\),\(88\)where‖h‖∞<∞\\\|h\\\|\_\{\\infty\}<\\inftysincehhis a continuous function on a compact space\. Given that2​γ−1<02\\gamma\-1<0, this error is exponentially small asε\\varepsilonapproaches0\. Therefore, for anyη\>0\\eta\>0and for all sufficiently smallε\\varepsilon, we have

I​\(\[x\]\)=1\(π​ε\)q/2​∫Bexp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\)​h​\(\[y\]\)​𝑑V𝒩​\(\[y\]\)\+o​\(εη\)\.\\displaystyle\\begin\{split\}I\(\[x\]\)&=\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{B\}\\exp\\\!\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\)h\(\[y\]\)dV\_\{\\mathcal\{N\}\}\(\[y\]\)\+o\(\\varepsilon^\{\\eta\}\)\.\\end\{split\}\(89\)Finally, since1/4<γ1/4<\\gamma, the approximation of the minimum kernel in Eq\. \([85](https://arxiv.org/html/2607.08987#A3.E85)\) applies to all\[y\]∈B\[y\]\\in B\. Substituting Eq\. \([85](https://arxiv.org/html/2607.08987#A3.E85)\) into Eq\. \([89](https://arxiv.org/html/2607.08987#A3.E89)\) thus gives, for anyη\>0\\eta\>0and all sufficiently smallε\\varepsilon,

I​\(\[x\]\)=1\(π​ε\)q/2​∫Bexp⁡\(−d𝒩2​\(\[x\],\[y\]\)/ε\)​\(1\+O​\(d𝒩4​\(\[x\],\[y\]\)/ε\)\)​h​\(\[y\]\)​𝑑V𝒩​\(\[y\]\)\+o​\(εη\)\.\\displaystyle I\(\[x\]\)=\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{B\}\\exp\\\!\\left\(\-d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)/\\varepsilon\\right\)\\left\(1\+O\\\!\\left\(d\_\{\\mathcal\{N\}\}^\{4\}\(\[x\],\[y\]\)/\\varepsilon\\right\)\\right\)h\(\[y\]\)dV\_\{\\mathcal\{N\}\}\(\[y\]\)\+o\(\\varepsilon^\{\\eta\}\)\.\(90\)

### Step 4: Change to Geodesic Normal Coordinates

In this step, we use the exponential map and the corresponding geodesic normal coordinates to expressI​\(\[x\]\)I\(\[x\]\)as an integral overqq\-dimensional Euclidean space\. The exponential map at\[x\]∈𝒩\[x\]\\in\\mathcal\{N\}, denotedexp\[x\]\\exp\_\{\[x\]\}, provides a canonical smooth map from the tangent space at\[x\]\[x\]into the manifold𝒩\\mathcal\{N\}\. It maps straight lines through the origin inT\[x\]​𝒩T\_\{\[x\]\}\\mathcal\{N\}to geodesics in𝒩\\mathcal\{N\}through the point\[x\]\[x\]\. Moreover,exp\[x\]\\exp\_\{\[x\]\}is a local diffeomorphism, meaning that there exist open neighborhoodsUUof the origin inT\[x\]​𝒩T\_\{\[x\]\}\\mathcal\{N\}andVVof\[x\]\[x\]in𝒩\\mathcal\{N\}such thatexp\[x\]:U→V\\exp\_\{\[x\]\}\\colon U\\to Vis a diffeomorphism\. By fixing an orthonormal basis onT\[x\]​𝒩T\_\{\[x\]\}\\mathcal\{N\}and lettingu=\(u1,…,uq\)Tu=\(u\_\{1\},\\dots,u\_\{q\}\)^\{T\}denote coordinates onUUwith respect to this basis, we obtain the*geodesic normal coordinates*for𝒩\\mathcal\{N\}centered at\[x\]\[x\]\. In these coordinates, for each\[y\]∈V\[y\]\\in V, there existsu∈ℝqu\\in\\mathbb\{R\}^\{q\}such that

d𝒩2​\(\[x\],\[y\]\)=‖u‖2\.d^\{2\}\_\{\\mathcal\{N\}\}\(\[x\],\[y\]\)=\\\|u\\\|^\{2\}\.\(91\)In addition, given any functionf:𝒩→ℝf\\colon\\mathcal\{N\}\\to\\mathbb\{R\}, we letf~​\(u\)\\tilde\{f\}\(u\)denote the local coordinate expression forf​\(\[y\]\)f\(\[y\]\)\.

Note that there exists some fixedε0\>0\\varepsilon\_\{0\}\>0such that for allε<ε0\\varepsilon<\\varepsilon\_\{0\}the open ballB~\\tilde\{B\}of radiusεγ\\varepsilon^\{\\gamma\}with1/4<γ<1/21/4<\\gamma<1/2around the origin inT\[x\]​𝒩T\_\{\[x\]\}\\mathcal\{N\}is contained inUU\. In this case,exp\[x\]\\exp\_\{\[x\]\}mapsB~\\tilde\{B\}diffeomorphically toBB, the geodesic ball around\[x\]\[x\]of radiusεγ\\varepsilon^\{\\gamma\}\. From Eqs\. \([90](https://arxiv.org/html/2607.08987#A3.E90)\) and \([91](https://arxiv.org/html/2607.08987#A3.E91)\), we see thatI​\(\[x\]\)I\(\[x\]\)is given in geodesic normal coordinates by

I​\(\[x\]\)=1\(π​ε\)q/2​∫B~exp⁡\(−‖u‖2/ε\)​\(1\+O​\(‖u‖4/ε\)\)​h~​\(u\)​det\(gi​j\)​𝑑u\+o​\(εη\),\\displaystyle I\(\[x\]\)=\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}\\exp\\left\(\-\\\|u\\\|^\{2\}/\\varepsilon\\right\)\\left\(1\+O\\left\(\\\|u\\\|^\{4\}/\\varepsilon\\right\)\\right\)\\tilde\{h\}\(u\)\\sqrt\{\\det\(g\_\{ij\}\)\}du\+o\(\\varepsilon^\{\\eta\}\),\(92\)wheredet\(gi​j\)​d​u\\sqrt\{\\det\(g\_\{ij\}\)\}duis the volume form for𝒩\\mathcal\{N\}\. Following\(Smolyanovet al\.,[2007](https://arxiv.org/html/2607.08987#bib.bib62), Corollary 3\), we have the Taylor expansion

det\(gi​j\)=1−16​uT​R~​\(0\)​u\+O​\(‖u‖3\),\\displaystyle\\sqrt\{\\det\(g\_\{ij\}\)\}=1\-\\frac\{1\}\{6\}u^\{T\}\\tilde\{R\}\(0\)u\+O\(\\\|u\\\|^\{3\}\),\(93\)whereRRdenotes the Ricci curvature tensor of𝒩\\mathcal\{N\}\. Since𝒩\\mathcal\{N\}is a compact manifold, the elements ofRRare bounded, so that we can further approximate

det\(gi​j\)=1\+O​\(‖u‖2\)\.\\displaystyle\\sqrt\{\\det\(g\_\{ij\}\)\}=1\+O\(\\\|u\\\|^\{2\}\)\.\(94\)We will use both expressions fordet\(gi​j\)\\sqrt\{\\det\(g\_\{ij\}\)\}to find the desired approximation forI​\(\[x\]\)I\(\[x\]\)\.

### Step 5: Analysis in Euclidean Space

Next, consider the Taylor expansion ofh~​\(u\)\\tilde\{h\}\(u\)around0, which is given by

h~​\(u\)=h~​\(0\)\+∑i=1qui​∂h~​\(0\)∂ui\+12​∑i=1q∑j=1qui​uj​∂2h~​\(0\)∂ui​∂uj\+O​\(‖u‖3\)\.\\tilde\{h\}\(u\)=\\tilde\{h\}\(0\)\+\\sum\_\{i=1\}^\{q\}u\_\{i\}\\frac\{\\partial\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}\}\+\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{q\}\\sum\_\{j=1\}^\{q\}u\_\{i\}u\_\{j\}\\frac\{\\partial^\{2\}\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}\\partial u\_\{j\}\}\+O\(\\\|u\\\|^\{3\}\)\.\(95\)Putting Eqs\. \([92](https://arxiv.org/html/2607.08987#A3.E92)\)\-\([95](https://arxiv.org/html/2607.08987#A3.E95)\) together, integralI​\(\[x\]\)I\(\[x\]\)becomes

I​\(\[x\]\)\\displaystyle I\(\[x\]\)=h~​\(0\)​1\(π​ε\)q/2​∫B~exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1−16​uT​R~​\(0\)​u\+O​\(‖u‖3\)\)​𝑑u\\displaystyle=\\tilde\{h\}\(0\)\\,\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\-\\frac\{1\}\{6\}u^\{T\}\\tilde\{R\}\(0\)u\+O\(\\\|u\\\|^\{3\}\)\\right\)du\(96\)\+∑i=1q∂h~​\(0\)∂ui​1\(π​ε\)q/2​∫B~ui​exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1\+O​\(‖u‖2\)\)​𝑑u\\displaystyle\\qquad\+\\sum\_\{i=1\}^\{q\}\\frac\{\\partial\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}\}\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}u\_\{i\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\+O\(\\\|u\\\|^\{2\}\)\\right\)du\+12​∑i=1q∑j=1q∂2h~​\(0\)∂ui​∂uj​1\(π​ε\)q/2​∫B~ui​uj​exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1\+O​\(‖u‖2\)\)​𝑑u\\displaystyle\\qquad\+\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{q\}\\sum\_\{j=1\}^\{q\}\\frac\{\\partial^\{2\}\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}\\partial u\_\{j\}\}\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}u\_\{i\}u\_\{j\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\+O\(\\\|u\\\|^\{2\}\)\\right\)du\+1\(π​ε\)q/2​∫B~O​\(‖u‖3\)​exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1\+O​\(‖u‖2\)\)​𝑑u\+o​\(εη\)\.\\displaystyle\\qquad\+\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}O\(\{\\\|u\\\|^\{3\}\}\)\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\+O\(\\\|u\\\|^\{2\}\)\\right\)du\+o\(\\varepsilon^\{\\eta\}\)\.By the properties of a zero\-mean Gaussian distribution with diagonal covariance matrix, all terms of the form∫B~ui​exp⁡\(−‖u‖2/ε\)​𝑑u\\int\_\{\\tilde\{B\}\}u\_\{i\}\\exp\(\-\\\|u\\\|^\{2\}/\\varepsilon\)du,∫B~ui​uj​uk​exp⁡\(−‖u‖2/ε\)​𝑑u\\int\_\{\\tilde\{B\}\}u\_\{i\}u\_\{j\}u\_\{k\}\\exp\(\-\\\|u\\\|^\{2\}/\\varepsilon\)duand∫B~ui​uj​exp⁡\(−‖u‖2/ε\)​𝑑u\\int\_\{\\tilde\{B\}\}u\_\{i\}u\_\{j\}\\exp\(\-\\\|u\\\|^\{2\}/\\varepsilon\)duwithi≠ji\\neq jvanish\. Therefore,I​\(\[x\]\)I\(\[x\]\)simplifies to

I​\(\[x\]\)\\displaystyle I\(\[x\]\)=h~​\(0\)​1\(π​ε\)q/2​∫B~exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1−16​∑i=1qR~i​i​\(0\)​ui2\+O​\(‖u‖3\)\)​𝑑u\\displaystyle=\\tilde\{h\}\(0\)\\,\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\-\\frac\{1\}\{6\}\\sum\_\{i=1\}^\{q\}\\tilde\{R\}\_\{ii\}\(0\)u\_\{i\}^\{2\}\+O\(\\\|u\\\|^\{3\}\)\\right\)du\(97\)\+12​∑i=1q∂2h~​\(0\)∂ui2​1\(π​ε\)q/2​∫B~ui2​exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1\+O​\(‖u‖2\)\)​𝑑u\\displaystyle\\qquad\+\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{q\}\\frac\{\\partial^\{2\}\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}^\{2\}\}\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}u\_\{i\}^\{2\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\+O\(\\\|u\\\|^\{2\}\)\\right\)du\+1\(π​ε\)q/2​∫B~O​\(‖u‖3\)​exp⁡\(−‖u‖2ε\)​\(1\+O​\(‖u‖4ε\)\)​\(1\+O​\(‖u‖2\)\)​𝑑u\+o​\(εη\)\.\\displaystyle\\qquad\+\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\tilde\{B\}\}O\(\{\\\|u\\\|^\{3\}\}\)\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\left\(1\+O\\left\(\\frac\{\\\|u\\\|^\{4\}\}\{\\varepsilon\}\\right\)\\right\)\\left\(1\+O\(\\\|u\\\|^\{2\}\)\\right\)du\+o\(\\varepsilon^\{\\eta\}\)\.Due to the exponential decay ofexp⁡\(−‖u‖2/ε\)\\exp\(\-\\\|u\\\|^\{2\}/\\varepsilon\), the domains of integration can be extended toℝq\\mathbb\{R\}^\{q\}due to bounds similar to \([88](https://arxiv.org/html/2607.08987#A3.E88)\)\. This allows us to use the following properties of theqq\-dimensional Gaussian distribution:

1\(π​ε\)q/2​∫ℝqexp⁡\(−‖u‖2ε\)​𝑑u=1,1\(π​ε\)q/2​∫ℝqexp⁡\(−‖u‖2ε\)​‖u‖m​𝑑u=εm/2​Γ​\(q\+m2\)Γ​\(q2\),\\displaystyle\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathbb\{R\}^\{q\}\}\\exp\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)du=1,\\qquad\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathbb\{R\}^\{q\}\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\\|u\\\|^\{m\}\\,du=\\varepsilon^\{m/2\}\\,\\frac\{\\Gamma\\\!\\left\(\\frac\{q\+m\}\{2\}\\right\)\}\{\\Gamma\\\!\\left\(\\frac\{q\}\{2\}\\right\)\},\(98\)and, for eachuiu\_\{i\},

1\(π​ε\)q/2​∫ℝqexp⁡\(−‖u‖2ε\)​ui2​𝑑u=ε2,1\(π​ε\)q/2​∫ℝqui2​exp⁡\(−‖u‖2ε\)​‖u‖m​𝑑u=εm/2\+1q​Γ​\(q\+m\+22\)Γ​\(q2\)\.\\displaystyle\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathbb\{R\}^\{q\}\}\\exp\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)u\_\{i\}^\{2\}du=\\frac\{\\varepsilon\}\{2\},\\qquad\\frac\{1\}\{\(\\pi\\varepsilon\)^\{q/2\}\}\\int\_\{\\mathbb\{R\}^\{q\}\}u\_\{i\}^\{2\}\\exp\\\!\\left\(\-\\frac\{\\\|u\\\|^\{2\}\}\{\\varepsilon\}\\right\)\\,\\\|u\\\|^\{m\}\\,du=\\frac\{\\varepsilon^\{m/2\+1\}\}\{q\}\\,\\frac\{\\Gamma\\\!\\left\(\\frac\{q\+m\+2\}\{2\}\\right\)\}\{\\Gamma\\\!\\left\(\\frac\{q\}\{2\}\\right\)\}\.\(99\)From Eqs\. \([98](https://arxiv.org/html/2607.08987#A3.E98)\) and \([99](https://arxiv.org/html/2607.08987#A3.E99)\), we approximate Eq\. \([97](https://arxiv.org/html/2607.08987#A3.E97)\) \(choosingη=3/2\\eta=3/2\) up to orderε3/2\\varepsilon^\{3/2\}as follows:

I​\(\[x\]\)=h~​\(0\)\+ε4​\(\(−13​∑i=1qR~i​i​\(0\)\+O​\(1\)\)​h~​\(0\)\+∑i=1q∂2h~​\(0\)∂ui2\)\+O​\(ε3/2\)\.\\displaystyle I\(\[x\]\)=\\tilde\{h\}\(0\)\+\\frac\{\\varepsilon\}\{4\}\\left\(\\left\(\-\\frac\{1\}\{3\}\\sum\_\{i=1\}^\{q\}\\tilde\{R\}\_\{ii\}\(0\)\+O\(1\)\\right\)\\tilde\{h\}\(0\)\+\\sum\_\{i=1\}^\{q\}\\frac\{\\partial^\{2\}\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}^\{2\}\}\\right\)\+O\(\\varepsilon^\{3/2\}\)\.\(100\)
We briefly explain why the error term ofO​\(ϵ3/2\)O\(\\epsilon^\{3/2\}\)in Eq\. \([100](https://arxiv.org/html/2607.08987#A3.E100)\) can be improved, followingSinger\([2006](https://arxiv.org/html/2607.08987#bib.bib30)\)\. First note that the minimum kernel is smooth near the diagonal\. Indeed, sinceGGacts freely the orbitG⋅xG\\cdot xis a compact embedded submanifold ofℝD\\mathbb\{R\}^\{D\}\. By the tubular neighborhood theoremLee\([2003](https://arxiv.org/html/2607.08987#bib.bib14)\),G⋅xG\\cdot xhas an open neighborhood inℝD\\mathbb\{R\}^\{D\}on which every point has a unique nearest point inG⋅xG\\cdot xwhere the nearest\-point projection toG⋅xG\\cdot xis smoothFoote\([1984](https://arxiv.org/html/2607.08987#bib.bib114)\); hencey↦minα∈G⁡‖x−α⋅y‖2y\\mapsto\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}is smooth there\. From

K¯min​\(\[x\],\[y\]\)=exp⁡\(−minα∈G⁡‖x−α⋅y‖2/ε\),\\overline\{K\}\_\{\\mathrm\{min\}\}\(\[x\],\[y\]\)=\\exp\\left\(\-\\min\_\{\\alpha\\in G\}\\\|x\-\\alpha\\cdot y\\\|^\{2\}/\\varepsilon\\right\),and compactness,K¯min\\overline\{K\}\_\{\\min\}is smooth in an open neighborhood of the diagonal of𝒩×𝒩\\mathcal\{N\}\\times\\mathcal\{N\}\. This implies that the integrand in Eq\. \([92](https://arxiv.org/html/2607.08987#A3.E92)\) is smooth in geodesic normal coordinates, and proceeding as in\(Singer,[2006](https://arxiv.org/html/2607.08987#bib.bib30), Section 2\)only integer powers ofε\\varepsilonsurvive\. Therefore, the error term in Eq\. \([100](https://arxiv.org/html/2607.08987#A3.E100)\) can be improved toO​\(ε2\)O\(\\varepsilon^\{2\}\)\.

Finally, as shown byRosenberg\([1997](https://arxiv.org/html/2607.08987#bib.bib55)\), we have the relation

∑i=1q∂2h~​\(0\)∂ui2=−Δ𝒩​h​\(\[x\]\),\\displaystyle\\sum\_\{i=1\}^\{q\}\\frac\{\\partial^\{2\}\\tilde\{h\}\(0\)\}\{\\partial u\_\{i\}^\{2\}\}=\-\\Delta\_\{\\mathcal\{N\}\}h\(\[x\]\),\(101\)whereΔ𝒩\\Delta\_\{\\mathcal\{N\}\}is the Laplace–Beltrami operator on𝒩\\mathcal\{N\}\. We conclude that

I​\(\[x\]\)=h​\(\[x\]\)\+ε4​\(E​\(\[x\]\)​h​\(\[x\]\)−Δ𝒩​h​\(\[x\]\)\)\+O​\(ε2\),\\displaystyle I\(\[x\]\)=h\(\[x\]\)\+\\frac\{\\varepsilon\}\{4\}\\left\(E\(\[x\]\)h\(\[x\]\)\-\\Delta\_\{\\mathcal\{N\}\}h\(\[x\]\)\\right\)\+O\(\\varepsilon^\{2\}\),\(102\)whereE​\(\[x\]\)=−13​∑i=1qRi​i​\(\[x\]\)\+O​\(1\)E\(\[x\]\)=\-\\frac\{1\}\{3\}\\sum\_\{i=1\}^\{q\}R\_\{ii\}\(\[x\]\)\+O\(1\)is some function that depends on the curvature of𝒩\\mathcal\{N\}at\[x\]\[x\]and on the embedding ofℳ\\mathcal\{M\}intoℝD\\mathbb\{R\}^\{D\}\.

## Appendix DTable of Notation

NotationMeaningSpectral embedding𝒳=\{x1,…,xn\}\\mathcal\{X\}=\\\{x\_\{1\},\\dots,x\_\{n\}\\\}Sampled dataset, withxi∈ℳ⊆ℝDx\_\{i\}\\in\\mathcal\{M\}\\subseteq\\mathbb\{R\}^\{D\}\.K,KG,Kmin,Kint,KIFK,K\_\{G\},K\_\{\\mathrm\{min\}\},K\_\{\\mathrm\{int\}\},K\_\{\\mathrm\{IF\}\}Base affinity kernel \(e\.g\.exp⁡\(−‖x−y‖2/ε\)\\exp\(\-\\\|x\-y\\\|^\{2\}/\\varepsilon\)\), genericGG\-invariant kernel, minimum kernel, integral kernel, and invariant\-features kernel \(Section[3\.1](https://arxiv.org/html/2607.08987#S3.SS1)\)\.ε\\varepsilonKernel bandwidth parameter\.W,DW,DWeight and degree matrices used to define the graph Laplacian\.LR​WL\_\{RW\}Random\-walk normalized graph Laplacian,LR​W=I−D−1​WL\_\{RW\}=I\-D^\{\-1\}W\.λi,φi\\lambda\_\{i\},\\varphi\_\{i\}Eigenvalues and eigenvectors of the graph Laplacian \(used as embedding coordinates\)\.xix\_\{i\}Embedded coordinate vector ofxix\_\{i\}in spectral embedding\.Spaces, geometry, and group actionsℳ⊆ℝD\\mathcal\{M\}\\subseteq\\mathbb\{R\}^\{D\}Data manifold\. A compact Riemannian submanifold without boundary\.d=dim\(ℳ\)d=\\dim\(\\mathcal\{M\}\)Intrinsic dimension of the data manifold\.GGCompact Lie group acting onℳ\\mathcal\{M\}\.p=dim\(G\)p=\\dim\(G\)Dimension of the Lie group\.α,β∈G\\alpha,\\beta\\in GGeneric group elements\.α⋅x\\alpha\\cdot xAction ofα∈G\\alpha\\in Gon a pointx∈ℳx\\in\\mathcal\{M\}\.θ:G×ℳ→ℳ\\theta\\colon G\\times\\mathcal\{M\}\\to\\mathcal\{M\}Group action map, withθ​\(α,x\)=α⋅x\\theta\(\\alpha,x\)=\\alpha\\cdot x\.\[x\]=G⋅x=\{α⋅x∣α∈G\}\[x\]=G\\cdot x=\\\{\\alpha\\cdot x\\mid\\alpha\\in G\\\}Orbit ofxxunder the action ofGG; equivalently, the quotient point represented byxx\.𝒩=ℳ/G\\mathcal\{N\}=\\mathcal\{M\}/GQuotient manifold, or orbit space, obtained by identifying points in the sameGG\-orbit\.q=dim\(𝒩\)=d−pq=\\dim\(\\mathcal\{N\}\)=d\-pDimension of the quotient manifold\.π:ℳ→𝒩\\pi\\colon\\mathcal\{M\}\\to\\mathcal\{N\}Quotient map,π​\(x\)=\[x\]\\pi\(x\)=\[x\]\.d​Vℳ,d​V𝒩,d​VG⋅xdV\_\{\\mathcal\{M\}\},\\ dV\_\{\\mathcal\{N\}\},\\ dV\_\{G\\cdot x\}Riemannian volume measures onℳ\\mathcal\{M\},𝒩\\mathcal\{N\}and the orbitG⋅xG\\cdot x\.d​ηd\\etaHaar measure onGG, normalized so that∫G1​𝑑η=1\\int\_\{G\}1\\,d\\eta=1\.δ,δ¯\\delta,\\overline\{\\delta\}Orbit\-volume density onℳ\\mathcal\{M\}\(defined byd​VG⋅x=δ​d​ηdV\_\{G\\cdot x\}=\\delta\\,d\\eta\) and its induced function on𝒩\\mathcal\{N\}\.Functions and operatorsf,f¯f,\\overline\{f\}SmoothGG\-invariant function onℳ\\mathcal\{M\}and its induced function on𝒩\\mathcal\{N\}\.h:𝒩→ℝh\\colon\\mathcal\{N\}\\to\\mathbb\{R\}Generic smooth test function on𝒩\\mathcal\{N\}\.ρ,ρ~\\rho,\\tilde\{\\rho\}Sampling density onℳ\\mathcal\{M\}for non\-uniform data, and its orbit average on𝒩\\mathcal\{N\}\.ϕ:ℝD→ℝE\\phi\\colon\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{E\},ϕ¯:𝒩→im​ϕ\\bar\{\\phi\}\\colon\\mathcal\{N\}\\to\\mathrm\{im\}\\phiGG\-invariant feature map and its induced map on𝒩\\mathcal\{N\}\.pϕp\_\{\\phi\}Density of the pushforward measureϕ∗​\(d​Vℳ\)\\phi\_\{\\ast\}\(dV\_\{\\mathcal\{M\}\}\)onim​ϕ\\mathrm\{im\}\\phi\.Δℳ,Δ𝒩,Δim​ϕ\\Delta\_\{\\mathcal\{M\}\},\\Delta\_\{\\mathcal\{N\}\},\\Delta\_\{\\mathrm\{im\}\\phi\}Laplace–Beltrami operators onℳ\\mathcal\{M\},𝒩\\mathcal\{N\}, andim​ϕ\\mathrm\{im\}\\phi\.∇ℳ,∇𝒩,∇im​ϕ\\nabla\_\{\\mathcal\{M\}\},\\nabla\_\{\\mathcal\{N\}\},\\nabla\_\{\\mathrm\{im\}\\phi\}Riemannian gradients onℳ\\mathcal\{M\},𝒩\\mathcal\{N\}, andim​ϕ\\mathrm\{im\}\\phi\.P​\(Δℳ\)P\(\\Delta\_\{\\mathcal\{M\}\}\)Projection of the Laplace–Beltrami operator onℳ\\mathcal\{M\}to an operator on𝒩\\mathcal\{N\}\.DDLimiting second\-order differential operator on𝒩\\mathcal\{N\}\.DρD\_\{\\rho\}Non\-uniform\-sampling version of the limiting operatorDD\.π∗,ϕ¯∗,ϕ∗\\pi^\{\\ast\},\\bar\{\\phi\}^\{\\ast\},\\phi\_\{\\ast\}Pullbacks byπ\\piandϕ¯\\bar\{\\phi\}, and pushforward byϕ\\phi\.