Gromov-Monge Flow Matching for Equivariant Graph Generation

# Gromov–Monge Flow Matching for Equivariant Graph Generation
Source: [https://arxiv.org/html/2608.26961](https://arxiv.org/html/2608.26961)
Moritz PieningAffiliation:Institute of Mathematics, Technische Universität BerlinAffiliation:Berlin, GermanyEmail:[piening@math\.tu\-berlin\.de](mailto:)Christian WaldAffiliation:Institut Camille Jordan, INSA LyonAffiliation:Villeurbanne, FranceEmail:[christian\.wald@insa\-lyon\.fr](mailto:)

###### Abstract

Graphs are invariant under node permutations, motivating the use of permutation\-equivariant architectures in generative models\. In flow matching, however, symmetry may also enter the source–target coupling: once graph pairs are compared up to node relabeling, the natural Wasserstein geometry is that of the graph quotient space\. The Euclidean quotient metric of this space coincides with the Gromov–Monge distance, obtained by optimally relabeling the nodes\. We develop this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow\-matching minimizers, including for categorical endpoint prediction\. In practice, exact Gromov–Monge alignment is intractable, so we construct minibatch couplings using efficient Gromov–Wasserstein\-type relaxations and lower bounds for the inner node alignment, optionally combined with an outer assignment between graphs\. The resulting procedure changes only the training coupling and is compatible with standard permutation\-equivariant architectures\. Across continuous graph and categorical molecular generation, these structure\-aware couplings substantially improve sample quality at small integration budgets, while our scaled\-up molecular models remain competitive under conventional many\-step sampling\.

## 1Introduction

Flow matching learns a velocity field that transports a tractable source law to data using simulation\-free regression and ordinary differential equation \(ODE\) sampling\([Albergo and Vanden\-Eijnden, 2023](https://arxiv.org/html/2608.26961#bib.bib17);[Lipman et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib16);[Liu et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib18)\)\. Its conditional paths are determined by a source–target coupling\. Although all such couplings have the same marginals at start and end time, transport\-informed choices can produce shorter, straighter trajectories and simplify numerical integration\([Chemseddine et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib23);[Tong et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib20);[Pooladian et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib19)\)\. The correct notion of transport is less obvious when data are defined only up to symmetry\([Klein et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib26);[Köhler et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib39)\)\.

For graphs the symmetry is node relabelling\. We encode edge channels and diagonal node features of anNN\-node graph byE∈ℝN×N×CE\\in\\mathbb\{R\}^\{N\\times N\\times C\}\. Because a graph has no distinguished vertex order, a permutationσ∈𝔖N\\sigma\\in\\mathfrak\{S\}\_\{N\}acts simultaneously on both node indices according to

\(ρNgraph​\(σ\)​E\)i​j​c≔Eσ−1​\(i\)​σ−1​\(j\)​c,1≤i,j≤N\.\\bigl\(\\rho\_\{N\}^\{\\rm graph\}\(\\sigma\)E\\bigr\)\_\{ijc\}\\coloneqq E\_\{\\sigma^\{\-1\}\(i\)\\sigma^\{\-1\}\(j\)c\},\\qquad 1\\leq i,j\\leq N\.The unordered graph is the orbit

\[E\]∈Q≔ℝN2​C/GNgraph,GNgraph=ρNgraph​\(𝔖N\)⊂O⁡\(N2​C\)\.\[E\]\\in Q\\coloneqq\\mathbb\{R\}^\{N^\{2\}C\}/G\_\{N\}^\{\\rm graph\},\\qquad G\_\{N\}^\{\\rm graph\}=\\rho\_\{N\}^\{\\rm graph\}\(\\mathfrak\{S\}\_\{N\}\)\\subset O\(N^\{2\}C\)\.Equivariant architectures employed in graph generative models respect this symmetry\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31);[Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34);[Vignac et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib32)\), but many models do not by themselves choose which labelled representatives to connect during flow matching\. Each conditional bridge fromEEto an unordered target\[F\]\[F\]may select someσ⋅F\\sigma\\cdot F\. Minimizing its length is a*Gromov–Monge*node\-alignment problem, the hard\-assignment counterpart of Gromov–Wasserstein transport\([Bauer et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib36);[Mémoli, 2011](https://arxiv.org/html/2608.26961#bib.bib3);[Mémoli and Needham, 2024](https://arxiv.org/html/2608.26961#bib.bib2)\)\.

We connect this representative choice to equivariant flow matching on general quotientsℝD/G\\mathbb\{R\}^\{D\}/G\([Klein et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib26);[Köhler et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib39);[Song et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib40)\)\. Our theory shows that optimal quotient couplings lift without extra cost and that symmetrizing a coupling between chosen representatives yields an equivariant minimizer of the flow\-matching objective, including for categorical endpoint prediction\. For graphs, we approximate the resulting Gromov–Monge alignment with Gromov–Wasserstein solvers and inexpensive lower bounds\([Mémoli, 2011](https://arxiv.org/html/2608.26961#bib.bib3);[Bauer et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib36);[Piening and Beinert, 2025](https://arxiv.org/html/2608.26961#bib.bib13)\), optionally followed by an outer minibatch assignment\. The aligned pairs train the same equivariant velocity or categorical endpoint models as standard flow matching; only the coupling changes\.

##### Relation to prior work

Graph generators commonly enforce node\-relabeling symmetry through equivariant architectures, including molecular and discrete diffusion\([Hoogeboom et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib35);[Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34);[Vignac et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib32)\)and variational flow matching\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31)\)\. Recent graph flow methods additionally optimize the source–target graph pairing using minibatch optimal transport\([Hou et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib45);[Wijesinghe et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib51)\)\. Our method complements these approaches by explicitly choosing aligned representatives within each graph orbit, while treating the outer graph assignment as optional, see Appendix[C](https://arxiv.org/html/2608.26961#A3)for a more detailed comparison\.

Optimal\-transport couplings have previously been used to straighten Euclidean flow\-matching paths\([Chemseddine et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib23);[Pooladian et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib19);[Tong et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib20)\)\.When the underlying sample space itself carries an optimal\-transport geometry, this naturally leads to nested transport formulations, as recently employed for point\-cloud generative models\([Haviv et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib12);[Piening et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib37);[Piening and Beinert, 2026](https://arxiv.org/html/2608.26961#bib.bib15)\)\. For graphs, the analogous construction must additionally account for node correspondences, leading to a Gromov–Monge ground cost\. Gromov–Wasserstein methods provide a natural relaxation because they compare relational structure and have long been used for graph and structured\-data alignment\([Beier et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib7);[Chowdhury et al\., 2021](https://arxiv.org/html/2608.26961#bib.bib10);[Peyré et al\., 2016](https://arxiv.org/html/2608.26961#bib.bib9);[Vayer et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib6)\)\. We use the resulting plans to select a node permutation for each target graph inside its conditional flow\-matching bridge, rather than to generate graphs directly\.

Our quotient\-space analysis builds on equivariant generative flows\([Klein et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib26);[Köhler et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib39);[Song et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib40)\)\. Beyond particle permutations by linear assignment\([Klein et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib26);[Haviv et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib12);[Hui et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib25);[Piening et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib37)\), graph relabeling acts on both node indices and yields the quadratic Gromov–Monge problem\. We connect this graph\-specific alignment to quotient optimal transport and representative lifts\([Mémoli and Needham, 2024](https://arxiv.org/html/2608.26961#bib.bib2)\)\.

Our contributions are:

- •We show that optimal quotient couplings lift to Euclidean couplings with the same quadratic cost, whose linear interpolations project to constant\-speed Wasserstein geodesics\.
- •We show that diagonal symmetrization preserves the flow\-matching objective on equivariant fields and yields an equivariant minimizer, including for categorical endpoint prediction\.
- •A practical outer–inner Gromov–Wasserstein alignment constructs minibatch couplings for continuous and categorical graph flows\.
- •Across continuous and categorical graph\-generation benchmarks, structure\-aware couplings improve sample quality when the learned flow is sampled using few Euler integration steps, while scaled\-up molecular models remain competitive under conventional many\-step sampling\.

## 2Background on transport and flow matching

In flow matching generative models, curves of probability measures are constructed that connect an easy\-to\-sample source measure with a target measure which is available only through samples\([Lipman et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib16);[Wald and Steidl, 2025](https://arxiv.org/html/2608.26961#bib.bib11)\)\. This construction is not limited to probability measures on Euclidean spaces\([Chen and Lipman, 2024](https://arxiv.org/html/2608.26961#bib.bib27)\)\. After a short recap on Wasserstein geometry\([Ambrosio et al\., 2005](https://arxiv.org/html/2608.26961#bib.bib24);[Santambrogio, 2015](https://arxiv.org/html/2608.26961#bib.bib5);[Wald and Steidl, 2025](https://arxiv.org/html/2608.26961#bib.bib11)\), we will see how the same construction can be formulated on quotient spaces induced by orthogonal group actions\.

### 2\.1Wasserstein geometry and dynamic transport

Let\(X,dX\)\(X,d\_\{X\}\)be a Polish metric space and𝒫2​\(X\)\\mathcal\{P\}\_\{2\}\(X\)the set of Borel probability measures onXXwith finite second moments, i\.e\.∫XdX​\(x,x0\)2​𝑑μ​\(x\)<∞\\int\_\{X\}d\_\{X\}\(x,x\_\{0\}\)^\{2\}\\,\\,\\mathrm\{d\}\\mu\(x\)<\\inftyfor somex0∈Xx\_\{0\}\\in X\. The set𝒫2​\(X\)\\mathcal\{P\}\_\{2\}\(X\)becomes a complete metric space with the*Wasserstein distance*W2,X\\operatorname\{W\}\_\{2,X\}\([Santambrogio, 2015](https://arxiv.org/html/2608.26961#bib.bib5);[Villani, 2009](https://arxiv.org/html/2608.26961#bib.bib30)\)which is given for anyμ,ν∈𝒫2​\(X\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(X\)by

W2,X2⁡\(μ,ν\)≔min⁡∫X×Xπ∈cX​\(μ,ν\)⁡dX​\(x,x′\)2​𝑑π​\(x,x′\)\.\\operatorname\{W\}\_\{2,X\}^\{2\}\(\\mu,\\nu\)\\coloneqq\\min\_\{\\pi\\in\\mathrm\{c\}\_\{X\}\(\\mu,\\nu\)\}\\int\_\{X\\times X\}d\_\{X\}\(x,x^\{\\prime\}\)^\{2\}\\,\\,\\mathrm\{d\}\\pi\(x,x^\{\\prime\}\)\.\(1\)The*couplings*or*transport plans*are given by

cX\(μ,ν\)=\{π∈𝒫2\(X×X\):proj♯0π=μ,proj♯1π=ν\},\\mathrm\{c\}\_\{X\}\(\\mu,\\nu\)=\\\{\\pi\\in\\mathcal\{P\}\_\{2\}\(X\\times X\):\\mathrm\{proj\}^\{0\}\_\{\\sharp\}\\pi=\\mu,\\mathrm\{proj\}^\{1\}\_\{\\sharp\}\\pi=\\nu\\\},withproji:X×X→X\\mathrm\{proj\}^\{i\}:X\\times X\\to X,\(x0,x1\)↦xi\(x\_\{0\},x\_\{1\}\)\\mapsto x\_\{i\}\. BycXopt​\(μ,ν\)\\mathrm\{c\}\_\{X\}^\{\{\\rm opt\}\}\(\\mu,\\nu\)we denote*optimal*couplings, where the minimum in \([1](https://arxiv.org/html/2608.26961#S2.E1)\) is attained\. Throughout, letI=\[0,1\]I=\[0,1\]\.

ForX=ℝDX=\\mathbb\{R\}^\{D\}, let\(μt\)t∈I\(\\mu\_\{t\}\)\_\{t\\in I\}be narrowly continuous and letv:I×ℝD→ℝDv:I\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{D\}be Borel\. The pair\(μt,vt\)\(\\mu\_\{t\},v\_\{t\}\)satisfies the continuity equation if

∂tμt\+div⁡\(vt​μt\)=0\\partial\_\{t\}\\mu\_\{t\}\+\\operatorname\{div\}\(v\_\{t\}\\mu\_\{t\}\)=0\(2\)in the sense of distributions\. If, in addition,

∫I∫ℝD‖vt​\(x\)‖2​d​μt​\(x\)​𝑑t<∞,\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\\|v\_\{t\}\(x\)\\\|^\{2\}\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(x\)\\,\\,\\mathrm\{d\}t<\\infty,then\(μt\)t∈I\(\\mu\_\{t\}\)\_\{t\\in I\}is absolutely continuous with respect toW2,ℝD\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}\. Conversely, curves that are absolutely continuous with respect toW2,ℝD\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}and have square\-integrable speed admit a Borel fieldvvsatisfying \([2](https://arxiv.org/html/2608.26961#S2.E2)\) and the displayed integrability condition\([Ambrosio et al\., 2005](https://arxiv.org/html/2608.26961#bib.bib24), Chapter 8\)\. The definitions and bounds used below are recalled in Appendix[A](https://arxiv.org/html/2608.26961#A1)\. Ifvvis sufficiently regular, the characteristic ODE

γ˙​\(t,x\)=v⁡\(t,γ⁡\(t,x\)\),γ⁡\(0,x\)=x,\\dot\{\\gamma\}\(t,x\)=v\(t,\\gamma\(t,x\)\),\\qquad\\gamma\(0,x\)=x,\(3\)transportsμ0\\mu\_\{0\}toμt\\mu\_\{t\}, i\.e\.,μt=γt,♯​μ0\\mu\_\{t\}=\\gamma\_\{t,\\sharp\}\\mu\_\{0\}\([Ambrosio et al\., 2005](https://arxiv.org/html/2608.26961#bib.bib24), Proposition 8\.1\.8\)\.

### 2\.2Euclidean flow matching

Flow matching turns the dynamic description above into mean\-squared\-error \(MSE\) vector\-field regression\([Lipman et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib16);[Wald and Steidl, 2025](https://arxiv.org/html/2608.26961#bib.bib11), see, e\.g\.,\)\. Forμ,ν∈𝒫2​\(ℝD\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\)andπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\), defineμt≔proj♯t​π\\mu\_\{t\}\\coloneqq\\mathrm\{proj\}^\{t\}\_\{\\sharp\}\\piwithprojt​\(x,x′\)≔\(1−t\)​x\+t​x′\\mathrm\{proj\}^\{t\}\(x,x^\{\\prime\}\)\\coloneqq\(1\-t\)x\+tx^\{\\prime\}\. Then\(μt,vtπ\)\(\\mu\_\{t\},v\_\{t\}^\{\\pi\}\)solves \([2](https://arxiv.org/html/2608.26961#S2.E2)\), wherevπv^\{\\pi\}minimizes

Jπ​\(v\)≔∫I∫ℝD×ℝD‖vt​\(projt​\(x,x′\)\)−\(x′−x\)‖2​𝑑π​\(x,x′\)​𝑑tJ\_\{\\pi\}\(v\)\\coloneqq\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\bigl\\\|v\_\{t\}\(\\mathrm\{proj\}^\{t\}\(x,x^\{\\prime\}\)\)\-\(x^\{\\prime\}\-x\)\\bigr\\\|^\{2\}\\,\\,\\mathrm\{d\}\\pi\(x,x^\{\\prime\}\)\\,\\,\\mathrm\{d\}t\(4\)over jointly Borel vector fieldsvv\. Equivalently, for\(X,X′\)∼π\(X,X^\{\\prime\}\)\\sim\\pi, we have that

vtπ​\(z\)=𝔼π​\[X′−X∣projt​\(X,X′\)=z\]v\_\{t\}^\{\\pi\}\(z\)=\\mathbb\{E\}\_\{\\pi\}\\bigl\[X^\{\\prime\}\-X\\mid\\mathrm\{proj\}^\{t\}\(X,X^\{\\prime\}\)=z\\bigr\]\(5\)ford​t​d​μt​\(z\)\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\-a\.e\.\(t,z\)\(t,z\)\. Ifπ∈cℝDopt​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}^\{\\rm opt\}\(\\mu,\\nu\), then the kinetic energy fulfills∫I‖vtπ​\(z\)‖L2​\(μt\)2​𝑑t=W2,ℝD2⁡\(μ,ν\)\.\\int\_\{I\}\\\|v\_\{t\}^\{\\pi\}\(z\)\\\|^\{2\}\_\{L^\{2\}\(\\mu\_\{t\}\)\}\\,\\,\\mathrm\{d\}t\\ =\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}^\{2\}\(\\mu,\\nu\)\.

##### Categorical endpoint prediction

Letμ,ν∈𝒫2​\(ℝD\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\)andπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)\. Suppose that the target is coordinatewise categorical\. Write\[m\]≔\{1,…,m\}\[m\]\\coloneqq\\\{1,\\ldots,m\\\}for a positive integermm, and fix a positive integerMM\. For everyd∈\[D\]d\\in\[D\], fix distinct valuesad,1,…,ad,M∈ℝa\_\{d,1\},\\ldots,a\_\{d,M\}\\in\\mathbb\{R\}such thatν⁡\(∏d=1D\{ad,1,…,ad,M\}\)=1\.\\nu\\\!\\left\(\\prod\_\{d=1\}^\{D\}\\\{a\_\{d,1\},\\ldots,a\_\{d,M\}\\\}\\right\)=1\.For\(X,X′\)∼π\(X,X^\{\\prime\}\)\\sim\\pi, setZt≔\(1−t\)​X\+t​X′Z\_\{t\}\\coloneqq\(1\-t\)X\+tX^\{\\prime\}andμt≔Law⁡\(Zt\)\\mu\_\{t\}\\coloneqq\\operatorname\{Law\}\(Z\_\{t\}\)\. Letκd​\(x′\)∈\[M\]\\kappa\_\{d\}\(x^\{\\prime\}\)\\in\[M\]denote the unique index such thatxd′=ad,κd​\(x′\)x^\{\\prime\}\_\{d\}=a\_\{d,\\kappa\_\{d\}\(x^\{\\prime\}\)\}\. Choosing jointly Borel versions, define the conditional coordinate probabilities

qt∗,d​\(n∣z\)≔ℙπ​\(Xd′=ad,n∣Zt=z\),d∈\[D\],n∈\[M\]\.q\_\{t\}^\{\\ast,d\}\(n\\mid z\)\\coloneqq\\mathbb\{P\}\_\{\\pi\}\\\!\\left\(X^\{\\prime\}\_\{d\}=a\_\{d,n\}\\mid Z\_\{t\}=z\\right\),\\qquad d\\in\[D\],\\quad n\\in\[M\]\.For a\.e\.t<1t<1andμt\\mu\_\{t\}\-a\.e\.zz, \([5](https://arxiv.org/html/2608.26961#S2.E5)\) yields

\(vtπ​\(z\)\)d=11−t​\(∑n=1Mad,n​qt∗,d​\(n∣z\)−zd\)\.\\bigl\(v\_\{t\}^\{\\pi\}\(z\)\\bigr\)\_\{d\}=\\frac\{1\}\{1\-t\}\\left\(\\sum\_\{n=1\}^\{M\}a\_\{d,n\}q\_\{t\}^\{\\ast,d\}\(n\\mid z\)\-z\_\{d\}\\right\)\.\(6\)
Moreover,\(q∗,d\)d=1D\(q^\{\\ast,d\}\)\_\{d=1\}^\{D\}minimizes the coordinatewise cross\-entropy objective

Jπcoord\(\(qd\)d=1D\)≔−∫I∫ℝD×ℝD∑d=1Dlogqtd\(κd\(x′\)∣\(1−t\)x\+tx′\)dπ\(x,x′\)dtJ\_\{\\pi\}^\{\\rm coord\}\\bigl\(\(q^\{d\}\)\_\{d=1\}^\{D\}\\bigr\)\\coloneqq\-\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\sum\_\{d=1\}^\{D\}\\log q\_\{t\}^\{d\}\\\!\\left\(\\kappa\_\{d\}\(x^\{\\prime\}\)\\mid\(1\-t\)x\+tx^\{\\prime\}\\right\)\\,\\,\\mathrm\{d\}\\pi\(x,x^\{\\prime\}\)\\,\\,\\mathrm\{d\}t\(7\)over jointly Borel kernelsqd:I×ℝD→ΔMq^\{d\}:I\\times\\mathbb\{R\}^\{D\}\\to\\Delta\_\{M\},d∈\[D\]d\\in\[D\], whereΔM≔\{p∈\[0,1\]M:∑n=1Mpn=1\}\\Delta\_\{M\}\\coloneqq\\\{p\\in\[0,1\]^\{M\}:\\sum\_\{n=1\}^\{M\}p\_\{n\}=1\\\}and−log⁡0≔\+∞\-\\log 0\\coloneqq\+\\infty\. Thus, although the joint endpoint may take up toMDM^\{D\}values, it suffices to trainDDdistributions overMMclasses using \([7](https://arxiv.org/html/2608.26961#S2.E7)\) and recover the velocity through \([6](https://arxiv.org/html/2608.26961#S2.E6)\) without any conditional independence assumption\([Chemseddine et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib38);[Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31)\)\.

## 3Flow matching on quotient spaces

LetG⊂O⁡\(D\)G\\subset O\(D\)be compact, letQ≔ℝD/GQ\\coloneqq\\mathbb\{R\}^\{D\}/G, and write𝔮⁡\(x\)=\[x\]\\mathfrak\{q\}\(x\)=\[x\]for the quotient map\. The quotient metric is

dQ​\(\[x\],\[y\]\)≔ming∈G⁡‖x−g​y‖,d\_\{Q\}\(\[x\],\[y\]\)\\coloneqq\\min\_\{g\\in G\}\\\|x\-gy\\\|,andW2,Q\\operatorname\{W\}\_\{2,Q\}denotes the induced Wasserstein distance\. For𝝁∈𝒫2​\(Q\)\\boldsymbol\{\\mu\}\\in\\mathcal\{P\}\_\{2\}\(Q\), write

\[𝝁\]≔\{μ∈𝒫2​\(ℝD\):𝔮♯​μ=𝝁\}\[\\boldsymbol\{\\mu\}\]\\coloneqq\\\{\\mu\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\):\\mathfrak\{q\}\_\{\\sharp\}\\mu=\\boldsymbol\{\\mu\}\\\}for its representative laws\. In the graph setting, simultaneous node relabeling gives the following construction\. LetZ≔ℝCZ\\coloneqq\\mathbb\{R\}^\{C\}withdZ​\(u,v\)≔‖u−v‖2d\_\{Z\}\(u,v\)\\coloneqq\\\|u\-v\\\|\_\{2\}\. ForE,F∈ZN×NE,F\\in Z^\{N\\times N\},

GM2,Z2⁡\(\[E\],\[F\]\)≔min⁡∑i,j=1Nσ∈𝔖N⁡dZ2​\(Ei​j,Fσ⁡\(i\)​σ​\(j\)\)\.\\operatorname\{GM\}\_\{2,Z\}^\{2\}\(\[E\],\[F\]\)\\coloneqq\\min\_\{\\sigma\\in\\mathfrak\{S\}\_\{N\}\}\\sum\_\{i,j=1\}^\{N\}d\_\{Z\}^\{2\}\\bigl\(E\_\{ij\},F\_\{\\sigma\(i\)\\sigma\(j\)\}\\bigr\)\.\(8\)Thus, choosing representatives of graph orbits is precisely the hard Gromov–Monge node\-alignment problem, studied in optimal transport literature\([Mémoli and Needham, 2024](https://arxiv.org/html/2608.26961#bib.bib2)\)\.

###### Theorem 3\.1\(Representative lifts and quotient geodesics\)\.

Let𝛍,𝛎∈𝒫2​\(Q\)\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\\in\\mathcal\{P\}\_\{2\}\(Q\), fixμ∈\[𝛍\]\\mu\\in\[\\boldsymbol\{\\mu\}\], and letΓ∈cQ​\(𝛍,𝛎\)\\Gamma\\in\\mathrm\{c\}\_\{Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\. There existνΓ∈\[𝛎\]\\nu\_\{\\Gamma\}\\in\[\\boldsymbol\{\\nu\}\]andγΓ∈cℝD​\(μ,νΓ\)\\gamma\_\{\\Gamma\}\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\_\{\\Gamma\}\)such that\(𝔮,𝔮\)♯​γΓ=Γ\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\gamma\_\{\\Gamma\}=\\Gammaand

∫ℝD×ℝD‖x−y‖2​d​γΓ​\(x,y\)=∫Q×QdQ​\(𝒙,𝒚\)2​𝑑Γ​\(𝒙,𝒚\)\.\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\\|x\-y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\gamma\_\{\\Gamma\}\(x,y\)=\\int\_\{Q\\times Q\}d\_\{Q\}\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)^\{2\}\\,\\,\\mathrm\{d\}\\Gamma\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)\.IfΓ\\Gammais optimal, thenγΓ\\gamma\_\{\\Gamma\}is optimal between its representative marginals\. Moreover, in this case, withprojt​\(x,y\)=\(1−t\)​x\+t​y\\mathrm\{proj\}^\{t\}\(x,y\)=\(1\-t\)x\+tyandμt≔proj♯t​γΓ\\mu\_\{t\}\\coloneqq\\mathrm\{proj\}^\{t\}\_\{\\sharp\}\\gamma\_\{\\Gamma\}, the projected curve𝛍t≔𝔮♯​μt\\boldsymbol\{\\mu\}\_\{t\}\\coloneqq\\mathfrak\{q\}\_\{\\sharp\}\\mu\_\{t\}is a constant\-speedW2,Q\\operatorname\{W\}\_\{2,Q\}\-geodesic, and the flow\-matching field associated with the lifted couplingγΓ\\gamma\_\{\\Gamma\}satisfies

∫I∫ℝD‖vtγΓ​\(z\)‖2​d​μt​\(z\)​𝑑t=W2,Q2⁡\(𝝁,𝝂\)\.\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\\|v\_\{t\}^\{\\gamma\_\{\\Gamma\}\}\(z\)\\\|^\{2\}\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\\,\\,\\mathrm\{d\}t=\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.

The proof is given in Appendix[A](https://arxiv.org/html/2608.26961#A1)\.

The theorem shows that any quotient coupling can be lifted through aligned representatives without increasing its cost\. In particular, an optimal quotient coupling admits a representative\-space lift whose quadratic cost equals the quotient Wasserstein distance\. The resulting liftγΓ\\gamma\_\{\\Gamma\}, however, need not be invariant under applying the same group action to both endpoints, because selecting representatives can break the symmetry\. This matters because the velocity or endpoint network is constrained to beGG\-equivariant\. For any representative couplingπ\\pi—in particular, forγΓ\\gamma\_\{\\Gamma\}—we therefore build a symmetrized coupling by applying the same group element to both endpoints and averaging overGG\. For a lawη\\etaand a couplingπ\\pi, set

ηG≔𝔼g​\[g♯​η\],πG≔𝔼g​\[\(g,g\)♯​π\],\\eta^\{G\}\\coloneqq\\mathbb\{E\}\_\{g\}\[g\_\{\\sharp\}\\eta\],\\qquad\\pi^\{G\}\\coloneqq\\mathbb\{E\}\_\{g\}\[\(g,g\)\_\{\\sharp\}\\pi\],where the expectation denotes integration against the normalized Haar measure onGG\. For a finite group such as𝔖N\\mathfrak\{S\}\_\{N\}, this is the average over all group elements\. The resulting coupling is diagonallyGG\-invariant, meaning that\(h,h\)♯​πG=πG\(h,h\)\_\{\\sharp\}\\pi^\{G\}=\\pi^\{G\}for everyh∈Gh\\in G\. It couplesμG\\mu^\{G\}andνG\\nu^\{G\}and represents the same quotient coupling asπ\\pi:\(𝔮,𝔮\)♯​πG=\(𝔮,𝔮\)♯​π\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\pi^\{G\}=\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\pi\. We next show that, on theGG\-equivariant model class, training withπ\\piis equivalent to training withπG\\pi^\{G\}\.

###### Theorem 3\.2\(Equivariant flow matching via symmetrization\)\.

Letπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)\. Then:

1. 1\.For everyGG\-equivariant fieldvv,Jπ​\(v\)=JπG​\(v\)J\_\{\\pi\}\(v\)=J\_\{\\pi^\{G\}\}\(v\)\. The fieldvπGv^\{\\pi^\{G\}\}can be chosenGG\-equivariant and minimizesJπJ\_\{\\pi\}over allGG\-equivariant fields\.
2. 2\.If the ODE of aGG\-equivariant field has a unique flowγt\\gamma\_\{t\}, then it induces a well\-defined quotient flowγ¯t​\(\[x\]\)≔\[γt​\(x\)\]\.\\bar\{\\gamma\}\_\{t\}\(\[x\]\)\\coloneqq\[\\gamma\_\{t\}\(x\)\]\.For everyη∈𝒫2​\(ℝD\)\\eta\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\),𝔮♯​γt,♯​η=γ¯t,♯​𝔮♯​η\.\\mathfrak\{q\}\_\{\\sharp\}\\gamma\_\{t,\\sharp\}\\eta=\\bar\{\\gamma\}\_\{t,\\sharp\}\\mathfrak\{q\}\_\{\\sharp\}\\eta\.Hence the projected curve depends only on the quotient law𝔮♯​η\\mathfrak\{q\}\_\{\\sharp\}\\eta, not on the representative lift chosen at time00\.
3. 3\.IfG⊂𝔖DG\\subset\\mathfrak\{S\}\_\{D\}permutes coordinates andν\\nuis supported on𝒜D\\mathcal\{A\}^\{D\},𝒜=\{a1,…,aM\}\\mathcal\{A\}=\\\{a\_\{1\},\\ldots,a\_\{M\}\\\}, let qt∗,πG,d​\(n∣z\)≔ℙπG​\(Xd′=an∣Zt=z\)\.q\_\{t\}^\{\\ast,\\pi^\{G\},d\}\(n\\mid z\)\\coloneqq\\mathbb\{P\}\_\{\\pi^\{G\}\}\(X^\{\\prime\}\_\{d\}=a\_\{n\}\\mid Z\_\{t\}=z\)\.Forg∈Gg\\in G, letg⋅dg\\cdot dbe the coordinate determined by\(g​z\)g⋅d=zd\(gz\)\_\{g\\cdot d\}=z\_\{d\}\. These coordinate posteriors can be chosen equivariantly ford∈\[D\]d\\in\[D\]andn∈\[M\]n\\in\[M\]: qt∗,πG,g⋅d​\(n∣g​z\)=qt∗,πG,d​\(n∣z\)\.q\_\{t\}^\{\\ast,\\pi^\{G\},g\\cdot d\}\(n\\mid gz\)=q\_\{t\}^\{\\ast,\\pi^\{G\},d\}\(n\\mid z\)\.They minimizeJπcoordJ\_\{\\pi\}^\{\\rm coord\}over equivariant coordinate kernels and recovervπGv^\{\\pi^\{G\}\}through \([6](https://arxiv.org/html/2608.26961#S2.E6)\)\.

The full proof is given in Appendix[B](https://arxiv.org/html/2608.26961#A2)\. Together, Theorems[3\.1](https://arxiv.org/html/2608.26961#S3.Thmtheorem1)and[3\.2](https://arxiv.org/html/2608.26961#S3.Thmtheorem2)connect the practical recipe used below to quotient transport: align representatives and train an equivariant Euclidean model whose flow, when well posed, descends to unordered graphs

## 4Constructing graph couplings via Gromov–Monge approximation

ForC=Ce\+CvC=C\_\{\\rm e\}\+C\_\{\\rm v\}, encode a graph byE∈\(ℝC\)N×NE\\in\(\\mathbb\{R\}^\{C\}\)^\{N\\times N\}with entries

Ei​k=\(λedgeei​kE,𝟏\{i=k\}λnodeCvfiE\),E\_\{ik\}=\\left\(\\sqrt\{\\lambda\_\{\\rm edge\}\}\\,e^\{E\}\_\{ik\},\\mathbf\{1\}\_\{\\\{i=k\\\}\}\\sqrt\{\\frac\{\\lambda\_\{\\rm node\}\}\{C\_\{\\rm v\}\}\}\\,f\_\{i\}^\{E\}\\right\),whereei​kE∈ℝCee^\{E\}\_\{ik\}\\in\\mathbb\{R\}^\{C\_\{\\rm e\}\}contains edge features andfiE∈ℝCvf\_\{i\}^\{E\}\\in\\mathbb\{R\}^\{C\_\{\\rm v\}\}contains node features on the diagonal\. Hence edge and node\-feature discrepancies enter the squared Euclidean cost with weightsλedge\\lambda\_\{\\rm edge\}andλnode\\lambda\_\{\\rm node\}, respectively, as in fused Gromov–Wasserstein transport\([Vayer et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib6)\)\. We useλedge=λnode=12\\lambda\_\{\\rm edge\}=\\lambda\_\{\\rm node\}=\\tfrac\{1\}\{2\}in all experiments\. Because the same permutation acts on both node indices, the quotient cost \([8](https://arxiv.org/html/2608.26961#S3.E8)\) compares all pairwise graph relations after a single consistent node alignment\. Its exact evaluation is an NP\-hard quadratic assignment, so we combine an inner Gromov–Wasserstein relaxation\([Bauer et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib36)\)with an optional outer assignment between graphs in a minibatch\.

##### Choosing a target representative with GW – inner solvers

ForE,F∈\(ℝC\)N×NE,F\\in\(\\mathbb\{R\}^\{C\}\)^\{N\\times N\}, we use a generalized version\([Bauer et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib36)\)of the discrete Gromov–Wasserstein relaxation\([Mémoli, 2011](https://arxiv.org/html/2608.26961#bib.bib3);[Peyré et al\., 2016](https://arxiv.org/html/2608.26961#bib.bib9)\), allowing for vector\-valued edge features:

GW2,Z2⁡\(\[E\],\[F\]\)≔min⁡∑i,j,k,l=1NT∈𝒰N⁡‖Ei​k−Fj​l‖2​Ti​j​Tk​l\.\\operatorname\{GW\}\_\{2,Z\}^\{2\}\(\[E\],\[F\]\)\\coloneqq\\min\_\{T\\in\\mathcal\{U\}\_\{N\}\}\\sum\_\{i,j,k,l=1\}^\{N\}\\\|E\_\{ik\}\-F\_\{jl\}\\\|^\{2\}T\_\{ij\}T\_\{kl\}\.\(9\)where𝒰N=\{T≥0:T​𝟏=T⊤​𝟏=N−1​𝟏\}\\mathcal\{U\}\_\{N\}=\\\{T\\geq 0:T\\mathbf\{1\}=T^\{\\top\}\\mathbf\{1\}=N^\{\-1\}\\mathbf\{1\}\\\}\. RestrictingTTto scaled permutation matrices recovers the Gromov–Monge objective, soGW2,Z2≤GM2,Z2/N2\\operatorname\{GW\}\_\{2,Z\}^\{2\}\\leq\\operatorname\{GM\}\_\{2,Z\}^\{2\}/N^\{2\}\. We approximately solve the non\-convex problem by Frank–Wolfe iterations\([Peyré et al\., 2016](https://arxiv.org/html/2608.26961#bib.bib9)\): each iteration solves an exact linearized transport problem and updates the soft plan by an exact line search\. From the final planTT, we recover the node permutation

σ^∈arg​maxσ∈𝔖N∑i=1NTi,σ⁡\(i\),\\widehat\{\\sigma\}\\in\\operatorname\*\{arg\\,max\}\_\{\\sigma\\in\\mathfrak\{S\}\_\{N\}\}\\sum\_\{i=1\}^\{N\}T\_\{i,\\sigma\(i\)\},using the Hungarian algorithm\([Kuhn, 1955](https://arxiv.org/html/2608.26961#bib.bib48)\)\. Equivalently,N−1​Pσ^N^\{\-1\}P\_\{\\widehat\{\\sigma\}\}is the scaled permutation matrix closest toTTin Frobenius norm, wherePσP\_\{\\sigma\}denotes the permutation matrix ofσ\\sigma\. The GW plan tells us how to relabelFF\. Using the recovered permutation, define the aligned representative by

Fi​jaligned≔Fσ^​\(i\)​σ^​\(j\)\.F^\{\\mathrm\{aligned\}\}\_\{ij\}\\coloneqq F\_\{\\widehat\{\\sigma\}\(i\)\\widehat\{\\sigma\}\(j\)\}\.We then train flow matching on\(E,Faligned\)\(E,F^\{\\mathrm\{aligned\}\}\)\. Since GW is a relaxation, this representative need not be the best solution of the original hard alignment problem\.

As a cheaper alternative, leteccE⁡\(i\)=\(N−1​∑k‖Ei​k‖2\)1/2\\operatorname\{ecc\}\_\{E\}\(i\)=\(N^\{\-1\}\\sum\_\{k\}\\\|E\_\{ik\}\\\|^\{2\}\)^\{1/2\}, and leteccE↑⁡\(i\)\\operatorname\{ecc\}\_\{E\}^\{\\uparrow\}\(i\)denote theii\-th smallest value among node eccentricities ofEE\. This gives the first lower bound \(FLB\)\([Bauer et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib36)\)

FLB2,Z2⁡\(\[E\],\[F\]\)≔1N​∑i\|eccE↑⁡\(i\)−eccF↑⁡\(i\)\|2≤GW2,Z2⁡\(\[E\],\[F\]\)\.\\operatorname\{FLB\}\_\{2,Z\}^\{2\}\(\[E\],\[F\]\)\\coloneqq\\frac\{1\}\{N\}\\sum\_\{i\}\|\\operatorname\{ecc\}\_\{E\}^\{\\uparrow\}\(i\)\-\\operatorname\{ecc\}\_\{F\}^\{\\uparrow\}\(i\)\|^\{2\}\\leq\\operatorname\{GW\}\_\{2,Z\}^\{2\}\(\[E\],\[F\]\)\.This is the squared22\-Wasserstein distance between the empirical eccentricity distributions\. We sort the nodes ofEEandFFby their eccentricities and pair theii\-th node in one ordering with theii\-th node in the other, breaking ties arbitrarily\. These pairs define a permutation\. Computing the eccentricities takesO⁡\(N2​C\)O\(N^\{2\}C\)operations, and sorting takesO⁡\(N​log⁡N\)O\(N\\log N\)\. ThusGWretains detailed pairwise structure at higher computational cost, whereasFLBprovides an inexpensive inner alignment and is cheap to evaluate for allB2B^\{2\}candidate pairs in the outer minibatch assignment\. Their empirical cost–quality trade\-off is reported in Appendix[G](https://arxiv.org/html/2608.26961#A7)\.

##### Outer–inner minibatch coupling

For a graph pair\(E,F\)\(E,F\), letc^​\(\[E\],\[F\]\)\\widehat\{c\}\(\[E\],\[F\]\)denote the approximate quotient cost returned by the selected inner solver, and letσ^​\(E,F\)∈𝔖N\\widehat\{\\sigma\}\(E,F\)\\in\\mathfrak\{S\}\_\{N\}denote its corresponding hard node alignment\. Thusσ^​\(E,F\)−1⋅F\\widehat\{\\sigma\}\(E,F\)^\{\-1\}\\cdot Fis the target representative aligned toEE\. For independently sampled batches\(Ea\)a=1B\(E^\{a\}\)\_\{a=1\}^\{B\}and\(Fb\)b=1B\(F^\{b\}\)\_\{b=1\}^\{B\}, either setτ^​\(a\)=a\\widehat\{\\tau\}\(a\)=aor solve

τ^∈arg​minτ∈𝔖B∑ac^\(\[Ea\],\[Fτ⁡\(a\)\]\),\\widehat\{\\tau\}\\in\\argmin\_\{\\tau\\in\\mathfrak\{S\}\_\{B\}\}\\sum\_\{a\}\\widehat\{c\}\(\[E^\{a\}\],\[F^\{\\tau\(a\)\}\]\),wherec^\\widehat\{c\}is the GW or FLB cost\. For each selected pair, setσ^a=σ^​\(Ea,Fτ^​\(a\)\)\\widehat\{\\sigma\}\_\{a\}=\\widehat\{\\sigma\}\(E^\{a\},F^\{\\widehat\{\\tau\}\(a\)\}\), defining

π^B≔1B​∑aδ\(Ea,σ^a−1⋅Fτ^​\(a\)\)\.\\widehat\{\\pi\}\_\{B\}\\coloneqq\\frac\{1\}\{B\}\\sum\_\{a\}\\delta\_\{\(E^\{a\},\\widehat\{\\sigma\}^\{\-1\}\_\{a\}\\cdot F^\{\\widehat\{\\tau\}\(a\)\}\)\}\.Thus inner alignment chooses graph representatives, while the optional outer assignment chooses which graphs to connect\. These are distinct decisions: even with independent outer pairing, an inner alignment can substantially shorten the interpolation between each selected pair\. Algorithm[1](https://arxiv.org/html/2608.26961#alg1)summarizes the complete construction\.

Algorithm 1Outer–inner minibatch graph alignment0:Source batch

\(Ea\)a=1B\(E^\{a\}\)\_\{a=1\}^\{B\}, target batch

\(Fb\)b=1B\(F^\{b\}\)\_\{b=1\}^\{B\}, outer cost

c^\\widehat\{c\}
1:ifthe outer coupling is independentthen

2:

τ^​\(a\)←a\\widehat\{\\tau\}\(a\)\\leftarrow afor all

a=1,…,Ba=1,\\ldots,B
3:else

4:

Ca​b←c^​\(\[Ea\],\[Fb\]\)C\_\{ab\}\\leftarrow\\widehat\{c\}\(\[E^\{a\}\],\[F^\{b\}\]\)for all

a,ba,b
5:

τ^←arg​minτ∈𝔖B∑aCa,τ⁡\(a\)\\widehat\{\\tau\}\\leftarrow\\argmin\_\{\\tau\\in\\mathfrak\{S\}\_\{B\}\}\\sum\_\{a\}C\_\{a,\\tau\(a\)\}
6:endif

7:for

a=1,…,Ba=1,\\ldots,Bdo

8:

Ga←Fτ^​\(a\)G^\{a\}\\leftarrow F^\{\\widehat\{\\tau\}\(a\)\}
9:Obtain

σ^a\\widehat\{\\sigma\}\_\{a\}using GW, FLB, or random reshuffling

10:

F~a←σ^a−1⋅Ga\\widetilde\{F\}^\{a\}\\leftarrow\\widehat\{\\sigma\}^\{\-1\}\_\{a\}\\cdot G^\{a\}
11:endfor

12:return

\(Ea,F~a\)a=1B\(E^\{a\},\\widetilde\{F\}^\{a\}\)\_\{a=1\}^\{B\}

##### Training

WriteF~a=σ^a−1⋅Fτ^​\(a\)\\widetilde\{F\}^\{a\}=\\widehat\{\\sigma\}^\{\-1\}\_\{a\}\\cdot F^\{\\widehat\{\\tau\}\(a\)\}, sampleta∼𝒰⁡\[0,1\]t^\{a\}\\sim\\mathcal\{U\}\[0,1\], and setEtaa=\(1−ta\)​Ea\+ta​F~aE\_\{t^\{a\}\}^\{a\}=\(1\-t^\{a\}\)E^\{a\}\+t^\{a\}\\widetilde\{F\}^\{a\}\. The continuous and categorical minibatch losses are, respectively,

JB​\(vθ\)≔1B​∑a=1B‖vθ​\(ta,Etaa\)−\(F~a−Ea\)‖2\.J\_\{B\}\(v^\{\\theta\}\)\\coloneqq\\frac\{1\}\{B\}\\sum\_\{a=1\}^\{B\}\\left\\\|v^\{\\theta\}\(t^\{a\},E\_\{t^\{a\}\}^\{a\}\)\-\(\\widetilde\{F\}^\{a\}\-E^\{a\}\)\\right\\\|^\{2\}\.\(10\)JBcoord\(\(qθd\)d=1D\)≔−1B∑a=1B∑d=1Dlogqθ,tad\(κd\(F~a\)∣Etaa\)\.J\_\{B\}^\{\\rm coord\}\\bigl\(\(q\_\{\\theta\}^\{d\}\)\_\{d=1\}^\{D\}\\bigr\)\\coloneqq\-\\frac\{1\}\{B\}\\sum\_\{a=1\}^\{B\}\\sum\_\{d=1\}^\{D\}\\log q\_\{\\theta,t^\{a\}\}^\{d\}\\\!\\left\(\\kappa\_\{d\}\(\\widetilde\{F\}^\{a\}\)\\mid E\_\{t^\{a\}\}^\{a\}\\right\)\.\(11\)The categorical velocity follows from \([6](https://arxiv.org/html/2608.26961#S2.E6)\)\. For datasets containing graphs with different numbers of nodes, one graph transformer is shared across all node counts: it maps anNN\-node input to anNN\-node prediction, whileNNmay vary between batches\. We form each training batch from graphs with the sameNN\. At inference, we sampleNNfrom the empirical distribution of node counts in the training set\. The graph transformer is permutation\-equivariant, ensuring that relabelling the source relabels the entire predicted trajectory\. Architecture and solver details are in Appendix[F](https://arxiv.org/html/2608.26961#A6)\.

## 5Numerical experiments

We evaluate the proposed coupling constructions in a permutation\-equivariant graph flow\-matching model\. We compare random node relabelling \(Random\), eccentricity sorting \(FLB\), and a 10\-iteration Frank–Wolfe GW solver \(GW\); “\+out\+\\mathrm\{out\}” additionally reorders same\-size graphs in sub\-batches of at most eight\. As a permutation\-blind reference,MinibatchOTrelabels each target at random and then assigns pairs over the full minibatch by squared Euclidean distance‖E−F‖2\\\|E\-F\\\|^\{2\}, with no inner alignment\. All models share the same equivariant graph\-transformer backbone and differ only in their training coupling\. We consider several Euler budgets to assess whether structure\-aware alignment improves few\-step generation\. Complete architecture, optimization, data, solver, and metric specifications are given in Appendix[F](https://arxiv.org/html/2608.26961#A6)\.

For variable\-size datasets, the population construction first draws the node countN∼pNN\\sim p\_\{N\}from its empirical distribution\. Conditional onNN,Randomuses the product couplingμN⊗νN\\mu\_\{N\}\\otimes\\nu\_\{N\}, whereμN\\mu\_\{N\}andνN\\nu\_\{N\}denote the source and target laws forNNnodes\. InnerFLB/GWchanges only the representative of each selected target graph, whereas the outer variants additionally optimize the source–target pairing within the fixed\-NNminibatch\. Hence, improvements from inner alignment measure the value of node correspondence, while differences between inner\-only and outer variants measure the additional effect of minibatch transport\.

### 5\.1Illustrative target via translated cycle graphs

Inspired by\([Piening et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib37)\), each graph is a1212\-node ring drawn as a regular1212\-gon of radius11centred at\(h,y\)\(h,y\), with the nodes randomly relabelled\. The representationE∈ℝ12×12×3E\\in\\mathbb\{R\}^\{12\\times 12\\times 3\}holds one binary adjacency channel and the planar 2D node positions on the diagonal\. Source and target differ only by a translation \(h∼𝒰⁡\[−6,6\]h\\sim\\mathcal\{U\}\[\-6,6\]and an independent rotation on each side, aty=0y=0andy=8y=8\), so both sides have the same edge law and any deformation of the ring is caused by the coupling alone\. Figure[1](https://arxiv.org/html/2608.26961#S5.F1)shows the learned trajectories att=0t=0\(bottom\),t=0\.5t=0\.5\(center\), andt=1t=1\(top\)\. InnerGWpairs corresponding nodes between two rings\. This results in a fixed adjacency channel along the interpolation\. Under random inner relabelling, the two sets of edges disagree, so the endpoints are still correct, but the intermediate graphs in between are much denser, carrying half\-present edges in place of crisp circular ones\. InnerGWremoves the edge ambiguity and keeps the ring at full size midway, but leaves the paths curved\. The outer assignment straightens them\. The inner alignment thus governs the shape of the transported object, the outer assignment the geometry of its path\.

![Refer to caption](https://arxiv.org/html/2608.26961v1/toy_circle_example/traj_random_gray.png)

\(a\)Random
![Refer to caption](https://arxiv.org/html/2608.26961v1/toy_circle_example/traj_gw_gray.png)

\(b\)GW
![Refer to caption](https://arxiv.org/html/2608.26961v1/toy_circle_example/traj_gw_outer_gw_gray.png)

\(c\)GW\+\+GWout

Figure 1:Learned trajectories from cycle graphs \(bottom\) to vertically offset cycle graphs \(top\), on a shared scale\. Colours mark source graphs, grey lines node paths, and edge opacity equals edge value\. The ring collapses att=1/2t=1/2in[1\(a\)](https://arxiv.org/html/2608.26961#S5.F1.sf1), stays crisp but on curved paths in[1\(b\)](https://arxiv.org/html/2608.26961#S5.F1.sf2), and is straightened in[1\(c\)](https://arxiv.org/html/2608.26961#S5.F1.sf3)\.
### 5\.2Continuous target via stochastic block models

##### Data

As a controlled continuous testbed, we use an SBM onN=10N=10nodes with two channels: one symmetric weighted\-edge channel in\[0,1\]\[0,1\]and one scalar node feature on the diagonal\. ForK∈\{1,…,5\}K\\in\\\{1,\\ldots,5\\\}, nodes are assigned to near\-balanced blocks; for everyi<ji<j, we sample and mirrorei​jE=ej​iEe^\{E\}\_\{ij\}=e^\{E\}\_\{ji\}fromBeta⁡\(6,2\)\\mathrm\{Beta\}\(6,2\)within a block andBeta⁡\(2,6\)\\mathrm\{Beta\}\(2,6\)across blocks\. A node in blockk∈\{0,…,K−1\}k\\in\\\{0,\\ldots,K\-1\\\}receives a Beta feature with concentration88and meanμk=\(2​k\+1\)/\(2​K\)\\mu\_\{k\}=\(2k\+1\)/\(2K\), so the diagonal channel also records community structure\. We randomize node order, making each tensor an arbitrary representative of its graph orbit\. The source has independent𝒰⁡\[0,1\]\\mathcal\{U\}\[0,1\]upper\-triangular edges, mirrored across the diagonal, and independent𝒰⁡\[0,1\]\\mathcal\{U\}\[0,1\]node features\. The main experiment pools allKK\. Appendix[D](https://arxiv.org/html/2608.26961#A4)instead suppliesKKto the model and aligns only within minibatches containing a single value ofKKto showcase conditional instead of unconditional generation\.

##### Metrics

For 100 real and 100 generated graphs, we estimate maximum mean discrepancies \(MMDs\) of degree, clustering, and four\-node graphlet\-orbit counts, which distinguish the roles a node can occupy in induced four\-node subgraphs\([Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34);[You et al\., 2018](https://arxiv.org/html/2608.26961#bib.bib33)\)\. All three require unweighted graphs, obtained by thresholding real and generated edges alike atei​jE\>1/2e^\{E\}\_\{ij\}\>1/2\. To compare the full attributed graphs including their node features, we also report fused\-GW nearest\-neighbor accuracy \(FGW–NNA\)\. This statistic builds on the popular OT\-NNA statistic used to compare real and generated sets of point clouds\([Yang et al\., 2019](https://arxiv.org/html/2608.26961#bib.bib14)\)by replacing pairwise Wasserstein distance with pairwise FGW\. Given 100 generated and 100 real graphs, we pool both samples and label each graph by its origin\. We then classify every graph by the label of its nearest neighbor under fused GW, excluding the graph itself\. Because fused GW compares attributed graphs up to node relabelling, this is a two\-sample test directly on graph orbits rather than on selected descriptors\. If real and generated graphs follow the same law, the accuracy approaches0\.50\.5\. Larger values mean that the samples are more easily separated\. Results are reported over three training runs with different seeds and evaluated at5/25/1255/25/125Euler steps\.

##### Results

At five steps \(Table[1](https://arxiv.org/html/2608.26961#S5.T1)\), GW with outer assignment reduces FGW–NNA from0\.7960\.796to0\.5680\.568, degree MMD from0\.1140\.114to0\.0180\.018, and clustering MMD from0\.1790\.179to0\.0340\.034\. The gap narrows by 125 steps, as expected because all matching should result in valid interpolations\. FLB captures part of the low\-step gain at much lower cost, while outer assignment adds a smaller improvement\. The permutation\-blindMinibatchOTalready matches or exceeds theFLBvariants on most statistics, yet remains clearly behind the GW variants\. The conditional experiment likewise favors GW alignment \(Appendix[D](https://arxiv.org/html/2608.26961#A4)\)\. Notably, at five steps the two GW variants are also the only conditions that bring FGW–NNA close to its ideal value of0\.50\.5\. Thus, the improvement is visible not merely in selected graph statistics, but in a two\-sample test using the underlying relational geometry\.

Table 1:Continuous SBM,*unconditional*mixture overK∈\{1,…,5\}K\\in\\\{1,\\dots,5\\\}\(N=10N=10\), at5/25/1255/25/125Euler steps\. Rows are meanstd\{\}\_\{\\text\{std\}\}over33training seeds \(55evaluation repeats each\)\. Best value per column is bold \(MMDs \(degree, clustering, graphlet orbit\)↓\\downarrow, Best FGW–NNA:∼0\.5\\sim 0\.5\.

### 5\.3Categorical target via molecular graph generation

##### Data

We use QM9 \(N≤9N\\leq 9\)\([Ramakrishnan et al\., 2014](https://arxiv.org/html/2608.26961#bib.bib49)\)and ZINC250k \(N≤38N\\leq 38\)\([Jin et al\., 2018](https://arxiv.org/html/2608.26961#bib.bib50)\)in the categorical CatFlow/DiGress representation\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31);[Vignac et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib32)\)\. Edges are one\-hot over\{none,single,double,triple\}\\\{\\text\{none\},\\text\{single\},\\text\{double\},\\text\{triple\}\\\}, while each diagonal node feature concatenates a one\-hot atom type with a formal\-charge one\-hot over\{−1,0,\+1\}\\\{\-1,0,\+1\\\}\. QM9 uses atom types\{C,N,O,F\}\\\{\\mathrm\{C\},\\mathrm\{N\},\\mathrm\{O\},\\mathrm\{F\}\\\}, givingC=11C=11; ZINC250k uses\{C,N,O,F,Br,Cl,I,P,S\}\\\{\\mathrm\{C\},\\mathrm\{N\},\\mathrm\{O\},\\mathrm\{F\},\\mathrm\{Br\},\\mathrm\{Cl\},\\mathrm\{I\},\\mathrm\{P\},\\mathrm\{S\}\\\}, givingC=16C=16\. The network predicts the coordinatewise categorical endpoint and minimises a blockwise softmax analogue of \([11](https://arxiv.org/html/2608.26961#S4.E11)\)\. Its velocity is recovered through \([6](https://arxiv.org/html/2608.26961#S2.E6)\)\. We group minibatches by node count and apply the same outer–inner alignment as for the continuous experiment\.

##### Metrics

Following the GDSS/CatFlow evaluation protocols\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31);[Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34)\), we report uncorrected molecular validity—the fraction of generated graphs that RDKit sanitizes without valence correction or other postprocessing—and Fréchet ChemNet Distance \(FCD\); higher is better for validity and lower is better for FCD\. We omit novelty because QM9 is a near\-exhaustive enumeration of small molecules: a high novelty score can therefore reward atypical or lower\-quality structures rather than faithful modelling\. Uniqueness is close to100%100\\%for all structure\-aware conditions, while additional invalid or atypical outputs can increase the apparent uniqueness ofRandom; reporting it would therefore favor the weakest coupling\. Validity and FCD capture chemical correctness and distributional fidelity, respectively; complete definitions are in Appendix[F](https://arxiv.org/html/2608.26961#A6)\.

#### 5\.3\.1Limited\-budget sweep of the endpoint\-induced velocity field

We train the shared 2\.8M\-parameter backbone for 100 epochs under each coupling and evaluate 10,000 samples at5/25/1255/25/125Euler steps; complete settings are in Appendix[F](https://arxiv.org/html/2608.26961#A6)\.

Table 2:Molecular short\-budget sweep \(100100training epochs\), evaluated at5/25/1255/25/125Euler steps\. Rows are meanstd\{\}\_\{\\text\{std\}\}over33evaluation repeats\. Best per column is bold \(Validity↑\\uparrow, FCD↓\\downarrow\)\.##### Results

At five steps \(Table[2](https://arxiv.org/html/2608.26961#S5.T2)\), inner GW raises validity from0\.88490\.8849to0\.93560\.9356on QM9 and from0\.56410\.5641to0\.64310\.6431on ZINC250k, while reducing FCD from1\.7311\.731to1\.2781\.278and from18\.44818\.448to15\.02415\.024\. Outer assignment further improves validity, whereas its FCD gain is dataset dependent: it is uniformly best on QM9 and gives the highest ZINC250k validity, while inner\-only GW has marginally better ZINC250k FCD\.MinibatchOTmatches inner GW on QM9 validity but not on FCD; on ZINC250k it sits strictly betweenRandomand both GW variants\. Again, differences narrow with more steps\.

#### 5\.3\.2Full\-budget molecular generation

We train larger GW\-aligned endpoint models without outer alignment and evaluate the resultingGW\-CatFlowmodel with 500 Euler steps\. The complete protocol is in Appendix[E](https://arxiv.org/html/2608.26961#A5)\.

##### Results

Table[3](https://arxiv.org/html/2608.26961#S5.T3)gives99\.34%99\.34\\%validity and0\.1150\.115FCD on QM9, comparable to DeFoG, and99\.01%99\.01\\%validity with0\.9660\.966FCD on ZINC250k, the lowest FCD listed\. Because baselines follow their published protocols and our full\-budget model includes additional architectural changes, this table establishes competitiveness rather than isolating the effect of alignment\. Together with the controlled short\-budget sweep, it shows that the proposed coupling improves the regime it is designed for without preventing the model from reaching strong quality at a conventional, large integration budget\.

Table 3:Molecular long\-budget generation with GW inner alignment\. Ours:500500Euler steps,10,00010\{,\}000samples, meanstdover three repeats\. Baselines follow their published protocols; GraphAF, MoFlow, and DiGress are quoted from\([Hou et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib45)\), DeFoG from\([Xiong et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib42)\)\. Validity↑\\uparrow, FCD↓\\downarrow\.

## 6Conclusion

We specialise equivariant flow matching to graphs via Gromov–Wasserstein alignments: the inner alignment reorders each target’s nodes, the outer assignment reorders the minibatch\. Both simplify trajectories and improve few\-step generation over random assignment or permutation\-blind minibatch OT\. GW solvers remain the limitation: only approximately Gromov–Monge and expensive, though absent at inference\. Future work may include accelerated GW approximations\([Beier et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib1);[Chowdhury et al\., 2021](https://arxiv.org/html/2608.26961#bib.bib10);[Piening and Beinert, 2025](https://arxiv.org/html/2608.26961#bib.bib13);[Jin et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib4);[Scetbon et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib8)\)and semi\-discrete formulations beyond minibatches\([Kong et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib22);[Mousavi\-Hosseini et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib21)\)\.

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## Appendix AProof of representative lifts and quotient geodesics

This section proves Theorem[3\.1](https://arxiv.org/html/2608.26961#S3.Thmtheorem1)\. We first recall the metric notions used in the geodesic part of the proof\. A curve\(μt\)t∈I\(\\mu\_\{t\}\)\_\{t\\in I\}belongs toA​CI2​\(𝒫2​\(X\)\)AC\_\{I\}^\{2\}\(\\mathcal\{P\}\_\{2\}\(X\)\)if there existsm∈L2​\(I\)m\\in L^\{2\}\(I\)such that

W2,X⁡\(μs,μt\)≤∫stm⁡\(r\)​𝑑rfor all​0≤s≤t≤1\.\\operatorname\{W\}\_\{2,X\}\(\\mu\_\{s\},\\mu\_\{t\}\)\\leq\\int\_\{s\}^\{t\}m\(r\)\\,\\,\\mathrm\{d\}r\\qquad\\text\{for all \}0\\leq s\\leq t\\leq 1\.
LetG⊂O⁡\(D\)G\\subset O\(D\)be compact and define

Q≔ℝD/G,\[x\]≔\{g​x:g∈G\},Q\\coloneqq\\mathbb\{R\}^\{D\}/G,\\qquad\[x\]\\coloneqq\\\{gx:g\\in G\\\},with quotient map

𝔮:ℝD→Q,𝔮⁡\(x\)=\[x\]\.\\mathfrak\{q\}:\\mathbb\{R\}^\{D\}\\to Q,\\qquad\\mathfrak\{q\}\(x\)=\[x\]\.The quotient is equipped with the metric

dQ​\(\[x\],\[y\]\)≔ming∈G⁡‖x−g​y‖\.d\_\{Q\}\(\[x\],\[y\]\)\\coloneqq\\min\_\{g\\in G\}\\\|x\-gy\\\|\.SinceGGis compact and acts by isometries, the minimum is attained anddQd\_\{Q\}defines a metric onQQ\. SinceGGis compact and acts isometrically,\(Q,dQ\)\(Q,d\_\{Q\}\)is Polish, and𝔮\\mathfrak\{q\}is11\-Lipschitz\.

For𝝁,𝝂∈𝒫2​\(Q\)\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\\in\\mathcal\{P\}\_\{2\}\(Q\), the corresponding Wasserstein distance is

W2,Q2⁡\(𝝁,𝝂\)≔min⁡∫Q×QΓ∈cQ​\(𝝁,𝝂\)⁡dQ​\(𝒙,𝒚\)2​𝑑Γ​\(𝒙,𝒚\),\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\\coloneqq\\min\_\{\\Gamma\\in\\mathrm\{c\}\_\{Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\}\\int\_\{Q\\times Q\}d\_\{Q\}\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)^\{2\}\\,\\,\\mathrm\{d\}\\Gamma\(\\boldsymbol\{x\},\\boldsymbol\{y\}\),\(12\)where

cQ\(𝝁,𝝂\)≔\{Γ∈𝒫2\(Q×Q\):proj♯0Γ=𝝁,proj♯1Γ=𝝂\}\.\\mathrm\{c\}\_\{Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\\coloneqq\\\{\\Gamma\\in\\mathcal\{P\}\_\{2\}\(Q\\times Q\):\\mathrm\{proj\}^\{0\}\_\{\\sharp\}\\Gamma=\\boldsymbol\{\\mu\},\\mathrm\{proj\}^\{1\}\_\{\\sharp\}\\Gamma=\\boldsymbol\{\\nu\}\\\}\.BycQopt​\(𝝁,𝝂\)\\mathrm\{c\}\_\{Q\}^\{\{\\rm opt\}\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)we denote the couplings which minimize \([12](https://arxiv.org/html/2608.26961#A1.E12)\)\.

###### Proposition A\.1\(Representative lifting of quotient couplings\)\.

Let𝛍,𝛎∈𝒫2​\(Q\)\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\\in\\mathcal\{P\}\_\{2\}\(Q\), fixμ∈\[𝛍\]\\mu\\in\[\\boldsymbol\{\\mu\}\], and letΓ∈cQ​\(𝛍,𝛎\)\\Gamma\\in\\mathrm\{c\}\_\{Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\. Then

∫Q×QdQ​\(𝒙,𝒚\)2​𝑑Γ​\(𝒙,𝒚\)=min⁡∫ℝD×ℝDν∈\[𝝂\],γ∈cℝD​\(μ,ν\)\(𝔮,𝔮\)♯​γ=Γ⁡‖x−y‖2​𝑑γ​\(x,y\)\.\\int\_\{Q\\times Q\}d\_\{Q\}\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)^\{2\}\\,\\,\\mathrm\{d\}\\Gamma\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)=\\min\_\{\\begin\{subarray\}\{c\}\\nu\\in\[\\boldsymbol\{\\nu\}\],\\,\\gamma\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)\\\\ \(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\gamma=\\Gamma\\end\{subarray\}\}\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\\|x\-y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\gamma\(x,y\)\.In particular, ifΓ∈cQopt​\(𝛍,𝛎\)\\Gamma\\in\\mathrm\{c\}\_\{Q\}^\{\\rm opt\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)and\(νΓ,γΓ\)\(\\nu\_\{\\Gamma\},\\gamma\_\{\\Gamma\}\)attains the minimum on the right\-hand side, thenγΓ∈cℝDopt​\(μ,νΓ\)\\gamma\_\{\\Gamma\}\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}^\{\\rm opt\}\(\\mu,\\nu\_\{\\Gamma\}\)andW2,ℝD2⁡\(μ,νΓ\)=W2,Q2⁡\(𝛍,𝛎\)\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}^\{2\}\(\\mu,\\nu\_\{\\Gamma\}\)=\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.

###### Proof\.

Letν∈\[𝝂\]\\nu\\in\[\\boldsymbol\{\\nu\}\]andγ∈cℝD​\(μ,ν\)\\gamma\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)satisfy\(𝔮,𝔮\)♯​γ=Γ\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\gamma=\\Gamma\. Since

dQ​\(𝔮⁡\(x\),𝔮⁡\(y\)\)≤‖x−y‖,d\_\{Q\}\(\\mathfrak\{q\}\(x\),\\mathfrak\{q\}\(y\)\)\\leq\\\|x\-y\\\|,we have

∫Q×QdQ​\(𝒙,𝒚\)2​𝑑Γ​\(𝒙,𝒚\)≤∫ℝD×ℝD‖x−y‖2​𝑑γ​\(x,y\)\.\\int\_\{Q\\times Q\}d\_\{Q\}\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)^\{2\}\\,\\,\\mathrm\{d\}\\Gamma\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)\\leq\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\\|x\-y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\gamma\(x,y\)\.
Conversely, disintegrate

μ=∫Qμ𝒙​𝑑𝝁​\(𝒙\),Γ=∫Qδ𝒙⊗Γ𝒙​𝑑𝝁​\(𝒙\),\\mu=\\int\_\{Q\}\\mu\_\{\\boldsymbol\{x\}\}\\,\\,\\mathrm\{d\}\\boldsymbol\{\\mu\}\(\\boldsymbol\{x\}\),\\qquad\\Gamma=\\int\_\{Q\}\\delta\_\{\\boldsymbol\{x\}\}\\otimes\\Gamma\_\{\\boldsymbol\{x\}\}\\,\\,\\mathrm\{d\}\\boldsymbol\{\\mu\}\(\\boldsymbol\{x\}\),whereμ𝒙\\mu\_\{\\boldsymbol\{x\}\}is supported on𝔮−1​\(𝒙\)\\mathfrak\{q\}^\{\-1\}\(\\boldsymbol\{x\}\), and defineη∈𝒫⁡\(ℝD×Q\)\\eta\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{D\}\\times Q\)by

η⁡\(A×B\)≔∫Qμ𝒙​\(A\)​Γ𝒙​\(B\)​𝑑𝝁​\(𝒙\)\.\\eta\(A\\times B\)\\coloneqq\\int\_\{Q\}\\mu\_\{\\boldsymbol\{x\}\}\(A\)\\Gamma\_\{\\boldsymbol\{x\}\}\(B\)\\,\\,\\mathrm\{d\}\\boldsymbol\{\\mu\}\(\\boldsymbol\{x\}\)\.Then\(𝔮,id\)♯​η=Γ\(\\mathfrak\{q\},\\mathrm\{id\}\)\_\{\\sharp\}\\eta=\\Gamma\.

SinceGGis compact, the corresponding argmin relation is closed with nonempty compact sections\. Hence, a measurable selection theorem\[[Kechris, 2012](https://arxiv.org/html/2608.26961#bib.bib41), Theorem 18\.18\]yields a Borel mapY⁡\(x,𝒚\)∈𝔮−1​\(𝒚\)Y\(x,\\boldsymbol\{y\}\)\\in\\mathfrak\{q\}^\{\-1\}\(\\boldsymbol\{y\}\)satisfying

‖x−Y⁡\(x,𝒚\)‖=dQ​\(𝔮⁡\(x\),𝒚\)\.\\\|x\-Y\(x,\\boldsymbol\{y\}\)\\\|=d\_\{Q\}\(\\mathfrak\{q\}\(x\),\\boldsymbol\{y\}\)\.SetγΓ≔\(id,Y\)♯​η\\gamma\_\{\\Gamma\}\\coloneqq\(\\mathrm\{id\},Y\)\_\{\\sharp\}\\eta, and letνΓ\\nu\_\{\\Gamma\}be its second marginal\. Then\(𝔮,𝔮\)♯​γΓ=Γ\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\gamma\_\{\\Gamma\}=\\Gammaand𝔮♯​νΓ=𝝂\.\\mathfrak\{q\}\_\{\\sharp\}\\nu\_\{\\Gamma\}=\\boldsymbol\{\\nu\}\.Moreover, sinceG⊂O⁡\(D\)G\\subset O\(D\), we get

∫ℝD‖y‖2​d​νΓ​\(y\)=∫QdQ​\(𝒚,\[0\]\)2​𝑑𝝂​\(𝒚\)<∞,\\int\_\{\\mathbb\{R\}^\{D\}\}\\\|y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\nu\_\{\\Gamma\}\(y\)=\\int\_\{Q\}d\_\{Q\}\(\\boldsymbol\{y\},\[0\]\)^\{2\}\\,\\,\\mathrm\{d\}\\boldsymbol\{\\nu\}\(\\boldsymbol\{y\}\)<\\infty,and henceνΓ∈\[𝝂\]\\nu\_\{\\Gamma\}\\in\[\\boldsymbol\{\\nu\}\]\. Finally,

∫ℝD×ℝD‖x−y‖2​d​γΓ​\(x,y\)=∫Q×QdQ​\(𝒙,𝒚\)2​𝑑Γ​\(𝒙,𝒚\)\.\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\\|x\-y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\gamma\_\{\\Gamma\}\(x,y\)=\\int\_\{Q\\times Q\}d\_\{Q\}\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)^\{2\}\\,\\,\\mathrm\{d\}\\Gamma\(\\boldsymbol\{x\},\\boldsymbol\{y\}\)\.Thus the minimum is attained\. If, in addition,Γ∈cQopt​\(𝝁,𝝂\)\\Gamma\\in\\mathrm\{c\}\_\{Q\}^\{\\rm opt\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\), then

W2,Q2⁡\(𝝁,𝝂\)≤W2,ℝD2⁡\(μ,νΓ\)≤∫ℝD×ℝD‖x−y‖2​d​γΓ​\(x,y\)=W2,Q2⁡\(𝝁,𝝂\)\.\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\\leq\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}^\{2\}\(\\mu,\\nu\_\{\\Gamma\}\)\\leq\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\\|x\-y\\\|^\{2\}\\,\\,\\mathrm\{d\}\\gamma\_\{\\Gamma\}\(x,y\)=\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.Hence all inequalities are equalities, which proves the final claim\. ∎

###### Geodesic and energy claims in Theorem[3\.1](https://arxiv.org/html/2608.26961#S3.Thmtheorem1)\.

LetΓ\\Gammabe optimal and let\(νΓ,γΓ\)\(\\nu\_\{\\Gamma\},\\gamma\_\{\\Gamma\}\)be the pair constructed in Proposition[A\.1](https://arxiv.org/html/2608.26961#A1.Thmtheorem1)\. The proposition gives

γΓ∈cℝDopt​\(μ,νΓ\),W2,ℝD2⁡\(μ,νΓ\)=W2,Q2⁡\(𝝁,𝝂\)\.\\gamma\_\{\\Gamma\}\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}^\{\\rm opt\}\(\\mu,\\nu\_\{\\Gamma\}\),\\qquad\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}^\{2\}\(\\mu,\\nu\_\{\\Gamma\}\)=\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.\(13\)For0≤s<t≤10\\leq s<t\\leq 1, the coupling\(projs,projt\)♯​γΓ\(\\mathrm\{proj\}^\{s\},\\mathrm\{proj\}^\{t\}\)\_\{\\sharp\}\\gamma\_\{\\Gamma\}and the fact that𝔮\\mathfrak\{q\}is11\-Lipschitz give

W2,Q⁡\(𝝁s,𝝁t\)≤W2,ℝD⁡\(μs,μt\)≤\(t−s\)​W2,Q⁡\(𝝁,𝝂\)\.\\operatorname\{W\}\_\{2,Q\}\(\\boldsymbol\{\\mu\}\_\{s\},\\boldsymbol\{\\mu\}\_\{t\}\)\\leq\\operatorname\{W\}\_\{2,\\mathbb\{R\}^\{D\}\}\(\\mu\_\{s\},\\mu\_\{t\}\)\\leq\(t\-s\)\\operatorname\{W\}\_\{2,Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.Thus\(𝝁t\)t∈I\(\\boldsymbol\{\\mu\}\_\{t\}\)\_\{t\\in I\}belongs toA​CI2​\(𝒫2​\(Q\)\)AC\_\{I\}^\{2\}\(\\mathcal\{P\}\_\{2\}\(Q\)\)\. Applying this bound on\[0,s\]\[0,s\],\[s,t\]\[s,t\], and\[t,1\]\[t,1\], and combining it with the triangle inequality between𝝁0=𝝁\\boldsymbol\{\\mu\}\_\{0\}=\\boldsymbol\{\\mu\}and𝝁1=𝝂\\boldsymbol\{\\mu\}\_\{1\}=\\boldsymbol\{\\nu\}, forces equality in each bound\. Hence

W2,Q⁡\(𝝁s,𝝁t\)=\(t−s\)​W2,Q⁡\(𝝁,𝝂\),\\operatorname\{W\}\_\{2,Q\}\(\\boldsymbol\{\\mu\}\_\{s\},\\boldsymbol\{\\mu\}\_\{t\}\)=\(t\-s\)\\operatorname\{W\}\_\{2,Q\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\),so\(𝝁t\)t∈I\(\\boldsymbol\{\\mu\}\_\{t\}\)\_\{t\\in I\}is a constant\-speed geodesic\.

Finally, the Euclidean optimal\-coupling identity in Section[2\.2](https://arxiv.org/html/2608.26961#S2.SS2)and \([13](https://arxiv.org/html/2608.26961#A1.E13)\) yield

∫I∫ℝD‖vtγΓ​\(z\)‖2​d​μt​\(z\)​𝑑t=W2,Q2⁡\(𝝁,𝝂\)\.\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\\|v\_\{t\}^\{\\gamma\_\{\\Gamma\}\}\(z\)\\\|^\{2\}\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\\,\\,\\mathrm\{d\}t=\\operatorname\{W\}\_\{2,Q\}^\{2\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\nu\}\)\.∎

## Appendix BProof of equivariant flow matching via symmetrization

This section proves Theorem[3\.2](https://arxiv.org/html/2608.26961#S3.Thmtheorem2)\. WriteλG\\lambda\_\{G\}for the normalized Haar probability measure onGG\. We callπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)diagonallyGG\-invariant if\(g,g\)♯​π=π\(g,g\)\_\{\\sharp\}\\pi=\\pifor everyg∈Gg\\in G\.

###### Proposition B\.1\(Equivariance of the flow\-matching minimizer\)\.

Letπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)be diagonallyGG\-invariant and letvπv^\{\\pi\}denote the minimizer ofJπJ\_\{\\pi\}in \([4](https://arxiv.org/html/2608.26961#S2.E4)\)\. Thenvπv^\{\\pi\}admits aGG\-equivariant representative\. More precisely, there exists a jointly Borel fieldv¯:I×ℝD→ℝD\\bar\{v\}:I\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{D\}such that

v¯t=vtπμt​\-a\.e\. for a\.e\.​t∈I,\\bar\{v\}\_\{t\}=v\_\{t\}^\{\\pi\}\\qquad\\mu\_\{t\}\\text\{\-a\.e\. for a\.e\. \}t\\in I,and

v¯t​\(g​z\)=g​v¯t​\(z\)for every​g∈G,z∈ℝD,t∈I\.\\bar\{v\}\_\{t\}\(gz\)=g\\bar\{v\}\_\{t\}\(z\)\\qquad\\text\{for every \}g\\in G,\\ z\\in\\mathbb\{R\}^\{D\},\\ t\\in I\.

###### Proof\.

Let\(X,Y\)∼π\(X,Y\)\\sim\\pi,Zt≔projt​\(X,Y\)Z\_\{t\}\\coloneqq\\mathrm\{proj\}^\{t\}\(X,Y\)andU≔Y−XU\\coloneqq Y\-X\. Diagonal invariance and linearity of the action give

\(Zt,U\)​=d​\(g​Zt,g​U\)for every​g∈G\.\(Z\_\{t\},U\)\\overset\{\\mathrm\{d\}\}\{=\}\(gZ\_\{t\},gU\)\\qquad\\text\{for every \}g\\in G\.In particular, eachμt\\mu\_\{t\}isGG\-invariant\. Forg∈Gg\\in G, define

vtπ,g​\(z\)≔g−1​vtπ​\(g​z\)\.v\_\{t\}^\{\\pi,g\}\(z\)\\coloneqq g^\{\-1\}v\_\{t\}^\{\\pi\}\(gz\)\.Diagonal invariance and orthogonality giveJπ​\(vπ,g\)=Jπ​\(vπ\)J\_\{\\pi\}\(v^\{\\pi,g\}\)=J\_\{\\pi\}\(v^\{\\pi\}\)\. By uniqueness of the minimizer inL2​\(d​t​d​μt\)L^\{2\}\(\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\),

g−1vtπ\(g⋅\)=vtπdtdμt\-a\.e\.g^\{\-1\}v\_\{t\}^\{\\pi\}\(g\\,\\cdot\)=v\_\{t\}^\{\\pi\}\\qquad\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\\text\{\-a\.e\.\}for everyg∈Gg\\in G\. Choose a jointly Borel representative ofvπv^\{\\pi\}and write

F⁡\(t,z,h\)≔h−1​vtπ​\(h​z\)\.F\(t,z,h\)\\coloneqq h^\{\-1\}v\_\{t\}^\{\\pi\}\(hz\)\.The action is continuous, soFFis jointly Borel\. Orthogonality ofhhandGG\-invariance of everyμt\\mu\_\{t\}give

∫I∫ℝD∫G‖F⁡\(t,z,h\)‖2​d​λG​\(h\)​d​μt​\(z\)​𝑑t\\displaystyle\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\int\_\{G\}\\\|F\(t,z,h\)\\\|^\{2\}\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\\,\\,\\mathrm\{d\}t=∫G∫I∫ℝD‖vtπ​\(h​z\)‖2​d​μt​\(z\)​dt​d​λG​\(h\)=‖vπ‖L2​\(d​t​d​μt\)2<∞\.\\displaystyle=\\int\_\{G\}\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\\|v\_\{t\}^\{\\pi\}\(hz\)\\\|^\{2\}\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\\,\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)=\\\|v^\{\\pi\}\\\|\_\{L^\{2\}\(\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\)\}^\{2\}<\\infty\.By Tonelli’s theorem, the innerL2​\(G\)L^\{2\}\(G\)integral is finite ford​t​d​μt\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\-almost every\(t,z\)\(t,z\)\. SinceλG\\lambda\_\{G\}is a probability measure, Cauchy–Schwarz then gives

∫G‖F⁡\(t,z,h\)‖​d​λG​\(h\)<∞\\int\_\{G\}\\\|F\(t,z,h\)\\\|\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)<\\inftyfor almost every\(t,z\)\(t,z\)\. Hence the vector\-valued Haar integral exists outside ad​t​d​μt\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\-null set\. Define

v¯t​\(z\)≔∫GF⁡\(t,z,h\)​d​λG​\(h\)\\bar\{v\}\_\{t\}\(z\)\\coloneqq\\int\_\{G\}F\(t,z,h\)\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)on this set and set it to zero otherwise\. Componentwise integration gives a jointly Borel fieldv¯\\bar\{v\}\.

It remains to show that this field represents the sameL2L^\{2\}minimizer\. Since the preceding equality holdsd​t​d​μt\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\-almost everywhere for every fixedh∈Gh\\in G, another application of Tonelli’s theorem gives

∫I∫ℝD∫G‖F⁡\(t,z,h\)−vtπ​\(z\)‖2​d​λG​\(h\)​d​μt​\(z\)​𝑑t=0\.\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\}\\int\_\{G\}\\bigl\\\|F\(t,z,h\)\-v\_\{t\}^\{\\pi\}\(z\)\\bigr\\\|^\{2\}\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)\\,\\,\\mathrm\{d\}\\mu\_\{t\}\(z\)\\,\\,\\mathrm\{d\}t=0\.Consequently, ford​t​d​μt\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\-almost every\(t,z\)\(t,z\), the integrand vanishes forλG\\lambda\_\{G\}\-almost everyhh, and hencev¯t​\(z\)=vtπ​\(z\)\\bar\{v\}\_\{t\}\(z\)=v\_\{t\}^\{\\pi\}\(z\)\. Thus the averaging changes only the representative of theL2​\(d​t​d​μt\)L^\{2\}\(\\,\\mathrm\{d\}t\\,\\,\\mathrm\{d\}\\mu\_\{t\}\)minimizer\.

It remains to check that the averaged representative is exactly equivariant\. Fork∈Gk\\in G, right invariance of Haar measure and the change of variablesr=h​kr=hkgive

v¯t​\(k​z\)\\displaystyle\\bar\{v\}\_\{t\}\(kz\)=∫Gh−1​vtπ​\(h​k​z\)​d​λG​\(h\)\\displaystyle=\\int\_\{G\}h^\{\-1\}v\_\{t\}^\{\\pi\}\(hkz\)\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(h\)=∫Gk​r−1​vtπ​\(r​z\)​d​λG​\(r\)=k​v¯t​\(z\)\.\\displaystyle=\\int\_\{G\}kr^\{\-1\}v\_\{t\}^\{\\pi\}\(rz\)\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(r\)=k\\bar\{v\}\_\{t\}\(z\)\.The set where the Haar integral is not finite is itselfGG\-invariant by the same change of variables\. Therefore the equality also holds there under the zero convention, andv¯t​\(k​z\)=k​v¯t​\(z\)\\bar\{v\}\_\{t\}\(kz\)=k\\bar\{v\}\_\{t\}\(z\)for everyk∈Gk\\in G,z∈ℝDz\\in\\mathbb\{R\}^\{D\}, andt∈It\\in I\.

∎

Forη∈𝒫2​\(ℝD\)\\eta\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\)andπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\), define

ηG≔∫Gg♯​η​d​λG​\(g\),πG≔∫G\(g,g\)♯​π​d​λG​\(g\),\\eta^\{G\}\\coloneqq\\int\_\{G\}g\_\{\\sharp\}\\eta\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(g\),\\qquad\\pi^\{G\}\\coloneqq\\int\_\{G\}\(g,g\)\_\{\\sharp\}\\pi\\,\\,\\mathrm\{d\}\\lambda\_\{G\}\(g\),Then

πG∈cℝD​\(μG,νG\),\(𝔮,𝔮\)♯​πG=\(𝔮,𝔮\)♯​π\.\\pi^\{G\}\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu^\{G\},\\nu^\{G\}\),\\qquad\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\pi^\{G\}=\(\\mathfrak\{q\},\\mathfrak\{q\}\)\_\{\\sharp\}\\pi\.Haar invariance gives\(h,h\)♯​πG=πG\(h,h\)\_\{\\sharp\}\\pi^\{G\}=\\pi^\{G\}for everyh∈Gh\\in G\. Thus diagonal symmetrization preserves the coupling on the quotient, andπG∈cℝD​\(μ,ν\)\\pi^\{G\}\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)whenever both marginals areGG\-invariant\.

###### Proposition B\.2\(Minimization over equivariant fields\)\.

Letπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)be arbitrary\. Then the fieldvπGv^\{\\pi^\{G\}\}associated withπG\\pi^\{G\}minimizesJπJ\_\{\\pi\}over allGG\-equivariant fields:

vπG∈arg​minv:I×ℝD→ℝD​jointly​Borelvt​\(𝑔𝑥\)=𝑔𝑣t​\(x\)​∀t,g,xJπ\(v\)\.v^\{\\pi^\{G\}\}\\in\\argmin\_\{\\begin\{subarray\}\{c\}v:I\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{D\}\\ \\mathrm\{jointly\\ Borel\}\\\\ v\_\{t\}\(gx\)=gv\_\{t\}\(x\)\\ \\forall\\,t,g,x\\end\{subarray\}\}J\_\{\\pi\}\(v\)\.In particular, the constrained problem is the ordinary flow\-matching problem for the diagonally symmetrized couplingπG\\pi^\{G\}\.

###### Proof\.

For everyGG\-equivariant fieldvvand everyg∈Gg\\in G, orthogonality of the action gives

J\(g,g\)♯​π​\(v\)=∫I∫ℝD×ℝD‖vt​\(g​projt​\(x,y\)\)−g⁡\(y−x\)‖2​𝑑π​\(x,y\)​𝑑t=Jπ​\(v\)\.J\_\{\(g,g\)\_\{\\sharp\}\\pi\}\(v\)=\\int\_\{I\}\\int\_\{\\mathbb\{R\}^\{D\}\\times\\mathbb\{R\}^\{D\}\}\\bigl\\\|v\_\{t\}\(g\\mathrm\{proj\}^\{t\}\(x,y\)\)\-g\(y\-x\)\\bigr\\\|^\{2\}\\,\\,\\mathrm\{d\}\\pi\(x,y\)\\,\\,\\mathrm\{d\}t=J\_\{\\pi\}\(v\)\.Averaging overGGtherefore yieldsJπG​\(v\)=Jπ​\(v\)J\_\{\\pi^\{G\}\}\(v\)=J\_\{\\pi\}\(v\)for every equivariantvv\. SinceπG\\pi^\{G\}is diagonallyGG\-invariant, its unrestricted minimizervπGv^\{\\pi^\{G\}\}admits aGG\-equivariant representative\. AsvπGv^\{\\pi^\{G\}\}minimizesJπGJ\_\{\\pi^\{G\}\}over all fields, it also minimizesJπJ\_\{\\pi\}over the equivariant ones\. ∎

###### Proposition B\.3\(Equivariant flows on the quotient\)\.

Letv:I×ℝD→ℝDv:I\\times\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{D\}beGG\-equivariant and assume that \([3](https://arxiv.org/html/2608.26961#S2.E3)\) admits a unique flowγt\\gamma\_\{t\}\. Thenγt\\gamma\_\{t\}induces the well\-defined quotient flow

γ¯t​\(\[x\]\)≔\[γt​\(x\)\]\.\\bar\{\\gamma\}\_\{t\}\(\[x\]\)\\coloneqq\[\\gamma\_\{t\}\(x\)\]\.For everyη∈𝒫2​\(ℝD\)\\eta\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\)andt∈It\\in I,

𝔮♯​γt,♯​η=γ¯t,♯​𝔮♯​η\.\\mathfrak\{q\}\_\{\\sharp\}\\gamma\_\{t,\\sharp\}\\eta=\\bar\{\\gamma\}\_\{t,\\sharp\}\\mathfrak\{q\}\_\{\\sharp\}\\eta\.In particular, if𝔮♯​η=𝔮♯​η~\\mathfrak\{q\}\_\{\\sharp\}\\eta=\\mathfrak\{q\}\_\{\\sharp\}\\widetilde\{\\eta\}, then

𝔮♯​γt,♯​η=𝔮♯​γt,♯​η~,\\mathfrak\{q\}\_\{\\sharp\}\\gamma\_\{t,\\sharp\}\\eta=\\mathfrak\{q\}\_\{\\sharp\}\\gamma\_\{t,\\sharp\}\\widetilde\{\\eta\},so the projected curve is independent of the representative lift at time00\.

###### Proof\.

Equivariance ofvvand uniqueness of the ODE implyγt​\(g​x\)=g​γt​\(x\)\\gamma\_\{t\}\(gx\)=g\\gamma\_\{t\}\(x\)\. Thusγ¯t\\bar\{\\gamma\}\_\{t\}is well\-defined and𝔮∘γt=γ¯t∘𝔮\\mathfrak\{q\}\\circ\\gamma\_\{t\}=\\bar\{\\gamma\}\_\{t\}\\circ\\mathfrak\{q\}, which gives

𝔮♯​γt,♯​η=γ¯t,♯​𝔮♯​η\.\\mathfrak\{q\}\_\{\\sharp\}\\gamma\_\{t,\\sharp\}\\eta=\\bar\{\\gamma\}\_\{t,\\sharp\}\\mathfrak\{q\}\_\{\\sharp\}\\eta\.The final claim follows by applying this identity to two measures with the same quotient push\-forward\. ∎

The coordinatewise categorical formulation in Section[2\.2](https://arxiv.org/html/2608.26961#S2.SS2.SSS0.Px1)is directly compatible with coordinate permutation actions\.

###### Corollary B\.4\(Coordinatewise equivariant endpoint prediction\)\.

Assume thatG⊂𝔖DG\\subset\\mathfrak\{S\}\_\{D\}acts by coordinate permutations, whereg⋅dg\\cdot dis determined by\(g​x\)g⋅d=xd\(gx\)\_\{g\\cdot d\}=x\_\{d\}, and let𝒜=\{a1,…,aM\}⊂ℝ\.\\mathcal\{A\}=\\\{a\_\{1\},\\ldots,a\_\{M\}\\\}~\\subset~\\mathbb\{R\}\.Letμ,ν∈𝒫2​\(ℝD\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{D\}\)be arbitrary, assume thatν\\nuis supported on𝒜D\\mathcal\{A\}^\{D\}, and letπ∈cℝD​\(μ,ν\)\\pi\\in\\mathrm\{c\}\_\{\\mathbb\{R\}^\{D\}\}\(\\mu,\\nu\)\. Then the conditional coordinate probabilitiesq∗,πG,dq^\{\\ast,\\pi^\{G\},d\}associated withπG\\pi^\{G\}admit a representative satisfying

qt∗,πG,g⋅d​\(n∣g​z\)=qt∗,πG,d​\(n∣z\)q\_\{t\}^\{\\ast,\\pi^\{G\},g\\cdot d\}\(n\\mid gz\)=q\_\{t\}^\{\\ast,\\pi^\{G\},d\}\(n\\mid z\)for everyt∈It\\in I,g∈Gg\\in G,d∈\[D\]d\\in\[D\],n∈\[M\]n\\in\[M\], andz∈ℝDz\\in\\mathbb\{R\}^\{D\}, and minimizeJπcoordJ\_\{\\pi\}^\{\\rm coord\}over all such equivariant coordinate kernels:

\(q∗,πG,d\)d=1D∈arg​minqd:I×ℝD→ΔM​jointly​Borelqtg⋅d​\(n∣𝑔𝑧\)=qtd​\(n∣z\)Jπcoord\(\(qd\)d=1D\)\.\\bigl\(q^\{\\ast,\\pi^\{G\},d\}\\bigr\)\_\{d=1\}^\{D\}\\in\\argmin\_\{\\begin\{subarray\}\{c\}q^\{d\}:I\\times\\mathbb\{R\}^\{D\}\\to\\Delta\_\{M\}\\ \\mathrm\{jointly\\ Borel\}\\\\ q\_\{t\}^\{g\\cdot d\}\(n\\mid gz\)=q\_\{t\}^\{d\}\(n\\mid z\)\\end\{subarray\}\}J\_\{\\pi\}^\{\\rm coord\}\\bigl\(\(q^\{d\}\)\_\{d=1\}^\{D\}\\bigr\)\.WritingμtG≔proj♯t​πG\\mu\_\{t\}^\{G\}\\coloneqq\\mathrm\{proj\}^\{t\}\_\{\\sharp\}\\pi^\{G\}, for a\.e\.t<1t<1andμtG\\mu\_\{t\}^\{G\}\-a\.e\.zz,

\(vtπG​\(z\)\)d=11−t​\(∑n=1Man​qt∗,πG,d​\(n∣z\)−zd\)\.\\bigl\(v\_\{t\}^\{\\pi^\{G\}\}\(z\)\\bigr\)\_\{d\}=\\frac\{1\}\{1\-t\}\\left\(\\sum\_\{n=1\}^\{M\}a\_\{n\}q\_\{t\}^\{\\ast,\\pi^\{G\},d\}\(n\\mid z\)\-z\_\{d\}\\right\)\.In particular, this defines aGG\-equivariant representative ofvπGv^\{\\pi^\{G\}\}, without requiringμ\\muorν\\nuto beGG\-invariant\.

###### Proof\.

SinceG⊂𝔖DG\\subset\\mathfrak\{S\}\_\{D\}is finite, diagonal invariance ofπG\\pi^\{G\}allows us to choose the conditional coordinate probabilities equivariantly\. If\(X,Y\)∼πG\(X,Y\)\\sim\\pi^\{G\}andZt=projt​\(X,Y\)Z\_\{t\}=\\mathrm\{proj\}^\{t\}\(X,Y\), then\(Zt,Y\)​=d​\(g​Zt,g​Y\)\(Z\_\{t\},Y\)\\overset\{\\mathrm\{d\}\}\{=\}\(gZ\_\{t\},gY\)for everyg∈Gg\\in G\. Consequently, starting from any jointly Borel versionqq, eachqtg⋅d​\(n∣g​z\)q\_\{t\}^\{g\\cdot d\}\(n\\mid gz\)is a version of the same conditional probability\. Define

q¯td​\(n∣z\)≔1\|G\|​∑g∈Gqtg⋅d​\(n∣g​z\)\.\\bar\{q\}\_\{t\}^\{d\}\(n\\mid z\)\\coloneqq\\frac\{1\}\{\|G\|\}\\sum\_\{g\\in G\}q\_\{t\}^\{g\\cdot d\}\(n\\mid gz\)\.This is another version of the same conditional probabilities and satisfies the displayed equivariance identity for everyg,d,n,t,zg,d,n,t,z\. We use this version below\.

For every equivariant coordinate kernelqq, relabelling the coordinates and summing overd∈\[D\]d\\in\[D\]gives

J\(g,g\)♯​πcoord​\(q\)=Jπcoord​\(q\)\.J\_\{\(g,g\)\_\{\\sharp\}\\pi\}^\{\\rm coord\}\(q\)=J\_\{\\pi\}^\{\\rm coord\}\(q\)\.HenceJπGcoord​\(q\)=Jπcoord​\(q\)J\_\{\\pi^\{G\}\}^\{\\rm coord\}\(q\)=J\_\{\\pi\}^\{\\rm coord\}\(q\)on the equivariant class\. By \([7](https://arxiv.org/html/2608.26961#S2.E7)\), these probabilities minimize the unrestricted coordinatewise objective forπG\\pi^\{G\}, and therefore the constrained objective forπ\\pi\.

For any equivariant coordinate kernelqq, define

vtq​\(z\)d≔11−t​\(∑n=1Man​qtd​\(n∣z\)−zd\)\.v\_\{t\}^\{q\}\(z\)\_\{d\}\\coloneqq\\frac\{1\}\{1\-t\}\\left\(\\sum\_\{n=1\}^\{M\}a\_\{n\}q\_\{t\}^\{d\}\(n\\mid z\)\-z\_\{d\}\\right\)\.Then

vtq​\(g​z\)g⋅d=∑n=1Man​qtg⋅d​\(n∣g​z\)−\(g​z\)g⋅d1−t=vtq​\(z\)d\.v\_\{t\}^\{q\}\(gz\)\_\{g\\cdot d\}=\\frac\{\\sum\_\{n=1\}^\{M\}a\_\{n\}q\_\{t\}^\{g\\cdot d\}\(n\\mid gz\)\-\(gz\)\_\{g\\cdot d\}\}\{1\-t\}=v\_\{t\}^\{q\}\(z\)\_\{d\}\.Thusvtq​\(g​z\)=g​vtq​\(z\)v\_\{t\}^\{q\}\(gz\)=gv\_\{t\}^\{q\}\(z\), independently of the invariance of the endpoint measures\. Takingqd=q∗,πG,dq^\{d\}=q^\{\\ast,\\pi^\{G\},d\}for everyd∈\[D\]d\\in\[D\]and using \([6](https://arxiv.org/html/2608.26961#S2.E6)\) proves the remaining claims\. ∎

## Appendix CRelated Work on Transport\-Based Graph Generation

Graph generators commonly enforce node\-relabeling symmetry through equivariant architectures\[[Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34),[Vignac et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib32),[Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26961#bib.bib31)\]\. Recent graph flow models additionally exploit transport geometry\. GGFlow uses minibatch OT for graph pairing\[[Hou et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib45)\], while BWFlow constructs paths from Bures–Wasserstein transport between graph representations\[[Jiang et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib29)\]\. Concurrent Flowette introduces a graph\-structured graphette prior and uses fused Gromov–Wasserstein distances for structure\-aware minibatch pairing\[[Wijesinghe et al\., 2026](https://arxiv.org/html/2608.26961#bib.bib51)\]\. Both GGFlow and Flowette therefore use transport primarily to choose which source and target graphs to pair\. In contrast, we use the GW plan itself to select a hard node relabeling before interpolation, explicitly separating this inner Gromov–Monge alignment from the optional outer graph assignment\. Our ablations show that much of the improvement already comes from the inner alignment, while outer graph matching provides a smaller or dataset\-dependent additional benefit\. This connects our construction to GW\-based graph matching and alignment\[[Peyré et al\., 2016](https://arxiv.org/html/2608.26961#bib.bib9),[Vayer et al\., 2020](https://arxiv.org/html/2608.26961#bib.bib6),[Xu et al\., 2019](https://arxiv.org/html/2608.26961#bib.bib52)\], while deriving both operations from transport on the permutation quotient\.

## Appendix DConditional continuous\-SBM experiment

We additionally condition the continuous SBM model on the community countKK, form minibatches containing only one value ofKK, and compare pooled generated and real graphs with matchingKK\-multisets\. All other settings and metrics agree with Section[5\.2](https://arxiv.org/html/2608.26961#S5.SS2)\. The conditional results in Table[4](https://arxiv.org/html/2608.26961#A4.T4)reproduce the unconditional ranking: GW alignment has the largest advantage at five Euler steps, and the gap narrows with the integration budget\.

Table 4:Continuous SBM,*class\-conditional*onK∈\{1,…,5\}K\\in\\\{1,\\dots,5\\\}\(N=10N=10\), evaluated by pooling generated and real graphs with the same ground\-truthKKmultiset\. Mean±\\pmstd over33training seeds \(55evaluation repeats each\) at5/25/1255/25/125Euler steps\. Descriptor MMDs: lower is better; FGW–NNA: closest to0\.50\.5is better\. Best per column in bold\.
## Appendix EFull\-budget molecular training protocol

##### Protocol

For the full\-budget evaluation, we retain the categorical endpoint objective and use GW inner alignment with independent outer pairing\. We increase the model capacities and training budgets toward those used by DeFoG\[[Qin et al\., 2025](https://arxiv.org/html/2608.26961#bib.bib28)\]\. In particular, we augment the transformer inputs with structural graph statistics and relative random\-walk probabilities \(RRWP\), and use self\-conditioning\. For our self\-conditioning implementation, on50%50\\%of optimization steps, the model is conditioned on a detached preliminary endpoint prediction\[[Chen et al\., 2023](https://arxiv.org/html/2608.26961#bib.bib46)\]\. During sampling, each Euler step receives the preceding step’s endpoint prediction\. As regularization during training, Gaussian noise with standard deviation0\.5​t​\(1−t\)0\.5\\,t\(1\-t\)is added to the interpolated network input while leaving the endpoint target unchanged\.

The categorical objective consists of a bond cross\-entropy term and a node\-feature term, the latter averaging the atom\-type and formal\-charge cross\-entropies\. These terms are weighted equally for QM9\. For ZINC250k, we multiply the bond term by55relative to the node\-feature term, following the stronger emphasis on edge prediction used by DeFoG and CatFlow\.

Both QM9 and ZINC250k are trained in two stages\. We first train without dropout, then restore the model and EMA weights and fine\-tune with dropout0\.10\.1using a new AdamW optimizer\. All reported full\-budget numbers in this section use the final EMA checkpoint after dropout fine\-tuning; the dropout\- free checkpoints are only used to initialize the second stage\.

The QM9 model has node, edge, and global widths256/64/64256/64/64, feed\-forward widths256/128/128256/128/128, depth99,88attention heads, and1212RRWP channels, for approximately6\.06\.0M parameters\. It is first trained for10001000epochs with batch size10241024without dropout, and then fine\-tuned for5050additional epochs with dropout0\.10\.1\. The ZINC250k model has widths256/64/128256/64/128, feed\-forward widths256/128/256256/128/256, depth1212,88heads, and2020RRWP channels, for approximately10\.810\.8M parameters\. It is first trained for300300epochs using microbatches of128128with two\-step gradient accumulation, giving an effective batch size of256256, and then fine\-tuned for3030additional epochs with dropout0\.10\.1\. The initial training runs use AdamW with learning rate2×10−42\\times 10^\{\-4\}, weight decay10−410^\{\-4\}, gradient\-norm clipping at1\.01\.0, cosine learning\-rate decay, and EMA decay0\.9990\.999\.

The dropout fine\-tuning stages restore both model and EMA weights, use a new AdamW optimizer at learning rate5×10−55\\times 10^\{\-5\}, and do not use cosine decay\. We report only the final fine\-tuned EMA checkpoints, evaluated using500500Euler steps\.

##### ZINC ablation\.

Table 5:Cumulative ZINC250k ablation at2525Euler steps and reduced training budget\. Higher validity is better and lower FCD is better\.To isolate the contribution of the main additions used in the full\-budget ZINC model, we run the cumulative ablation in Table[5](https://arxiv.org/html/2608.26961#A5.T5)on ZINC250k using the smaller 100\-epoch architecture from the base molecular experiments\. This model has node, edge, and global widths128/64/128128/64/128, depth66,88attention heads, and about2\.82\.8M parameters\. All rows use GW inner alignment, independent outer pairing, formal\-charge features, the categorical endpoint objective, batch size3232, and are trained for100100epochs\. We evaluate final EMA checkpoints with the standard molecular protocol using10,00010\{,\}000samples and33repeats, but only at2525Euler steps\. The expanded computational budget for the full run explains the remaining performance gap\.

## Appendix FReproducibility details

This appendix collects the architecture, data, noise, solver, and metric specifications needed to reproduce the experiments of Section[5](https://arxiv.org/html/2608.26961#S5)\. All coupling conditions within an experiment share every model and optimization setting listed here and differ only in their training coupling, namely the inner solverσ^a\\widehat\{\\sigma\}\_\{a\}and outer reorderingτ^\\widehat\{\\tau\}of Section[4](https://arxiv.org/html/2608.26961#S4), with MINIBATCHOT replacing both by its permutation\-blind minibatch assignment\.

##### Architecture and optimization

Every model is the sameGNgraphG\_\{N\}^\{\\mathrm\{graph\}\}\-equivariant graph transformer with node \(XX\), edge \(EE\), and global \(yy\) streams, following the XEy architecture of[Vignac et al\. \[2023\]](https://arxiv.org/html/2608.26961#bib.bib32)\. Its node, edge, and global widths aredX=128d\_\{X\}=128,dE=64d\_\{E\}=64, anddy=128d\_\{y\}=128, respectively, with feed\-forward widths256/128/256256/128/256, depth66,88attention heads, and∼2\.8\\sim\\\!2\.8M parameters\. Training uses AdamW \(weight decay10−410^\{\-4\}\), gradient\-norm clipping at1\.01\.0, a cosine learning\-rate schedule with base rate2×10−42\\times 10^\{\-4\}, and an exponential moving average of the weights with decay0\.9990\.999; the EMA weights are used for all evaluation and checkpointing\. The per\-experiment channel countCC, node countNN, batch size, epoch budget, and training head are summarized in Table[6](https://arxiv.org/html/2608.26961#A6.T6)\. The SBM models use an MSE velocity loss \([10](https://arxiv.org/html/2608.26961#S4.E10)\); the molecular models use the coordinatewise categorical endpoint head with the cross\-entropy objective \([11](https://arxiv.org/html/2608.26961#S4.E11)\), with the velocity recovered through \([6](https://arxiv.org/html/2608.26961#S2.E6)\)\. All experiments use at most a single NVIDIA GeForce RTX 5090 GPU with 32 GB of VRAM\.

Table 6:Per\-experiment settings for the controlled coupling comparisons, shared across all coupling conditions\. Architecture, optimizer, EMA, gradient clipping, and learning\-rate schedule are identical throughout \(see text\); only the entries below vary\. The larger\-model protocol is specified separately in Appendix[E](https://arxiv.org/html/2608.26961#A5)\. “Head” is the flow\-matching parametrization: MSE on the velocity field, or cross\-entropy on the predicted categorical endpoint\.
##### Cycle\-graph data

Both the source and the target are cycle graphs onN=12N=12nodes, so this experiment replaces the noise source of the other datasets by a structured one\. A sample places twelve nodes at equispaced angles, offset by a rotation drawn uniformly from\[0,2​π\)\[0,2\\pi\), on the circle of radius11centred at\(h,y\)\(h,y\), and joins two nodes exactly when they are neighbours along that circle\. Nodes are then relabelled by a uniformly random permutation, so the index carries no information and recovering the circular order is precisely the task of the inner alignment\. The tensorE∈ℝ12×12×3E\\in\\mathbb\{R\}^\{12\\times 12\\times 3\}has adjacency in\{0,1\}\\\{0,1\\\}as its first channel, while the remaining two channels store the planar node positions on the diagonal,Ei​i,2:3∈ℝ2E\_\{ii,2:3\}\\in\\mathbb\{R\}^\{2\}, matching the diagonal node\-feature convention used throughout\. Positions are stored divided by88, which puts the position and adjacency channels on comparable scales in both the GW cost and the MSE loss\. The source law takesy=0y=0and the target lawy=8y=8, withh∼𝒰⁡\[−6,6\]h\\sim\\mathcal\{U\}\[\-6,6\]drawn independently on each side; the two therefore agree up to a translation, and in particular have identical edge laws\. We draw40004000target graphs, useλedge=λnode=0\.5\\lambda\_\{\\mathrm\{edge\}\}=\\lambda\_\{\\mathrm\{node\}\}=0\.5as in the other experiments, and train for200200epochs at batch size1616with the velocity \(MSE\) head\.

##### Continuous SBM data

A graph onN=10N=10nodes withKKcommunities assigns the nodes to fixed near\-balanced blocks \(remainder placed in the first blocks, e\.g\.K=3→4/3/3K=3\\to 4/3/3\) in a randomized order\. ParametrizingBeta⁡\(a,b\)\\mathrm\{Beta\}\(a,b\)by its meanμ\\muand concentrationssviaa=s​μa=s\\mu,b=s⁡\(1−μ\)b=s\(1\-\\mu\), every unordered pairi<ki<kcarries an edge weightBeta⁡\(se​μin,se​\(1−μin\)\)\\mathrm\{Beta\}\(s\_\{e\}\\mu\_\{\\mathrm\{in\}\},s\_\{e\}\(1\-\\mu\_\{\\mathrm\{in\}\}\)\)wheni,ki,kshare a block andBeta⁡\(se​μout,se​\(1−μout\)\)\\mathrm\{Beta\}\(s\_\{e\}\\mu\_\{\\mathrm\{out\}\},s\_\{e\}\(1\-\\mu\_\{\\mathrm\{out\}\}\)\)otherwise, withμin=0\.75\\mu\_\{\\mathrm\{in\}\}=0\.75,μout=0\.25\\mu\_\{\\mathrm\{out\}\}=0\.25, andse=8s\_\{e\}=8\(i\.e\.Beta⁡\(6,2\)\\mathrm\{Beta\}\(6,2\)andBeta⁡\(2,6\)\\mathrm\{Beta\}\(2,6\)\)\. The diagonal node feature of a node in blockkkisBeta⁡\(sn​μk,sn​\(1−μk\)\)\\mathrm\{Beta\}\(s\_\{n\}\\mu\_\{k\},s\_\{n\}\(1\-\\mu\_\{k\}\)\)with evenly spaced meansμk=\(2​k\+1\)/\(2​K\)\\mu\_\{k\}=\(2k\+1\)/\(2K\)andsn=8s\_\{n\}=8; forK=2K=2this reproduces the binary SBM feature model exactly\. In the unconditional variant graphs withK∈\{1,…,5\}K\\in\\\{1,\\dots,5\\\}are pooled; in the conditional variantKKis the class label\. In the unconditional variant,20002000graphs are drawn for eachK∈\{1,…,5\}K\\in\\\{1,\\dots,5\\\}and pooled \(10 00010\\,000in total\)\. In the conditional variant, the same per\-class sets are used, withKKsupplied to the model as the class label\.

##### Molecular data

Molecules are encoded from their RDKit representation after kekulization, so bond orders are integer\-valued\. A molecule onNNheavy atoms becomes a tensorE∈ℝN×N×CE\\in\\mathbb\{R\}^\{N\\times N\\times C\}whose edge channel is a one\-hot over\{none,single,double,triple\}\\\{\\text\{none\},\\text\{single\},\\text\{double\},\\text\{triple\}\\\}and whose diagonal node feature concatenates an atom\-type one\-hot with a formal\-charge one\-hot over\{−1,0,\+1\}\\\{\-1,0,\+1\\\}\. QM9 uses heavy atomsN≤9N\\leq 9over\{\\\{C,N,O,F\}\\\}, givingC=11C=11; ZINC250k usesN≤38N\\leq 38over\{\\\{C,N,O,F,Br,Cl,I,P,S\}\\\}, givingC=16C=16\. For ZINC250k the2828\-way PyTorch Geometric atom vocabulary—which encodes charged species—is remapped to the nine element types listed above and to a signed formal charge in\{−1,0,\+1\}\\\{\-1,0,\+1\\\};31\.8%31\.8\\%of ZINC250k molecules carry a formal charge, and modelling it explicitly is required for those atoms to pass valence on decode\. QM9 uses the GDSS train/test split, matched on raw QM9 record identifiers \(valid\_idx\_qm9\.json\); ZINC250k uses the standard PyTorch Geometric split with220,011220\{,\}011training,24,44524\{,\}445validation, and5,0005\{,\}000test molecules\. FCD is evaluated against the respective test split\.

##### Variable node count

For the molecular datasets, one set of graph\-transformer parameters is shared across all node counts\. An input withNNnodes produces an output withNNnodes\. Source noise for each target is drawn at the target’s own node count, so no padding or masking is used\. During training, minibatches are formed by a sampler that groups dataset indices by exactNNand yields same\-NNbatches; this is required because the outer cost matrices of Section[4](https://arxiv.org/html/2608.26961#S4)compare same\-size tensors\. In the conditional SBM variant, each batch additionally contains only graphs with a single value ofKK, so graphs with different community counts are never coupled\. At generation timeNNis sampled from the empirical training\-set node\-count distribution and the Euler solver is initialized from noise of shape\(N,N,C\)\(N,N,C\)\.

##### Source noise

For the molecular datasets the source is a symmetric Gaussian\. For each edge channel and eachi<ji<j, we drawUi​j,Uj​i​∼iid​𝒩​\(0,σadj2\)U\_\{ij\},U\_\{ji\}\\overset\{\\mathrm\{iid\}\}\{\\sim\}\\mathcal\{N\}\(0,\\sigma\_\{\\rm adj\}^\{2\}\)and setAi​j=Aj​i=Ui​j\+Uj​i2,A\_\{ij\}=A\_\{ji\}=\\frac\{U\_\{ij\}\+U\_\{ji\}\}\{\\sqrt\{2\}\},so that each off\-diagonal edge entry satisfiesAi​j∼𝒩⁡\(0,σadj2\)\.A\_\{ij\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\rm adj\}^\{2\}\)\.The edge diagonal is zero\. Each diagonal node feature is drawn independently from𝒩⁡\(0,σnode2\)\\mathcal\{N\}\(0,\\sigma\_\{\\rm node\}^\{2\}\), withσadj=σnode=0\.5\\sigma\_\{\\rm adj\}=\\sigma\_\{\\rm node\}=0\.5\. For the continuous SBM the source is independent𝒰⁡\[0,1\]\\mathcal\{U\}\[0,1\]on both channels\. The upper\-triangular edge entries are mirrored across the diagonal, and the diagonal node features are drawn independently\. This matches the\(0,1\)\(0,1\)scale of the dense Beta target\. The edge diagonal is again zero\.

##### Coupling solvers

All fused solvers use the combined graph entries from Section[4](https://arxiv.org/html/2608.26961#S4), whose squared norms are

∥Ei​k∥2=λedge∥ei​kE∥2\+λnodeCv∥fiE∥21\{i=k\},λedge=λnode=12,\\lVert E\_\{ik\}\\rVert^\{2\}=\\lambda\_\{\\rm edge\}\\,\\lVert e^\{E\}\_\{ik\}\\rVert^\{2\}\+\\tfrac\{\\lambda\_\{\\rm node\}\}\{C\_\{\\rm v\}\}\\,\\lVert f^\{E\}\_\{i\}\\rVert^\{2\}\\,\\mathbf\{1\}\_\{\\\{i=k\\\}\},\\qquad\\lambda\_\{\\rm edge\}=\\lambda\_\{\\rm node\}=\\tfrac\{1\}\{2\},where node features enter only on the diagonal\. The inner Gromov–Wasserstein solver \(GW\) runs1010Frank–Wolfe iterations, each linearized by an exact optimal\-transport \(Hungarian\) step, and the final soft plan is projected to the Frobenius\-nearest scaled permutation matrix using the Hungarian assignment in Section[4](https://arxiv.org/html/2608.26961#S4)\. When GW is used for outer batch matching, we compute the pairwise batch\-cost matrix with55Frank–Wolfe iterations before applying the Hungarian assignment over the batch\. The first lower bound \(FLB\) ranks nodes by the root\-mean\-square eccentricityeccE⁡\(i\)=\(N−1​∑k∥Ei​k∥2\)1/2\\operatorname\{ecc\}\_\{E\}\(i\)=\(N^\{\-1\}\\sum\_\{k\}\\lVert E\_\{ik\}\\rVert^\{2\}\)^\{1/2\}defined in Section[4](https://arxiv.org/html/2608.26961#S4)\. It sorts the nodes in each graph by this value and pairs nodes at the same position in the two orderings\. Computing the eccentricities and sorting costsO⁡\(N2​C\+N​log⁡N\)O\(N^\{2\}C\+N\\log N\)per pair\. For outer reordering, each same\-NNtraining batch is partitioned into sub\-batches of at most eight graphs\. Within each sub\-batch, we build the pairwise FLB cost matrix and apply the Hungarian algorithm to permute the target graphs\.

##### Evaluation metrics

The descriptor MMDs follow the GraphRNN/GDSS conventions\[[Jo et al\., 2022](https://arxiv.org/html/2608.26961#bib.bib34),[You et al\., 2018](https://arxiv.org/html/2608.26961#bib.bib33)\]with the biased estimator and kernels of the formk\(x,y\)=exp\[−d\(x,y\)2/\(2σ2\)\]k\(x,y\)=\\exp\[\-d\(x,y\)^\{2\}/\(2\\sigma^\{2\}\)\]\. Degree uses earth mover’s distance \(EMD\) withσ=1\.0\\sigma=1\.0on per\-graph degree histograms; clustering uses EMD over100100bins on\[0,1\]\[0,1\]withσ=10\\sigma=10in raw\-bin units \(equivalentlyσ=0\.1\\sigma=0\.1in the\[0,1\]\[0,1\]\-normalized EMD units of GraphRNN/GDSS\)\. Graphlet orbit uses Euclidean distance in a Gaussian radial basis function kernel withσ=30\.0\\sigma=30\.0on the1515\-dimensional, per\-node\-averaged four\-node graphlet\-orbit counts computed with the ORCA graphlet\-counting program\[[Hočevar and Demšar, 2014](https://arxiv.org/html/2608.26961#bib.bib47)\]\. FGW–NNA pools thenngenerated andnnreal graphs and classifies each by the label of its nearest neighbor \(self excluded\) under the fused Gromov–Wasserstein distance withλedge=λnode=12\\lambda\_\{\\rm edge\}=\\lambda\_\{\\rm node\}=\\tfrac\{1\}\{2\}; under the null this accuracy tends to1/21/2\. For molecules, validity is the fraction of generated graphs that decode to a sanitizable RDKit molecule without valence correction or other postprocessing\. For FCD, we retain the largest connected fragment of each sanitizable generated molecule, canonicalize it, remove duplicates, and compare the ChemNet embeddings of these unique generated SMILES with the corresponding unique test\-split SMILES\.

## Appendix GSolver cost and bound tightness

Table[7](https://arxiv.org/html/2608.26961#A7.T7)quantifies the trade\-off between the 10\-iteration Frank–Wolfe Gromov–Wasserstein approximation \(GW\) and the first lower bound \(FLB\) on real data\. For each dataset we draw300300pairs of distinct real graphs at a single fixed node countNN\(chosen as the modal size with enough graphs:N=10N=10for the continuous SBM,N=9N=9for QM9,N=23N=23for ZINC250k, using the categorical molecular encoding of Section[5\.3](https://arxiv.org/html/2608.26961#S5.SS3)\), and for every pair we compute both theGWcost—the fused objective at the final1010\-iteration plan—and theFLBcostW22W\_\{2\}^\{2\}between the sorted node\-eccentricity distributions, withλedge=λnode=12\\lambda\_\{\\rm edge\}=\\lambda\_\{\\rm node\}=\\tfrac\{1\}\{2\}\. Timings are single\-threaded on CPU, so they reflect the per\-pair alignment cost rather than batched throughput\.

Table 7:Cost and tightness ofFLBrelative to computedGW, over300300same\-NNreal graph pairs per dataset\. Runtime is milliseconds per pair \(CPU, single thread\)\. Tightness ismean⁡\(FLB\)/mean⁡\(GW\)\\operatorname\{mean\}\(\\textsc\{FLB\}\)/\\operatorname\{mean\}\(\\textsc\{GW\}\); sinceFLBis a lower bound this is at most100%100\\%\. Correlation is the Pearson \(and, in parentheses, Spearman rank\) correlation between theGWandFLBcosts across the300300pairs\.FLBis cheaper thanGWby roughly21×21\\times\(N=10N=10\),34×34\\times\(N=9N=9\), and44×44\\times\(N=23N=23\), and the gap widens with the node count, as expected from theO⁡\(N2​C\+N​log⁡N\)O\(N^\{2\}C\+N\\log N\)versusO⁡\(N3×iters\)O\(N^\{3\}\\times\\text\{iters\}\)scaling\. However, it is numerically loose and only weakly correlated withGW\. Its empirical benefit should therefore be interpreted as that of a cheap alignment heuristic, rather than evidence that it accurately approximates or screens for theGWcost\. For further lower\-bound comparisons, see\[[Piening and Beinert, 2025](https://arxiv.org/html/2608.26961#bib.bib13)\]\.

Table[8](https://arxiv.org/html/2608.26961#A7.T8)lifts this comparison to the batch level, at the batch sizes actually used\. TheGWrows dominate the gradient step, since every aligned pair needs its own Frank–Wolfe solve\. They are pair\-separable, so a1212\-worker pool cuts the inner QM9 alignment to about2828ms and the reported shares are upper bounds\.

Table 8:Total alignment cost per training batch under the settings used in Section[5](https://arxiv.org/html/2608.26961#S5), in milliseconds \(CPU, single thread\), together with its approximate share of one gradient step\. Inner\-only rows pay forBBnode alignments; “\+out\+\\,\\mathrm\{out\}” rows additionally pay for the batch reordering, in sub\-batches of eight\.MinibatchOTuses no inner alignment and assigns over the full minibatch\. The share of a gradient step is approximate, obtained from model forward/backward times in the training logs\.