元测度的垂直切片瓦瑟斯坦距离

# Vertical Sliced Wasserstein Distancesfor Meta-Measures
Source: [https://arxiv.org/html/2610.20743](https://arxiv.org/html/2610.20743)
Ahmadreza MoradipariAffiliation:Toyota Motor North America, InfoTech LabsKyungtae HanAffiliation:Toyota Motor North America, InfoTech LabsHamed PirsiavashAffiliation:Department of Computer Science, University of California, Davis Yikun BaiAffiliation:Department of Mathematics, Purdue UniversityRocio Diaz MartinAffiliation:Department of Mathematics, Florida State UniversitySoheil KolouriAffiliation:elaheh\.akbari@vanderbilt\.eduAffiliation:College of Connected Computing, Vanderbilt University

###### Abstract

Many modern learning problems require comparing probability distributions whose individual observations are themselves probability measures, naturally giving rise to*meta\-measures*in𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. However, optimal transport distances on this space can be prohibitively expensive to evaluate\. We introduce the*vertical sliced Wasserstein*\(VSW\\mathrm\{VSW\}\) distance, a scalable metric on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. Rather than slicing along the transport geometry of𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\),VSW\\mathrm\{VSW\}projects each inner measure using continuous test functions and compares the resulting one\-dimensional distributions with one\-dimensional Wasserstein distance\. We prove thatVSW\\mathrm\{VSW\}is a valid metric on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), metrizes weak convergence, and admits dimension\-independent statistical convergence guarantees\. Empirically,VSW\\mathrm\{VSW\}provides high accuracy and efficiency compared with existing distances for meta\-measures\. We further derive a practical particle gradient for the empiricalVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}objective and demonstrate its use for gradient\-based transport of meta\-measures\.VSW\\mathrm\{VSW\}provides a simple and scalable alternative for comparing and optimizing distributions of distributions\.

## 1Introduction and Related Work

Many learning problems naturally involve*distributions of distributions*\(DoDs\): a dataset can be viewed as a distribution over class\-conditional distributions\([Alvarez\-Melis and Fusi, 2020](https://arxiv.org/html/2610.20743#bib.bib2)\), a collection of shapes as a distribution over point clouds where each point cloud is itself an empirical distribution\([Haviv et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib22)\), and documents, cell populations, or mixture models can similarly be represented as collections of probability measures\([Yurochkin et al\., 2019](https://arxiv.org/html/2610.20743#bib.bib48);[Nguyen and Mueller, 2026](https://arxiv.org/html/2610.20743#bib.bib33)\)\. Formally, these objects are*meta\-measures*ℙ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), i\.e\., probability measures whose observations are themselves probability measures onΩ\\Omega\. Comparing such objects requires accounting for variation both*within*each inner distribution and*across*the population of distributions\.

Wasserstein distances on meta\-measures\.A natural metric for this setting is Wasserstein\-over\-Wasserstein \(WoW\), which solves an outer optimal transport \(OT\) problem using Wasserstein distance between inner measures as its ground cost\. Related constructions appear in hierarchical OT\([Yurochkin et al\., 2019](https://arxiv.org/html/2610.20743#bib.bib48)\), generative modeling\([Dukler et al\., 2019](https://arxiv.org/html/2610.20743#bib.bib16)\), and the optimal transport dataset distance \(OTDD\)\([Alvarez\-Melis and Fusi, 2020](https://arxiv.org/html/2610.20743#bib.bib2)\)\. While geometrically appealing, WoW is expensive: comparing empirical meta\-measures requires many inner OT problems and inherits the statistical and computational difficulties of Wasserstein distance at both levels\([Dukler et al\., 2019](https://arxiv.org/html/2610.20743#bib.bib16);[Bonet et al\., 2025b](https://arxiv.org/html/2610.20743#bib.bib9)\)\. Additionally,[Bonet et al\. \(2025b\)](https://arxiv.org/html/2610.20743#bib.bib9)develop WoW gradient flows and introduce tractable MMD objectives with Sliced\-Wasserstein kernels\.

Sliced and scalable meta\-measure distances\.Recent methods therefore seek scalable alternatives through slicing\. Double\-Sliced Wasserstein \(DSW\)\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\)uses one\-dimensional Wasserstein projections and quantile representations, Busemann\-based approaches slice along geodesic directions of Wasserstein space\([Bonet et al\., 2025a](https://arxiv.org/html/2610.20743#bib.bib8)\), and s\-OTDD uses moment\-based projections for dataset comparison\([Nguyen et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib34)\)\. Although substantially cheaper than exact WoW, these approaches still construct their projections from the transport geometry of the inner measures or surrogates thereof\.

We introduce*Vertical Sliced Wasserstein*\(VSW\), a complementary approach that slices a measure through its action on continuous test functions\. For a test functionf∈C⁡\(Ω\)f\\in C\(\\Omega\), an inner measureμ\\muis represented by the scalarPf​\(μ\)=∫Ωf​𝑑μP\_\{f\}\(\\mu\)=\\int\_\{\\Omega\}f\\,\\,\\mathrm\{d\}\\mu\. Applying this projection to every inner measure converts a meta\-measure into a one\-dimensional distribution\. Then, one can compare efficientlyℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}in𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)using the one\-dimensionalpp\-Wasserstein distanceWp\\mathrm\{W\}\_\{p\}byVSWpp​\(ℙ,ℚ\):=𝔼f∼σ​\[Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)\],\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\):=\\mathbb\{E\}\_\{f\\sim\\sigma\}\[\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\],whereσ\\sigmais the law of a Gaussian process with universal covariance kernelκ\\kappa\. We call this construction*vertical*because it acts through the dual representation of each measure, in contrast to methods that slice along its transport geometry\.

This simple change has important practical consequences\. Each slice compresses an entire inner distribution to one scalar, requiring only test\-function evaluation and averaging before a single one\-dimensional OT computation across the outer distributions\. VSW therefore avoids inner OT, per\-distribution quantile representations, and Gaussian surrogates\. At the same time, the family of test\-function projections is rich enough to distinguish meta\-measures: we prove thatVSWp\\mathrm\{VSW\}\_\{p\}is a metric and metrizes weak convergence\. Forp=2p=2, VSW is additionally Hilbertian and closely connected to kernel mean embeddings, while retaining the outer Wasserstein comparison that distinguishes distributions beyond their mean measures\.

Empirically, this combination yields an accurate and efficient meta\-measure distance\. Across dataset comparison, point\-cloud and image\-patch distributions, and shape classification, VSW closely tracks transport\-based reference distances while consistently reducing runtime\. We further derive an explicit gradient formulation for the empiricalVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}objective, supporting parametric generative modeling\. Our main contributions are:

1. 1\.We introduceVertical Sliced Wasserstein \(VSW\), a new sliced distance for meta\-measures based on test\-function projections, and prove that it defines a metric that metrizes weak convergence\.
2. 2\.We establish statistical and computational scalability, including dimension\-independent two\-stage convergence rates and efficient Monte Carlo estimation without inner OT\.
3. 3\.We show thatVSW2\\mathrm\{VSW\}\_\{2\}is Hilbertian, induces universal characteristic kernels on meta\-measures, and admits, under discretization, an explicit gradient for optimization and generative modeling\.
4. 4\.We provide broad empirical validation across dataset comparison, point clouds, image patches, shape classification, discrete particle gradient flows, and hierarchical generation, where VSW is fast while maintaining strong agreement with transport\-based baselines\.

The closest computational baseline is DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), which first projects multivariate inner measures to one dimension, represents the resulting measures through quantile functions inL2​\(\[0,1\]\)L^\{2\}\(\[0,1\]\), and then applies functional slicing with Gaussian\-process directions\. VSW instead applies test functions directly to the original inner measures, reducing each to a single scalar per slice and avoiding both inner quantile construction and an additional Euclidean slicing stage\. A related test\-functional approach is the hierarchical IPM of[Catalano and Lavenant \(2024\)](https://arxiv.org/html/2610.20743#bib.bib10), which, unlike VSW, requires optimization over a class of test functions\.

## 2Background and standing assumptions

#### Spaces\.

Throughout,\(Ω,ρ\)\(\\Omega,\\rho\)is a compact metric space\. When gradients onΩ\\Omegaare needed \(Sections[3\.2](https://arxiv.org/html/2610.20743#S3.SS2)–[3\.4](https://arxiv.org/html/2610.20743#S3.SS4)\) we takeΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}\.C⁡\(Ω\)C\(\\Omega\)denotes the continuous real functions with the sup norm‖⋅‖∞\\left\\lVert\\cdot\\right\\rVert\_\{\\infty\}\. SinceΩ\\Omegais compact,\(C\(Ω\),∥⋅∥∞\)\(C\(\\Omega\),\\\|\\cdot\\\|\_\{\\infty\}\)is a separable Banach space\. By the Riesz–Markov representation theorem its dual is the space of finite signed Radon measuresℳ⁡\(Ω\)\\mathcal\{M\}\(\\Omega\)with the total variation norm, and𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\)is a convex subset of the unit ball ofℳ⁡\(Ω\)\\mathcal\{M\}\(\\Omega\)\. The weak \(narrow\) topology on𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\)is the restriction of the weak∗topology ofℳ⁡\(Ω\)=C​\(Ω\)∗\\mathcal\{M\}\(\\Omega\)=C\(\\Omega\)^\{\*\}\.

By Prokhorov’s theorem,𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\)is weakly compact and it is metrizable becauseC⁡\(Ω\)C\(\\Omega\)is separable\. Its Borelσ\\sigma\-algebra,ℬ⁡\(𝒫⁡\(Ω\)\)\\mathcal\{B\}\(\\mathcal\{P\}\(\\Omega\)\), is generated by the mapsPf:𝒫⁡\(Ω\)→ℝP\_\{f\}:\\mathcal\{P\}\(\\Omega\)\\to\\mathbb\{R\}

Pf:μ↦⟨f,μ⟩:=∫Ωf​𝑑μ,f∈C⁡\(Ω\),P\_\{f\}:\\mu\\mapsto\\langle f,\\mu\\rangle:=\\int\_\{\\Omega\}f\\,\\mathrm\{d\}\\mu,\\qquad f\\in C\(\\Omega\),\(1\)\([Kallenberg, 2017](https://arxiv.org/html/2610.20743#bib.bib23), Lem\. 4\.5 and Thm\. 4\.2\)\. Since𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\)is itself a compact metric space, the same reasoning applies one level up:𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), the space of Borel probability measures on𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\), is compact and metrizable in its weak topology\. Forℙ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)we writeμ∼ℙ\\mu\\sim\\mathbb\{P\}for a random inner measure\.

Note the two elementary but crucial algebraic properties of*vertical projections*defined by \([1](https://arxiv.org/html/2610.20743#S2.E1)\):

###### Lemma 2\.1\.

For allf,g∈C⁡\(Ω\)f,g\\in C\(\\Omega\),a,b∈ℝa,b\\in\\mathbb\{R\},μ,μ0,μ1∈𝒫⁡\(Ω\)\\mu,\\mu\_\{0\},\\mu\_\{1\}\\in\\mathcal\{P\}\(\\Omega\), andt∈\[0,1\]t\\in\[0,1\],

Pa​f\+b​g​\(μ\)=a​Pf​\(μ\)\+b​Pg​\(μ\),Pf​\(\(1−t\)​μ0\+t​μ1\)=\(1−t\)​Pf​\(μ0\)\+t​Pf​\(μ1\)\.P\_\{af\+bg\}\(\\mu\)=aP\_\{f\}\(\\mu\)\+bP\_\{g\}\(\\mu\),\\qquad P\_\{f\}\(\(1\-t\)\\mu\_\{0\}\+t\\mu\_\{1\}\)=\(1\-t\)P\_\{f\}\(\\mu\_\{0\}\)\+tP\_\{f\}\(\\mu\_\{1\}\)\.Thus,Pf​\(μ\)P\_\{f\}\(\\mu\)is linear inffand affine inμ\\mualong mixtures\.

Optimal transport between one\-dimensional measures\.Forp≥1p\\geq 1andα,β∈𝒫⁡\(ℝ\)\\alpha,\\beta\\in\\mathcal\{P\}\(\\mathbb\{R\}\), with finitepp\-moment,1≤p<∞1\\leq p<\\infty, letFα,FβF\_\{\\alpha\},F\_\{\\beta\}denote their corresponding Cumulative Distribution Functions \(CDFs\), andqα=Fα−1,qβ=Fβ−1q\_\{\\alpha\}=F\_\{\\alpha\}^\{\-1\},q\_\{\\beta\}=F\_\{\\beta\}^\{\-1\}be their quantile functions\. The optimal transport cost is:

Wpp​\(α,β\)=∫01\|qα​\(u\)−qβ​\(u\)\|p​𝑑u\.\\mathrm\{W\}\_\{p\}^\{p\}\(\\alpha,\\beta\)=\\int\_\{0\}^\{1\}\\big\|q\_\{\\alpha\}\(u\)\-q\_\{\\beta\}\(u\)\\big\|^\{p\}\\,\\mathrm\{d\}u\.\(2\)We useWpp​\(α,β\)≤\(diam⁡supp\)p−1​W1​\(α,β\)\\mathrm\{W\}\_\{p\}^\{p\}\(\\alpha,\\beta\)\\leq\(\\operatorname\{diam\}\\mathrm\{supp\}\)^\{p\-1\}\\mathrm\{W\}\_\{1\}\(\\alpha,\\beta\)for compactly supportedα,β\\alpha,\\beta, and the empirical rate: ifα\\alphais supported in an interval of length\|I\|\|I\|andα^N\\widehat\{\\alpha\}\_\{N\}is the empirical measure ofNNi\.i\.d\. samples, then

𝔼​W1​\(α^N,α\)≤1N​∫ℝFα​\(1−Fα\)≤\|I\|2​N,\\mathbb\{E\}\\,\\mathrm\{W\}\_\{1\}\(\\widehat\{\\alpha\}\_\{N\},\\alpha\)\\;\\leq\\;\\tfrac\{1\}\{\\sqrt\{N\}\}\\int\_\{\\mathbb\{R\}\}\\sqrt\{F\_\{\\alpha\}\(1\-F\_\{\\alpha\}\)\}\\;\\leq\\;\\tfrac\{\|I\|\}\{2\\sqrt\{N\}\},\(3\)where the first inequality follows from[Bobkov and Ledoux \(2019, Thm 3\.2\)](https://arxiv.org/html/2610.20743#bib.bib5)\.

Gaussian slicing measure\.We writeσ\\sigmafor a Borel probability measure onC⁡\(Ω\)C\(\\Omega\)\. A slicing direction is a realizationf∼σf\\sim\\sigma, andLLiidslices aref1,…,fL∼σf\_\{1\},\\dots,f\_\{L\}\{\\sim\}\\sigma\. This is akin to the sliced\-Wasserstein distance with Gaussian projections introduced by[Nadjahi et al\. \(2021\)](https://arxiv.org/html/2610.20743#bib.bib31), but for meta\-measures\.

###### Assumption 1\.

We consider a continuous, symmetric, positive definite kernel and universal “slicing kernel”κ:Ω×Ω→ℝ\\kappa:\\Omega\\times\\Omega\\to\\mathbb\{R\}, where universal means that the reproducing kernel Hilbert space \(RKHS\)ℋκ\\mathcal\{H\}\_\{\\kappa\}associated toκ\\kappais dense in\(C\(Ω\),∥⋅∥∞\)\(C\(\\Omega\),\\\|\\cdot\\\|\_\{\\infty\}\), and such that the centered Gaussian process with covariance kernelκ\\kappa,GP⁡\(0,κ\)\\mathrm\{GP\}\(0,\\kappa\), admits a version with almost surely continuous sample paths\.

Under the above assumption, we consider the model wheref∼σ=GP⁡\(0,κ\)f\\sim\\sigma=\\mathrm\{GP\}\(0,\\kappa\)\. For every finite collectionx1,…,xn∈Ωx\_\{1\},\\dots,x\_\{n\}\\in\\Omega, the random vector\(f⁡\(x1\),…,f⁡\(xn\)\)\(f\(x\_\{1\}\),\\dots,f\(x\_\{n\}\)\)has law𝒩⁡\(0,K\)\\mathcal\{N\}\(0,K\), whereKi​j=𝔼f∼GP⁡\(0,κ\)​\[f⁡\(xi\)​f​\(xj\)\]=κ⁡\(xi,xj\)K\_\{ij\}=\\mathbb\{E\}\_\{f\\sim\\mathrm\{GP\}\(0,\\kappa\)\}\[f\(x\_\{i\}\)f\(x\_\{j\}\)\]=\\kappa\(x\_\{i\},x\_\{j\}\)\. See extra details in Appendix[A](https://arxiv.org/html/2610.20743#A1)\.

Wasserstein\-over\-Wasserstein\.For a Borel metricDDon𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\),

𝕎p,Dp​\(ℙ,ℚ\):=infΓ∈Π⁡\(ℙ,ℚ\)∫D​\(μ,ν\)p​𝑑Γ​\(μ,ν\),\\mathbb\{W\}\_\{p,D\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\):=\\inf\_\{\\Gamma\\in\\Pi\(\\mathbb\{P\},\\mathbb\{Q\}\)\}\\int D\(\\mu,\\nu\)^\{p\}\\,\\,\\mathrm\{d\}\\Gamma\(\\mu,\\nu\),withΠ⁡\(ℙ,ℚ\)\\Pi\(\\mathbb\{P\},\\mathbb\{Q\}\)the set of couplings betweenℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. WhenD=WpD=\\mathrm\{W\}\_\{p\}, it recovers the standard WoW distance, which we simply denote by𝕎p\\mathbb\{W\}\_\{p\}\. Additional details of this section are available in Appendix[A](https://arxiv.org/html/2610.20743#A1)\.

## 3Our method: The Vertical Sliced Wasserstein distance

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

Forℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),p∈\[1,∞\)p\\in\[1,\\infty\), andσ=GP⁡\(0,κ\)\\sigma=\\mathrm\{GP\}\(0,\\kappa\)the Gaussian slicing measure under Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1), we define the Vertical Sliced Wasserstein discrepancy by

VSWp​\(ℙ,ℚ\):=\(∫C⁡\(Ω\)Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)​𝑑σ​\(f\)\)1/p,\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\;:=\\;\\Big\(\\int\_\{C\(\\Omega\)\}\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\\,\\mathrm\{d\}\\sigma\(f\)\\Big\)^\{1/p\},\(4\)where, for each fixedf∈C⁡\(Ω\)f\\in C\(\\Omega\), the pushforward\(Pf\)\#​ℙ∈𝒫⁡\(ℝ\)\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\\in\\mathcal\{P\}\(\\mathbb\{R\}\)is the law of the scalar vertical slicePf​\(μ\)P\_\{f\}\(\\mu\)whenμ∼ℙ\\mu\\sim\\mathbb\{P\}defined precisely by\(Pf\)\#​ℙ​\(B\)=ℙ⁡\(\{μ∈𝒫⁡\(Ω\):Pf​\(μ\)∈B\}\)\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\(B\)=\\mathbb\{P\}\(\\\{\\mu\\in\\mathcal\{P\}\(\\Omega\):P\_\{f\}\(\\mu\)\\in B\\\}\)for allB∈ℬ⁡\(ℝ\)B\\in\\mathcal\{B\}\(\\mathbb\{R\}\), which is supported on\[−‖f‖∞,‖f‖∞\]\[\-\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert f\\right\\rVert\_\{\\infty\}\]\(similarly for\(Pf\)\#​ℚ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\)\.

The*“vertical”*viewpoint reflects the affine structure of the space of measures: the slicers \([1](https://arxiv.org/html/2610.20743#S2.E1)\) are affine along mixtures\(1−t\)​μ0\+t​μ1\(1\-t\)\\mu\_\{0\}\+t\\mu\_\{1\}, as stated in Lemma[2\.1](https://arxiv.org/html/2610.20743#S2.Thmtheorem1)\. In contrast,*“horizontal”*or*transport geometry*describes the movement of mass withinΩ\\Omega, e\.g\., through displacement \(McCann\) interpolationμt=\(\(1−t\)​id\+t​T\)\#​μ0\\mu\_\{t\}=\(\(1\-t\)\\mathrm\{id\}\+tT\)\_\{\\\#\}\\mu\_\{0\}, for an optimal Monge mapTTfor the quadratic\-cost\. A related distinction between spatial transport and local mass variation appears in unbalanced OT\([Chizat et al\., 2018](https://arxiv.org/html/2610.20743#bib.bib12);[Liero et al\., 2018](https://arxiv.org/html/2610.20743#bib.bib27)\); here “vertical” specifically refers to affinity along probability mixtures\.

### 3\.1Well\-posedness, metricity, and topology of VSW

We show regularity of the slicing procedure to ensure thatVSWp\\mathrm\{VSW\}\_\{p\}in \([4](https://arxiv.org/html/2610.20743#S3.E4)\) is well\-defined \(Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\)\. Moreover,VSWp\\mathrm\{VSW\}\_\{p\}defines a true metric on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\(Theorem[3\.3](https://arxiv.org/html/2610.20743#S3.Thmtheorem3)\) which metrizes weak convergence \(Proposition[3\.5](https://arxiv.org/html/2610.20743#S3.Thmtheorem5)\), and\(𝒫⁡\(𝒫⁡\(Ω\)\),VSW2\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{2\}\)embeds isometrically into a Hilbert space \(Theorem[3\.6](https://arxiv.org/html/2610.20743#S3.Thmtheorem6)\)\.

###### Lemma 3\.2\.

\(i\) The mapP:C⁡\(Ω\)×𝒫⁡\(Ω\)→ℝP:C\(\\Omega\)\\times\\mathcal\{P\}\(\\Omega\)\\to\\mathbb\{R\},P⁡\(f,μ\):=⟨f,μ⟩P\(f,\\mu\):=\\langle f,\\mu\\rangle, is jointly continuous\. \(ii\) For fixedℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)the mapf↦Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)f\\mapsto\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)is22\-Lipschitz from\(C⁡\(Ω\),‖⋅‖∞\)\(C\(\\Omega\),\\left\\lVert\\cdot\\right\\rVert\_\{\\infty\}\)toℝ\\mathbb\{R\}; in particular it is continuous, and the integral in \([4](https://arxiv.org/html/2610.20743#S3.E4)\) is well defined and finite\.

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

VSWp\\mathrm\{VSW\}\_\{p\}is a metric on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)for everyp∈\[1,∞\)p\\in\[1,\\infty\)\.

###### Proposition 3\.5\.

For everyp∈\[1,∞\)p\\in\[1,\\infty\), the metricVSWp\\mathrm\{VSW\}\_\{p\}induces the weak topology on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. In particular\(𝒫⁡\(𝒫⁡\(Ω\)\),VSWp\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{p\}\)is a compact metric space\.

###### Theorem 3\.6\.

Letλ\\lambdabe Lebesgue measure on\(0,1\)\(0,1\)\. The map

T:𝒫⁡\(𝒫⁡\(Ω\)\)→L2​\(C⁡\(Ω\)×\(0,1\),σ⊗λ\),T⁡\(ℙ\)​\(f,u\):=q\(Pf\)\#​ℙ​\(u\),T:\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\\to L^\{2\}\(C\(\\Omega\)\\times\(0,1\),\\sigma\\otimes\\lambda\),\\qquad T\(\\mathbb\{P\}\)\(f,u\):=q\_\{\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\}\(u\),is well defined and satisfies

‖T⁡\(ℙ\)−T⁡\(ℚ\)‖L2​\(σ⊗λ\)=VSW2​\(ℙ,ℚ\)\.\\left\\lVert T\(\\mathbb\{P\}\)\-T\(\\mathbb\{Q\}\)\\right\\rVert\_\{L^\{2\}\(\\sigma\\otimes\\lambda\)\}=\\mathrm\{VSW\}\_\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\.Hence\(𝒫⁡\(𝒫⁡\(Ω\)\),VSW2\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{2\}\)embeds isometrically into a separable Hilbert space\. Moreover,VSW2β\\mathrm\{VSW\}\_\{2\}^\{\\beta\}is conditionally negative definite forβ∈\(0,2\]\\beta\\in\(0,2\], soexp⁡\(−γ​VSW2β\)\\exp\(\-\\gamma\\mathrm\{VSW\}\_\{2\}^\{\\beta\}\)is positive definite for allγ\>0\\gamma\>0\.

Further details are provided in Appendices[B](https://arxiv.org/html/2610.20743#A2)and[D](https://arxiv.org/html/2610.20743#A4), and all proofs of this section are in Appendix[C](https://arxiv.org/html/2610.20743#A3)\.

### 3\.2Computation

#### Sampling slices:

For translation\-invariantκ\\kappaonΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}with spectral measureτκ\\tau\_\{\\kappa\}\(Bochner\), anRR\-feature slice is

fR​\(x\)=2R​∑r=1Rar​cos⁡\(⟨ωr,x⟩\+br\),ar∼𝒩⁡\(0,1\),ωr∼τκ,br∼Unif⁡\[0,2​π\],f\_\{R\}\(x\)=\\sqrt\{\\tfrac\{2\}\{R\}\}\\sum\_\{r=1\}^\{R\}a\_\{r\}\\cos\\big\(\\langle\\omega\_\{r\},x\\rangle\+b\_\{r\}\\big\),\\qquad a\_\{r\}\\sim\\mathcal\{N\}\(0,1\),\\ \\omega\_\{r\}\\sim\\tau\_\{\\kappa\},\\ b\_\{r\}\\sim\\mathrm\{Unif\}\[0,2\\pi\],\(5\)where we normalizeκ⁡\(x,x\)=κ⁡\(0\)=1\\kappa\(x,x\)=\\kappa\(0\)=1so that the spectral measureτκ\\tau\_\{\\kappa\}is a probability distribution\. Conditionally on\(ω,b\)\(\\omega,b\),fRf\_\{R\}is a Gaussian process whose covariance converges uniformly toκ\\kappaasR→∞R\\to\\infty\([Rahimi and Recht, 2007](https://arxiv.org/html/2610.20743#bib.bib37)\)\. Alternatively, a truncated Karhunen–Loève expansion may be used \(see uniform approximation guarantees in Proposition[D\.2](https://arxiv.org/html/2610.20743#A4.Thmtheorem2)and Lemma[D\.3](https://arxiv.org/html/2610.20743#A4.Thmtheorem3)\)\. Gradients have the closed form∇fR\(x\)=−2/R∑rarsin\(⟨ωr,x⟩\+br\)ωr\\nabla f\_\{R\}\(x\)=\-\\sqrt\{2/R\}\\sum\_\{r\}a\_\{r\}\\sin\(\\langle\\omega\_\{r\},x\\rangle\+b\_\{r\}\)\\,\\omega\_\{r\}\.

Cost:The full VSW algorithm is shown in Algorithm[1](https://arxiv.org/html/2610.20743#alg1)\. LetNNdenote the maximum number of inner measures in either empirical meta\-measure, andnnthe maximum number of points per inner measure\. EvaluatingLLindependentRRfeature slices costsO⁡\(L​N​n​R​d\+L​N​log⁡N\)O\\\!\\left\(LNnRd\+LN\\log N\\right\)\. In practice, we share the Fourier basis\(ωr,br\)r=1R\(\\omega\_\{r\},b\_\{r\}\)\_\{r=1\}^\{R\}across slices and resample only the Gaussian coefficients \(conditional on the basis, this estimates Gaussian\-sliced Wasserstein between RFF mean embeddings inℝR\\mathbb\{R\}^\{R\}; see Appendix[C](https://arxiv.org/html/2610.20743#A3.SS0.SSS0.Px1)for the scaling identity and pseudometric caveat\)\. Computing and averaging theRRbasis features once for each inner measure, and reusing them across slices, reduces the cost toO⁡\(N​n​R​d\+L​N​R\+L​N​log⁡N\)O\\\!\\left\(NnRd\+LNR\+LN\\log N\\right\)\. Unlike prior sliced WoW constructions, VSW requires no per inner\-measure sorting or quantile representation\. Its runtime advantages are verified empirically in Section[4](https://arxiv.org/html/2610.20743#S4)\.

### 3\.3Statistical properties

We consider the two\-stage observation model of[Szabó et al\. \(2016\)](https://arxiv.org/html/2610.20743#bib.bib43): inner measuresμ1,…,μN∼iidℙ\\mu\_\{1\},\\dots,\\mu\_\{N\}\\stackrel\{\{\\scriptstyle\\text\{iid\}\}\}\{\{\\sim\}\}\\mathbb\{P\}; for eachii, pointsxi​1,…,xi​n∼iidμix\_\{i1\},\\dots,x\_\{in\}\\stackrel\{\{\\scriptstyle\\text\{iid\}\}\}\{\{\\sim\}\}\\mu\_\{i\}\(conditionally independent givenμi\\mu\_\{i\}\), empirical inner measuresμ^i=1n​∑jδxi​j\\widehat\{\\mu\}\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{j\}\\delta\_\{x\_\{ij\}\}, and the plug\-in meta\-measuresℙ^N=1N​∑i=1Nδμi,ℙ~N,n=1N​∑i=1Nδμ^i\.\\widehat\{\\mathbb\{P\}\}\_\{N\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\delta\_\{\\mu\_\{i\}\},~\\widetilde\{\\mathbb\{P\}\}\_\{N,n\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\delta\_\{\\widehat\{\\mu\}\_\{i\}\}\.

###### Theorem 3\.7\.

For everyp∈\[1,∞\)p\\in\[1,\\infty\), there existsCp<∞C\_\{p\}<\\infty, depending only onpp\(withCp=1C\_\{p\}=1forp∈\[1,2\]p\\in\[1,2\]\), such that

𝔼\[VSWpp\(ℙ,ℙ~N,n\)\]≤2p−1\(122pΛpN−1/2\+Cp2pΛpn−p/2\),\\mathbb\{E\}\\,\\left\[\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\widetilde\{\\mathbb\{P\}\}\_\{N,n\}\)\\right\]\\leq 2^\{p\-1\}\\Big\(\\tfrac\{1\}\{2\}\\,2^\{p\}\\Lambda\_\{p\}N^\{\-1/2\}\+C\_\{p\}\\,2^\{p\}\\Lambda\_\{p\}n^\{\-p/2\}\\Big\),whereΛp=𝔼σ​‖f‖∞p<∞\\Lambda\_\{p\}=\\mathbb\{E\}\_\{\\sigma\}\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{p\}<\\infty\. The rate exponents are dimension\-free\. Dependence on\(Ω,κ\)\(\\Omega,\\kappa\)enters only through the sup\-norm moment, which is finite but may grow with the size ofΩ\\Omegarelative to the kernel bandwidth\.

Finally, the Monte Carlo slice complexity reads as follows:

###### Proposition 3\.9\.

Letf1,…,fL∼iidσf\_\{1\},\\dots,f\_\{L\}\\stackrel\{\{\\scriptstyle\\text\{iid\}\}\}\{\{\\sim\}\}\\sigmaandVSW^Lp​\(ℙ,ℚ\):=1L​∑lWpp​\(\(Pfl\)\#​ℙ,\(Pfl\)\#​ℚ\)\\widehat\{\\mathrm\{VSW\}\}^\{p\}\_\{L\}\(\\mathbb\{P\},\\mathbb\{Q\}\):=\\frac\{1\}\{L\}\\sum\_\{l\}\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\_\{l\}\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\_\{l\}\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\. Then

𝔼​\|VSW^Lp−VSWpp\|≤Varσ⁡\[Wpp\]L≤2p​Λ2​p/L\.\\mathbb\{E\}\\Big\|\\widehat\{\\mathrm\{VSW\}\}^\{p\}\_\{L\}\-\\mathrm\{VSW\}\_\{p\}^\{p\}\\Big\|\\;\\leq\\;\\sqrt\{\\tfrac\{\\operatorname\{Var\}\_\{\\sigma\}\\big\[\\mathrm\{W\}\_\{p\}^\{p\}\\big\]\}\{L\}\}\\;\\leq\\;2^\{p\}\\sqrt\{\\Lambda\_\{2p\}/L\}\.

Following[Nadjahi et al\. \(2020\)](https://arxiv.org/html/2610.20743#bib.bib30), we examine empirical scaling on Gaussian\-mixture meta\-measures ford∈\{2,10,20,50,100\}d\\in\\\{2,10,20,50,100\\\}using finite Fourier features\. Figure[1](https://arxiv.org/html/2610.20743#S3.F1)shows anL−1/2L^\{\-1/2\}trend for Monte Carlo*distance*error against a finite\-slice reference, andVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}trends ofN−1/2N^\{\-1/2\}andn−1n^\{\-1\}at the outer and inner levels, with nearly unchanged slopes across dimensions\. This is an empirical scaling study: Theorem[3\.7](https://arxiv.org/html/2610.20743#S3.Thmtheorem7)assumes a compact domain, and Proposition[3\.9](https://arxiv.org/html/2610.20743#S3.Thmtheorem9)bounds the powered\-distance estimator\.

Figure 1:Statistical complexity analysis for empirical meta\-measuresX,YX,Y\. Setup in Appendix[F](https://arxiv.org/html/2610.20743#A6)\. The outer law has three atoms, so theN−1/2N^\{\-1/2\}reference forVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}corresponds to the atomic\-outer rate \(Remark[C\.2](https://arxiv.org/html/2610.20743#A3.Thmtheorem2)\)\. Left: distance error against a10510^\{5\}\-slice reference\. Right:X,YX,Yshare component labels to isolate inner sampling error\.Algorithm 1VSW^pp\\widehat\{\\mathrm\{VSW\}\}\_\{p\}^\{p\}between empirical meta\-measures1:clouds

\{Xi\}i≤N\\\{X\_\{i\}\\\}\_\{i\\leq N\},

\{Yk\}k≤M\\\{Y\_\{k\}\\\}\_\{k\\leq M\}; kernel

κ\\kappa; slices

LL; features

RR; order

pp
2:for

l=1,…,Ll=1,\\dots,Ldo

3:sample

flf\_\{l\}via \([5](https://arxiv.org/html/2610.20743#S3.E5)\) \(or KL truncation\)

4:

si←1\|Xi\|​∑x∈Xifl​\(x\)s\_\{i\}\\leftarrow\\frac\{1\}\{\|X\_\{i\}\|\}\\sum\_\{x\\in X\_\{i\}\}f\_\{l\}\(x\)for all

ii;

tk←1\|Yk\|​∑y∈Ykfl​\(y\)t\_\{k\}\\leftarrow\\frac\{1\}\{\|Y\_\{k\}\|\}\\sum\_\{y\\in Y\_\{k\}\}f\_\{l\}\(y\)for all

kk
5:

dl←Wpp​\(1N​∑iδsi,1M​∑kδtk\)d\_\{l\}\\leftarrow\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\\tfrac\{1\}\{N\}\\sum\_\{i\}\\delta\_\{s\_\{i\}\},\\,\\tfrac\{1\}\{M\}\\sum\_\{k\}\\delta\_\{t\_\{k\}\}\\big\)⊳\\trianglerightsort \+ quantile matching, closed form \([2](https://arxiv.org/html/2610.20743#S2.E2)\)

6:endfor

7:return

1L​∑ldl\\frac\{1\}\{L\}\\sum\_\{l\}d\_\{l\}

### 3\.4Gradient flows: Practical implementation

We optimize the empiricalVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}objective by moving the support points of its inner measures, thereby obtaining a finite\-dimensional particle\-level gradient flow\. Throughout this calculation, the target and the slicing law are fixed\. Fix a targetℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)and defineℱ:𝒫⁡\(𝒫⁡\(Ω\)\)→ℝ\\mathcal\{F\}:\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\\to\\mathbb\{R\}byℱ⁡\(ℙ\):=VSW22​\(ℙ,ℚ\)=𝔼f∼σ​\[W22​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)\]\\mathcal\{F\}\(\\mathbb\{P\}\)\\;:=\\;\\mathrm\{VSW\}\_\{2\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\;=\\;\\mathbb\{E\}\_\{f\\sim\\sigma\}\[\\mathrm\{W\}\_\{2\}^\{2\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\]\.

The discretization used in practice is as follows: Letℚ=1N​∑k=1Nδν^k\\mathbb\{Q\}=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\delta\_\{\\widehat\{\\nu\}\_\{k\}\},ν^k=1m​∑lδyk​l\\widehat\{\\nu\}\_\{k\}=\\frac\{1\}\{m\}\\sum\_\{l\}\\delta\_\{y\_\{kl\}\}, andℙX=1N​∑i=1Nδμ^i\\mathbb\{P\}\_\{X\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\delta\_\{\\widehat\{\\mu\}\_\{i\}\},μ^i=1n​∑jδxi​j\\widehat\{\\mu\}\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{j\}\\delta\_\{x\_\{ij\}\}, withX=\(xi​j\)∈\(ℝd\)N×nX=\(x\_\{ij\}\)\\in\(\\mathbb\{R\}^\{d\}\)^\{N\\times n\}\. For simplicity we take equal outer sizes\. The quantile formulation also handlesN≠MN\\neq M\. Define

ℱ⁡\(X\):=VSW22​\(ℙX,ℚ\)=𝔼f∼σ​\[1N​∑i=1N\(s\(i\)​\(f\)−t\(i\)​\(f\)\)2\],\\mathcal\{F\}\(X\)\\;:=\\;\\mathrm\{VSW\}\_\{2\}^\{2\}\\big\(\\mathbb\{P\}\_\{X\},\\mathbb\{Q\}\\big\)=\\mathbb\{E\}\_\{f\\sim\\sigma\}\\Big\[\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\big\(s\_\{\(i\)\}\(f\)\-t\_\{\(i\)\}\(f\)\\big\)^\{2\}\\Big\],\(6\)wheresi​\(f\)=1n​∑jf⁡\(xi​j\)s\_\{i\}\(f\)=\\frac\{1\}\{n\}\\sum\_\{j\}f\(x\_\{ij\}\),tk​\(f\)=1m​∑lf⁡\(yk​l\)t\_\{k\}\(f\)=\\frac\{1\}\{m\}\\sum\_\{l\}f\(y\_\{kl\}\), and\(⋅\)\(i\)\(\\cdot\)\_\{\(i\)\}denotes increasing rearrangement\. Here the projected lawsαf=1N​∑iδsi​\(f\)\\alpha\_\{f\}=\\frac\{1\}\{N\}\\sum\_\{i\}\\delta\_\{s\_\{i\}\(f\)\}andβf=1N​∑kδtk​\(f\)\\beta\_\{f\}=\\frac\{1\}\{N\}\\sum\_\{k\}\\delta\_\{t\_\{k\}\(f\)\}are atomic, so transport is determined by the sorting permutation rather than a continuous monotone map\. Lemma[C\.5](https://arxiv.org/html/2610.20743#A3.Thmtheorem5)ensures the required uniqueness\.

###### Proposition 3\.10\.

Assume Assumptions[1](https://arxiv.org/html/2610.20743#Thmassumption1)–[2](https://arxiv.org/html/2610.20743#Thmassumption2), with the smoothness and moment conditions holding on a compact neighborhood of the particle configurations\. If the source clouds and the target clouds are each pairwise distinct as measures, as in Lemma[C\.5](https://arxiv.org/html/2610.20743#A3.Thmtheorem5), thenℱ\\mathcal\{F\}is differentiable atXX, and

∇xi​jℱ\(X\)=2N​n𝔼f∼σ\[\(si\(f\)−tπf​\(i\)\(f\)\)∇f\(xi​j\)\]\.\\nabla\_\{x\_\{ij\}\}\\mathcal\{F\}\(X\)\\;=\\;\\frac\{2\}\{Nn\}\\;\\mathbb\{E\}\_\{f\\sim\\sigma\}\\Big\[\\big\(s\_\{i\}\(f\)\-t\_\{\\pi\_\{f\}\(i\)\}\(f\)\\big\)\\,\\nabla f\(x\_\{ij\}\)\\Big\]\.\(7\)

Consequently, the associated Euclidean particle gradient flow isx˙i​j​\(t\)=−∇xi​jℱ​\(X⁡\(t\)\)\\dot\{x\}\_\{ij\}\(t\)=\-\\nabla\_\{x\_\{ij\}\}\\mathcal\{F\}\(X\(t\)\), with the right\-hand side given by \([7](https://arxiv.org/html/2610.20743#S3.E7)\)\. The proof and corresponding algorithm are given in Appendix[C](https://arxiv.org/html/2610.20743#A3)\. In the ideal setting, replacing the expectation in Proposition[3\.10](https://arxiv.org/html/2610.20743#S3.Thmtheorem10)by an average over independent slicesf1,…,fL∼σf\_\{1\},\\ldots,f\_\{L\}\\sim\\sigmagives a Monte Carlo estimate of the particle gradient\. In practice, we represent the slices using finitely many random Fourier features, yielding the corresponding finite\-feature approximation of this gradient\. When the particles are produced by a differentiable generator, the particle gradients can be backpropagated through the generator by the chain rule\.

## 4Experiments

We evaluate VSW across several tasks, demonstrating both performance and runtime advantages\. For each experiment, we also optimize the baseline hyperparameters and implementations to ensure a fair comparison\. Additional ablations and implementation details are provided in Appendices[E](https://arxiv.org/html/2610.20743#A5)and[F](https://arxiv.org/html/2610.20743#A6)\.

Optimal Transport Dataset Distance\.Dataset distances provide a model\-agnostic way to quantify similarity between datasets, with applications such as transfer learning and dataset selection\([Alvarez\-Melis and Fusi, 2020](https://arxiv.org/html/2610.20743#bib.bib2);[Liu et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib28)\)\. Following previous work\([Nguyen et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib34);[Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), we evaluate VSW as an efficient approximation to the optimal transport dataset distance \(OTDD\), comparing against s\-OTDD\([Nguyen et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib34)\)and DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), using OTDD\([Alvarez\-Melis and Fusi, 2020](https://arxiv.org/html/2610.20743#bib.bib2)\)as the reference\. We construct random pairs of sub\-datasets from MNIST\([LeCun et al\., 1998](https://arxiv.org/html/2610.20743#bib.bib25)\), FashionMNIST\([Xiao et al\., 2017](https://arxiv.org/html/2610.20743#bib.bib46)\), and CIFAR\-10\([Krizhevsky and Hinton, 2009](https://arxiv.org/html/2610.20743#bib.bib24)\)and represent each dataset through its class\-conditional empirical distributions\. Following[Piening and Beinert \(2026\)](https://arxiv.org/html/2610.20743#bib.bib35), we use identical dataset pairs and feature representations across methods, remove the feature cost from OTDD so that it reduces to a Wasserstein\-over\-Wasserstein distance between class\-conditional distributions, and use s\-OTDD with its default implementation\.

We report Pearson and Spearman correlations with OTDD while varying the number of slices\. For DSW, the total projection budget isS=P​KS=PK, wherePPandKKdenote the numbers of outer and inner projections, respectively\. For each dataset, we use3030pairs of disjoint sub\-datasets drawn from the same training set\. The two sub\-datasets in each pair have equal size, with that size varying from5,0005\{,\}000to10,00010\{,\}000across pairs\. Figure[2](https://arxiv.org/html/2610.20743#S4.F2)shows that VSW achieves substantially stronger correlation in the low\-slice regime across all three datasets and remains competitive at larger slice budgets under this protocol\. Figure[3](https://arxiv.org/html/2610.20743#S4.F3)compares runtime as the dataset size increases for matched budgets of1​k1\\mathrm\{k\},5​k5\\mathrm\{k\}, and10​k10\\mathrm\{k\}slices\. VSW is consistently the fastest method across all three datasets and projection budgets under this resampling protocol\.

Figure 2:Pearson and Spearman correlations with exact OTDD versus the number of slices, computed over3030pairs per dataset\. Both sub\-datasets in a pair are drawn from the same training set and have equal size \(5,0005\{,\}000–10,00010\{,\}000samples\); sizes vary across pairs\.Comparing Distributions of Point Clouds\.Following the point\-cloud distribution experiment of DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), we evaluate whether VSW captures changes in distributions of 3D point clouds from ModelNet\-10\([Wu et al\., 2015](https://arxiv.org/html/2610.20743#bib.bib45)\)\. Starting from a fixed reference distribution, we independently vary \(i\) the number of shapes in the target distribution, \(ii\) the amount of Gaussian noise applied to the target points, and \(iii\) the number of points per target shape\. We compare VSW with OT\-NNA, exact WoW, and DSW\. To facilitate direct comparison between methods with different scales, we report the normalized metrics together in Figure[4](https://arxiv.org/html/2610.20743#S4.F4), with the reference configuration indicated by the dashed line\.

VSW exhibits trends consistent with the OT\-based baselines across all three perturbations\. In particular, the distance decreases as the target contains more shapes, increases with stronger Gaussian perturbations, and remains comparatively stable once the point\-cloud resolution is sufficiently high\. These results indicate that VSW preserves the behavior of established distribution\-comparison metrics while being substantially more efficient\. We additionally report the wall\-clock time in a realistic point cloud evaluation setting, comparing 662 ground\-truth and 662 generated ShapeNet chairs\([Chang et al\., 2015](https://arxiv.org/html/2610.20743#bib.bib11)\)from PVD\([Zhou et al\., 2021](https://arxiv.org/html/2610.20743#bib.bib49)\), each containing 2048 points\. The results are shown in Table[1](https://arxiv.org/html/2610.20743#S4.T1)\. VSW requires only0\.01250\.0125s per distance evaluation, compared with0\.07090\.0709s for DSW, corresponding to approximately a5\.7×5\.7\\timesreduction in runtime\. OT\-NNA and WoW are both substantially slower\.

![Refer to caption](https://arxiv.org/html/2610.20743v1/runtime_final_combined.png)Figure 3:Runtime \(s\) vs the dataset size \(each of 2 sets\)\. VSW remains fastest among all methods\.Table 1:Wall\-clock time of a single distance evaluation between the 662 ground\-truth ShapeNet chairs and 662 generated samples \(20482048points per cloud\)\.†denotes a single run\.Figure 4:Point cloud comparison metric trends\.Figure 5:Comparing image distributions via patch distributions\. VSW remains the fastest method, while achieving the same trend as baselines\.Comparing Image Distributions via Patch Distributions\.Following DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), we evaluate VSW for comparing image distributions through their patch distributions\. We use synthetic Perlin\-noise textures and independently vary the*lacunarity*and*persistence*of the target distribution relative to a fixed reference\. We compare VSW against DSW and exact Wasserstein\-over\-Wasserstein \(WoW\), and normalize each metric for visualization\.

As shown in Figure[5](https://arxiv.org/html/2610.20743#S4.F5), all three methods correctly attain their minimum near the reference parameters, while increasing as the target distribution deviates from the reference\. VSW closely follows the trends of DSW and WoW, indicating that it preserves meaningful sensitivity to changes in image\-patch distributions\. At the same time, VSW requires only6767ms per comparison, compared with298298ms for DSW and669669s for exact WoW, making it approximately4\.4×4\.4\\timesfaster than DSW and nearly four orders of magnitude faster than exact WoW\.

Gradient Flow on Natural Images\.We use ImageNet\([Deng et al\., 2009](https://arxiv.org/html/2610.20743#bib.bib14)\)to evaluate the VSW particle gradient on a high\-dimensional dataset\. We optimize five clouds of twenty noise\-initialized particles against a fixed target containing twenty training images from each of five classes\. Figure[6](https://arxiv.org/html/2610.20743#S4.F6)shows the particle trajectories in pixel space and in a two\-dimensional PCA projection of CLIP ViT\-L/14 embeddings\([Radford et al\., 2021](https://arxiv.org/html/2610.20743#bib.bib36)\), fitted to the target images\. The particles progressively resemble target images, with projected trajectories moving toward the target clusters\. Nearest\-neighbor ground\-truth images provide a qualitative reference for the final particles\. These visualizations illustrate optimization toward the fixed empirical target\.

![Refer to caption](https://arxiv.org/html/2610.20743v1/final_trajectory.png)Figure 6:Particle optimization toward fixed ImageNet targets in pixel space\(Left\), and in a 2D PCA projection of CLIP ViT\-L/14 embeddings\(Right\)\.Parametric Generative Modeling\.We next evaluate VSW in a parametric generative modeling setting on two 2D hierarchical datasets: a checkerboard distribution and an eight\-component Gaussian mixture\. Each dataset is represented as a distribution of inner distributions, with an outer latent variable selecting the component and inner noise generating samples within it\. We train a lightweight MLP using the corresponding objective\. We compare VSW with DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\)and MMDSW\([Bonet et al\., 2025b](https://arxiv.org/html/2610.20743#bib.bib9)\)\. As shown in Figure[7](https://arxiv.org/html/2610.20743#S4.F7), VSW and MMDSW recover the main structure of both targets, and VSW closely matches the held\-out distributions\. VSW also reduces the Wasserstein error substantially faster in wall\-clock time, reaching a low error earlier than DSW and MMDSW on both datasets\.

![Refer to caption](https://arxiv.org/html/2610.20743v1/2d_gen.png)Figure 7:Parametric generation on Checkerboard and 8\-Gaussians datasets\.Shape Classification\.Following the protocol of DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), we evaluate VSW on 2D Shapes\([Beier et al\., 2022](https://arxiv.org/html/2610.20743#bib.bib3)\), Animals\([Sumner and Popović, 2004](https://arxiv.org/html/2610.20743#bib.bib42)\), FAUST\-500/1000 \([Bogo et al\., 2014](https://arxiv.org/html/2610.20743#bib.bib7)\)\(500500and10001000vertices per shape\), and MNIST\-2000\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\)using local distance distributions\. For each distance, we precompute the pairwise distance matrix and perform 3\-NN classification over 1000 random 25%/75% train/test splits, reporting mean accuracy and CPU time per distance\. VSW, SQW, and STLB share one random projection basis across the distance matrix, while AE caches its per\-shape energy self\-terms\. TLB and GW are deterministic and do not include such amortization\.

Table[2](https://arxiv.org/html/2610.20743#S4.T2)shows competitive mean accuracies across these datasets, with VSW consistently faster than the other sliced baselines\. Compared with SQW, VSW is approximately2\.32\.3–2\.8×2\.8\\timesfaster across all datasets\. The advantage is especially clear for larger shapes, requiring only3\.763\.76ms on FAUST\-1000 and7\.487\.48ms on MNIST\-2000\. VSW provides a favorable accuracy–runtime trade\-off for this task\.

Table 2:Shape classification with KNN \(Acc\.,↑\\uparrow\) and runtime per distance computation \(Time\)\.
## 5Conclusion and Future work

We introduced Vertical Sliced Wasserstein \(VSW\), a scalable distance for meta\-measures based on slicing through continuous test\-function projections rather than the transport geometry of the inner measures\. We showed that VSW is a valid metric that metrizes weak convergence, and thatVSW2\\mathrm\{VSW\}\_\{2\}admits a Hilbertian embedding, yielding characteristic kernels on meta\-measures\. We further established dimension\-independent two\-stage convergence rates and derived a practical particle gradient for optimization\. Across dataset comparison, shape classification, point\-cloud and image\-distribution comparison, gradient flow and parametric generation, VSW provides competitive agreement with transport\-based distances while substantially reducing computation\.

Our analysis also leaves several limitations and open directions\. Promising directions include extending parametric generative modeling beyond low\-dimensional examples to higher\-dimensional and structured domains, such as point cloud and image generation\. For instance, a continuum WoW\-flow interpretation \(where, based on the differential geometry of[Bonet et al\. \(2025b\)](https://arxiv.org/html/2610.20743#bib.bib9), VSW defines the objective, while\(𝒫⁡\(𝒫⁡\(Ω\)\),𝕎2\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathbb\{W\}\_\{2\}\)defines the geometry of the descent\) together with a convergence of the discrete optimization schemes requires a separate theoretical analysis\.

## Acknowledgments

This work was supported in part by NSF CAREER Award No\. 233989, NSF DMS Award No\. 2603773, and a generous gift from Toyota Infotech Lab\. The authors also gratefully acknowledge the computing resources provided by the NVIDIA Academic Grant Program\.

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## Appendix ABackground

###### Proof of Lemma[2\.1](https://arxiv.org/html/2610.20743#S2.Thmtheorem1)\.

Both identities follow from linearity of the integral, in the integrand and in the measure respectively\. For the first,a​f\+b​g∈C⁡\(Ω\)af\+bg\\in C\(\\Omega\)and

Pa​f\+b​g​\(μ\)=∫Ω\(a​f\+b​g\)​𝑑μ=a​∫Ωf​𝑑μ\+b​∫Ωg​𝑑μ=a​Pf​\(μ\)\+b​Pg​\(μ\)\.P\_\{af\+bg\}\(\\mu\)=\\int\_\{\\Omega\}\(af\+bg\)\\,\\mathrm\{d\}\\mu=a\\int\_\{\\Omega\}f\\,\\mathrm\{d\}\\mu\+b\\int\_\{\\Omega\}g\\,\\mathrm\{d\}\\mu=a\\,P\_\{f\}\(\\mu\)\+b\\,P\_\{g\}\(\\mu\)\.For the second,μt:=\(1−t\)​μ0\+t​μ1∈𝒫⁡\(Ω\)\\mu\_\{t\}:=\(1\-t\)\\mu\_\{0\}\+t\\mu\_\{1\}\\in\\mathcal\{P\}\(\\Omega\), and for every BorelB⊆ΩB\\subseteq\\Omegawe haveμt​\(B\)=\(1−t\)​μ0​\(B\)\+t​μ1​\(B\)\\mu\_\{t\}\(B\)=\(1\-t\)\\mu\_\{0\}\(B\)\+t\\mu\_\{1\}\(B\); the identity∫Ωf​d​μt=\(1−t\)​∫Ωf​d​μ0\+t​∫Ωf​d​μ1\\int\_\{\\Omega\}f\\,\\mathrm\{d\}\\mu\_\{t\}=\(1\-t\)\\int\_\{\\Omega\}f\\,\\mathrm\{d\}\\mu\_\{0\}\+t\\int\_\{\\Omega\}f\\,\\mathrm\{d\}\\mu\_\{1\}therefore holds for simpleffand extends to allf∈C⁡\(Ω\)f\\in C\(\\Omega\)by uniform approximation and boundedness offfon the compactΩ\\Omega\. HencePf​\(μt\)=\(1−t\)​Pf​\(μ0\)\+t​Pf​\(μ1\)P\_\{f\}\(\\mu\_\{t\}\)=\(1\-t\)P\_\{f\}\(\\mu\_\{0\}\)\+tP\_\{f\}\(\\mu\_\{1\}\)\. ∎

Details on the Slicing Kernel Assumption

We provide additional details for Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1)\. Letf∼σ=GP⁡\(0,κ\)f\\sim\\sigma=\\mathrm\{GP\}\(0,\\kappa\), whereκ\\kappais continuous and universal on the compact spaceΩ\\Omega, i\.e\., its RKHSℋκ\\mathcal\{H\}\_\{\\kappa\}is dense inC⁡\(Ω\)C\(\\Omega\)\. A standard example is the Gaussian radial basis function \(RBF\) kernel on compactΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}\([Steinwart, 2001](https://arxiv.org/html/2610.20743#bib.bib41);[Micchelli et al\., 2006](https://arxiv.org/html/2610.20743#bib.bib29)\)\.

The canonical metric induced by the Gaussian process is

dκ​\(x,y\):=\(𝔼f∼σ​\[\(f⁡\(x\)−f⁡\(y\)\)2\]\)1/2=\(κ⁡\(x,x\)−2​κ​\(x,y\)\+κ⁡\(y,y\)\)1/2\.d\_\{\\kappa\}\(x,y\):=\\left\(\\mathbb\{E\}\_\{f\\sim\\sigma\}\\big\[\(f\(x\)\-f\(y\)\)^\{2\}\\big\]\\right\)^\{1/2\}=\\big\(\\kappa\(x,x\)\-2\\kappa\(x,y\)\+\\kappa\(y,y\)\\big\)^\{1/2\}\.For a metricdd, the covering number is defined as

N\(Ω,d,ε\):=min\{n∈ℕ:∃x1,…,xn∈Ωsuch thatΩ=⋃i=1nBd\(xi,ε\)\}\.N\(\\Omega,d,\\varepsilon\):=\\min\\left\\\{n\\in\\mathbb\{N\}:\\exists x\_\{1\},\\ldots,x\_\{n\}\\in\\Omega\\text\{ such that \}\\Omega=\\bigcup\_\{i=1\}^\{n\}B\_\{d\}\(x\_\{i\},\\varepsilon\)\\right\\\}\.The sample\-path continuity required by Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1)is guaranteed by Dudley’s entropy condition

∫01log⁡N⁡\(Ω,dκ,ε\)​𝑑ε<∞,\\int\_\{0\}^\{1\}\\sqrt\{\\log N\(\\Omega,d\_\{\\kappa\},\\varepsilon\)\}\\,\\,\\mathrm\{d\}\\varepsilon<\\infty,whereNNdenotes the covering number\. Under this condition, the Gaussian process admits a version with almost surely continuous sample paths\([Dudley, 1967](https://arxiv.org/html/2610.20743#bib.bib15);[Adler and Taylor, 2007](https://arxiv.org/html/2610.20743#bib.bib1)\)\. Universality remains a separate requirement onκ\\kappa\.

###### Lemma A\.1\.

Under Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1), \(i\) there isε\>0\\varepsilon\>0with𝔼σ​exp⁡\(ε​‖f‖∞2\)<∞\\mathbb\{E\}\_\{\\sigma\}\\exp\(\\varepsilon\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{2\}\)<\\infty; in particularΛr:=𝔼σ​‖f‖∞r<∞\\Lambda\_\{r\}:=\\mathbb\{E\}\_\{\\sigma\}\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{r\}<\\inftyfor allr≥1r\\geq 1\([Fernique, 1970](https://arxiv.org/html/2610.20743#bib.bib17);[Ledoux and Talagrand, 1991](https://arxiv.org/html/2610.20743#bib.bib26)\); \(ii\)supp⁡\(σ\)=ℋκ¯‖⋅‖∞=C⁡\(Ω\)\\mathrm\{supp\}\(\\sigma\)=\\overline\{\\mathcal\{H\}\_\{\\kappa\}\}^\{\\left\\lVert\\cdot\\right\\rVert\_\{\\infty\}\}=C\(\\Omega\)\([Bogachev, 1998](https://arxiv.org/html/2610.20743#bib.bib6), Thm\. 3\.6\.1\), see also\([van der Vaart and van Zanten, 2008](https://arxiv.org/html/2610.20743#bib.bib44)\)\.

###### Assumption 2\.

We assume that the underlying spaceΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}is compact and thatκ∈C2​\(Ω×Ω\)\\kappa\\in C^\{2\}\(\\Omega\\times\\Omega\)\. Then the derivative process∇f=\(∂1f,…,∂df\)\\nabla f=\(\\partial\_\{1\}f,\\dots,\\partial\_\{d\}f\)is a centered Gaussian field with covariance

𝔼σ​\[∂if⁡\(x\)​∂jf⁡\(y\)\]=∂2κ⁡\(x,y\)∂xi​∂yj,i,j=1,…,d,\\mathbb\{E\}\_\{\\sigma\}\\big\[\\partial\_\{i\}f\(x\)\\,\\partial\_\{j\}f\(y\)\\big\]=\\frac\{\\partial^\{2\}\\kappa\(x,y\)\}\{\\partial x\_\{i\}\\,\\partial y\_\{j\}\},\\qquad i,j=1,\\dots,d,and we assume that its canonical metric

d∇​\(x,y\):=\(𝔼σ​‖∇f​\(x\)−∇f​\(y\)‖2\)1/2d\_\{\\nabla\}\(x,y\):=\\big\(\\mathbb\{E\}\_\{\\sigma\}\\left\\lVert\\nabla f\(x\)\-\\nabla f\(y\)\\right\\rVert^\{2\}\\big\)^\{1/2\}satisfies Dudley’s entropy condition∫01log⁡N⁡\(Ω,d∇,ε\)​𝑑ε<∞\\int\_\{0\}^\{1\}\\sqrt\{\\log N\(\\Omega,d\_\{\\nabla\},\\varepsilon\)\}\\,\\,\\mathrm\{d\}\\varepsilon<\\infty\. Under these hypothesesf∈C1​\(Ω\)f\\in C^\{1\}\(\\Omega\)forσ\\sigma\-a\.e\.ff, i\.e\.σ​\(C1​\(Ω\)\)=1\\sigma\\big\(C^\{1\}\(\\Omega\)\\big\)=1, and the Borell–TIS inequality\([Adler and Taylor, 2007](https://arxiv.org/html/2610.20743#bib.bib1)\)gives

𝔼σ​\[supx∈Ω‖∇f​\(x\)‖r\]<∞for all​r≥1\.\\mathbb\{E\}\_\{\\sigma\}\\Big\[\\sup\_\{x\\in\\Omega\}\\left\\lVert\\nabla f\(x\)\\right\\rVert^\{r\}\\Big\]<\\infty\\qquad\\text\{for all \}r\\geq 1\.For instance, the Gaussian RBF kernel isC∞C^\{\\infty\}and satisfies these conditions, withC∞C^\{\\infty\}sample paths\.

#### Schoenberg theory\.

A metricddon a setXXis*Hilbertian*if it is realized by an isometric feature map: there exist a Hilbert spaceℋ\\mathcal\{H\}and a mapϕ:X→ℋ\\phi:X\\to\\mathcal\{H\}with

d⁡\(x,y\)=‖ϕ⁡\(x\)−ϕ⁡\(y\)‖ℋfor all​x,y∈X\.d\(x,y\)=\\left\\lVert\\phi\(x\)\-\\phi\(y\)\\right\\rVert\_\{\\mathcal\{H\}\}\\qquad\\text\{for all \}x,y\\in X\.A symmetric functionψ:X×X→ℝ\\psi:X\\times X\\to\\mathbb\{R\}withψ⁡\(x,x\)=0\\psi\(x,x\)=0is*conditionally negative definite*\(CND\) if

∑i,j=1nci​cj​ψ​\(xi,xj\)≤0whenever​n∈ℕ,x1,…,xn∈X,c∈ℝn,∑i=1nci=0,\\sum\_\{i,j=1\}^\{n\}c\_\{i\}c\_\{j\}\\,\\psi\(x\_\{i\},x\_\{j\}\)\\;\\leq\\;0\\qquad\\text\{whenever \}n\\in\\mathbb\{N\},\\ x\_\{1\},\\dots,x\_\{n\}\\in X,\\ c\\in\\mathbb\{R\}^\{n\},\\ \\sum\_\{i=1\}^\{n\}c\_\{i\}=0,that is, negative semi\-definiteness restricted to the hyperplane of zero\-sum coefficient vectors\. Schoenberg’s theorem states thatddis Hilbertian if and only ifd2d^\{2\}is CND, and that a nonnegative CND function stays CND under fractional powers:ψ\\psiCND impliesψα\\psi^\{\\alpha\}CND for everyα∈\(0,1\]\\alpha\\in\(0,1\]\. Applying this withψ=d2\\psi=d^\{2\}andα=β/2\\alpha=\\beta/2gives thatdβd^\{\\beta\}is CND for everyβ∈\(0,2\]\\beta\\in\(0,2\], and by the CND–PD correspondence

dβ​is CND⇔exp⁡\(−γ​dβ\)​is positive definite for every​γ\>0,d^\{\\beta\}\\text\{ is CND\}\\iff\\exp\(\-\\gamma d^\{\\beta\}\)\\text\{ is positive definite for every \}\\gamma\>0,we conclude that

Kγ,β​\(x,y\)=exp⁡\(−γ​d​\(x,y\)β\)K\_\{\\gamma,\\beta\}\(x,y\)=\\exp\\big\(\-\\gamma\\,d\(x,y\)^\{\\beta\}\\big\)is a positive definite kernel onXXfor allγ\>0\\gamma\>0andβ∈\(0,2\]\\beta\\in\(0,2\]\([Schoenberg, 1938](https://arxiv.org/html/2610.20743#bib.bib39);[Berg et al\., 1984](https://arxiv.org/html/2610.20743#bib.bib4)\)\.

## Appendix BOur method: Position between kernel and transport lifts

Writingmμ:=∫κ⁡\(⋅,x\)​𝑑μ​\(x\)∈ℋκm\_\{\\mu\}:=\\int\\kappa\(\\cdot,x\)\\,\\mathrm\{d\}\\mu\(x\)\\in\\mathcal\{H\}\_\{\\kappa\}, the Gaussian fieldμ↦Pf​\(μ\)\\mu\\mapsto P\_\{f\}\(\\mu\)has covariance𝔼σ​\[Pf​\(μ\)​Pf​\(ν\)\]=⟨mμ,mν⟩ℋκ\\mathbb\{E\}\_\{\\sigma\}\[P\_\{f\}\(\\mu\)P\_\{f\}\(\\nu\)\]=\\langle m\_\{\\mu\},m\_\{\\nu\}\\rangle\_\{\\mathcal\{H\}\_\{\\kappa\}\}\. Thus VSW uses Gaussian linear probes of the embedded meta\-measures, in the sense of an isonormal Gaussian process onℋκ\\mathcal\{H\}\_\{\\kappa\}\. This connects the construction to Hilbert\-space slicing\([Han, 2023](https://arxiv.org/html/2610.20743#bib.bib21)\)and kernel expected sliced Wasserstein\([Naslidnyk et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib32)\)\. It does not identify their slicing laws: our GP isC⁡\(Ω\)C\(\\Omega\)\-valued and need not beℋκ\\mathcal\{H\}\_\{\\kappa\}\-valued, whereas a Hilbert\-valued Gaussian requires trace\-class covariance\. Intrinsic sliced Wasserstein\([Rustamov and Majumdar, 2023](https://arxiv.org/html/2610.20743#bib.bib38)\)instead pushes each inner measure through Laplacian eigenfunctions to one\-dimensional laws; VSW takes scalar expectations and then compares their outer distributions\.

DefineMMDκ​\(μ,ν\)=‖∫κ⁡\(⋅,x\)​d​\(μ−ν\)​\(x\)‖ℋκ\\mathrm\{MMD\}\_\{\\kappa\}\(\\mu,\\nu\)=\\left\\lVert\\int\\kappa\(\\cdot,x\)\\,\\mathrm\{d\}\(\\mu\-\\nu\)\(x\)\\right\\rVert\_\{\\mathcal\{H\}\_\{\\kappa\}\}, which satisfies

MMDκ2​\(μ,ν\)=∬κ​d​\(μ−ν\)⊗d⁡\(μ−ν\)=𝔼f∼σ​\[⟨f,μ−ν⟩2\]\.\\mathrm\{MMD\}\_\{\\kappa\}^\{2\}\(\\mu,\\nu\)=\\iint\\kappa\\,\\,\\mathrm\{d\}\(\\mu\-\\nu\)\\otimes\\,\\mathrm\{d\}\(\\mu\-\\nu\)=\\mathbb\{E\}\_\{f\\sim\\sigma\}\\big\[\\langle f,\\mu\-\\nu\\rangle^\{2\}\\big\]\.\(8\)
Writecp:=𝔼​\|Z\|p=2p/2​Γ​\(p\+12\)/πc\_\{p\}:=\\mathbb\{E\}\|Z\|^\{p\}=2^\{p/2\}\\Gamma\\big\(\\tfrac\{p\+1\}\{2\}\\big\)/\\sqrt\{\\pi\}forZ∼𝒩⁡\(0,1\)Z\\sim\\mathcal\{N\}\(0,1\)\(soc2=1c\_\{2\}=1,c1=2/πc\_\{1\}=\\sqrt\{2/\\pi\}\)\.

###### Proposition B\.1\(Exact restriction to Diracs:VSW\\mathrm\{VSW\}extends MMD\)\.

For allμ,ν∈𝒫⁡\(Ω\)\\mu,\\nu\\in\\mathcal\{P\}\(\\Omega\)andp∈\[1,∞\)p\\in\[1,\\infty\),

VSWp​\(δμ,δν\)=cp1/p​MMDκ​\(μ,ν\)\.\\mathrm\{VSW\}\_\{p\}\\big\(\\delta\_\{\\mu\},\\delta\_\{\\nu\}\\big\)\\;=\\;c\_\{p\}^\{1/p\}\\,\\mathrm\{MMD\}\_\{\\kappa\}\(\\mu,\\nu\)\.In particular the embeddingμ↦δμ\\mu\\mapsto\\delta\_\{\\mu\}is, up to the constantcp1/pc\_\{p\}^\{1/p\}, an isometry from\(𝒫⁡\(Ω\),MMDκ\)\(\\mathcal\{P\}\(\\Omega\),\\mathrm\{MMD\}\_\{\\kappa\}\)into\(𝒫⁡\(𝒫⁡\(Ω\)\),VSWp\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{p\}\)\.

###### Proof of Proposition[B\.1](https://arxiv.org/html/2610.20743#A2.Thmtheorem1)\.

Underδμ\\delta\_\{\\mu\}the projected law is the point mass at⟨f,μ⟩\\langle f,\\mu\\rangle, so the slice value is\|⟨f,μ−ν⟩\|\|\\langle f,\\mu\-\\nu\\rangle\|\. The scalar⟨f,μ−ν⟩\\langle f,\\mu\-\\nu\\rangleis centered Gaussian with varianceMMDκ2​\(μ,ν\)\\mathrm\{MMD\}\_\{\\kappa\}^\{2\}\(\\mu,\\nu\)by \([8](https://arxiv.org/html/2610.20743#A2.E8)\); takingpp\-th absolute moments andpp\-th roots gives the claim\. ∎

###### Proposition B\.2\(Upper bound: Wasserstein over MMD, tight on Diracs\)\.

For allℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)andp∈\[1,∞\)p\\in\[1,\\infty\),

VSWp​\(ℙ,ℚ\)≤cp1/p​𝕎p,MMDκ​\(ℙ,ℚ\),\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\;\\leq\\;c\_\{p\}^\{1/p\}\\;\\mathbb\{W\}\_\{p,\\mathrm\{MMD\}\_\{\\kappa\}\}\(\\mathbb\{P\},\\mathbb\{Q\}\),with equality whenℙ,ℚ\\mathbb\{P\},\\mathbb\{Q\}are Dirac meta\-measures\. If in addition Assumption[2](https://arxiv.org/html/2610.20743#Thmassumption2)holds andLp:=𝔼σ​\[Lip​\(f\)p\]1/p<∞L\_\{p\}:=\\mathbb\{E\}\_\{\\sigma\}\[\\mathrm\{Lip\}\(f\)^\{p\}\]^\{1/p\}<\\infty, then alsoVSWp​\(ℙ,ℚ\)≤Lp​𝕎p,W1​\(ℙ,ℚ\)≤Lp​𝕎p,Wp​\(ℙ,ℚ\)\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq L\_\{p\}\\,\\mathbb\{W\}\_\{p,\\mathrm\{W\}\_\{1\}\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq L\_\{p\}\\,\\mathbb\{W\}\_\{p,\\mathrm\{W\}\_\{p\}\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\.

###### Proof of Proposition[B\.2](https://arxiv.org/html/2610.20743#A2.Thmtheorem2)\.

LetΓ∈Π⁡\(ℙ,ℚ\)\\Gamma\\in\\Pi\(\\mathbb\{P\},\\mathbb\{Q\}\)be arbitrary\. For eachff,\(Pf×Pf\)\#​Γ\(P\_\{f\}\\times P\_\{f\}\)\_\{\\\#\}\\Gammacouples the projected laws, soWpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)≤∫\|⟨f,μ−ν⟩\|p​𝑑Γ​\(μ,ν\)\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\\leq\\int\|\\langle f,\\mu\-\\nu\\rangle\|^\{p\}\\,\\mathrm\{d\}\\Gamma\(\\mu,\\nu\)\. Integrate inffand use Fubini:

VSWpp​\(ℙ,ℚ\)≤∫\(𝔼σ​\|⟨f,μ−ν⟩\|p\)​𝑑Γ​\(μ,ν\)=cp​∫MMDκ​\(μ,ν\)p​𝑑Γ​\(μ,ν\)\.\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\int\\Big\(\\mathbb\{E\}\_\{\\sigma\}\|\\langle f,\\mu\-\\nu\\rangle\|^\{p\}\\Big\)\\,\\mathrm\{d\}\\Gamma\(\\mu,\\nu\)=c\_\{p\}\\int\\mathrm\{MMD\}\_\{\\kappa\}\(\\mu,\\nu\)^\{p\}\\,\\mathrm\{d\}\\Gamma\(\\mu,\\nu\)\.Taking the infimum overΓ\\Gammaproves the bound\. ∎

###### Proposition B\.3\(Lower bound: intensities in MMD\)\.

For allℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)and allp∈\[1,∞\)p\\in\[1,\\infty\),

VSWp​\(ℙ,ℚ\)≥cp1/p​MMDκ​\(mℙ,mℚ\),\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\;\\geq\\;c\_\{p\}^\{1/p\}\\,\\mathrm\{MMD\}\_\{\\kappa\}\\big\(m\_\{\\mathbb\{P\}\},\\,m\_\{\\mathbb\{Q\}\}\\big\),wheremℙ,mℚm\_\{\\mathbb\{P\}\},m\_\{\\mathbb\{Q\}\}are the intensity measures, in particular,mℙ​\(B\)=𝔼μ∼ℙ​\[μ⁡\(B\)\],∀B∈ℬ⁡\(Ω\)m\_\{\\mathbb\{P\}\}\(B\)=\\mathbb\{E\}\_\{\\mu\\sim\\mathbb\{P\}\}\[\\mu\(B\)\],\\forall B\\in\\mathcal\{B\}\(\\Omega\)\.

###### Proof of Proposition[B\.3](https://arxiv.org/html/2610.20743#A2.Thmtheorem3)\.

For lawsα,β\\alpha,\\betaonℝ\\mathbb\{R\}, any coupling\(X,Y\)\(X,Y\)satisfies, by Jensen,𝔼​\|X−Y\|p≥\|𝔼​X−𝔼​Y\|p\\mathbb\{E\}\|X\-Y\|^\{p\}\\geq\|\\mathbb\{E\}X\-\\mathbb\{E\}Y\|^\{p\}; henceWp​\(α,β\)≥\|mean⁡\(α\)−mean⁡\(β\)\|\\mathrm\{W\}\_\{p\}\(\\alpha,\\beta\)\\geq\|\\operatorname\{mean\}\(\\alpha\)\-\\operatorname\{mean\}\(\\beta\)\|\. The mean of\(Pf\)\#​ℙ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}is𝔼μ∼ℙ​⟨f,μ⟩=⟨f,mℙ⟩\\mathbb\{E\}\_\{\\mu\\sim\\mathbb\{P\}\}\\langle f,\\mu\\rangle=\\langle f,m\_\{\\mathbb\{P\}\}\\rangleby Fubini\. ThereforeVSWpp≥𝔼σ​\|⟨f,mℙ−mℚ⟩\|p\\mathrm\{VSW\}\_\{p\}^\{p\}\\geq\\mathbb\{E\}\_\{\\sigma\}\|\\langle f,m\_\{\\mathbb\{P\}\}\-m\_\{\\mathbb\{Q\}\}\\rangle\|^\{p\}\. The scalar⟨f,mℙ−mℚ⟩\\langle f,m\_\{\\mathbb\{P\}\}\-m\_\{\\mathbb\{Q\}\}\\rangleis centered Gaussian with varianceMMDκ2​\(mℙ,mℚ\)\\mathrm\{MMD\}\_\{\\kappa\}^\{2\}\(m\_\{\\mathbb\{P\}\},m\_\{\\mathbb\{Q\}\}\)by \([8](https://arxiv.org/html/2610.20743#A2.E8)\)\. ∎

###### Example B\.4\(Strictness: identical intensities, positiveVSW\\mathrm\{VSW\}\)\.

Leta≠b∈Ωa\\neq b\\in\\Omegaand consider

ℙ=δ12​\(δa\+δb\)versusℚ=12​δδa\+12​δδb\.\\mathbb\{P\}=\\delta\_\{\\frac\{1\}\{2\}\(\\delta\_\{a\}\+\\delta\_\{b\}\)\}\\qquad\\text\{versus\}\\qquad\\mathbb\{Q\}=\\tfrac\{1\}\{2\}\\,\\delta\_\{\\delta\_\{a\}\}\+\\tfrac\{1\}\{2\}\\,\\delta\_\{\\delta\_\{b\}\}\.Both have intensitymℙ=mℚ=12​\(δa\+δb\)m\_\{\\mathbb\{P\}\}=m\_\{\\mathbb\{Q\}\}=\\frac\{1\}\{2\}\(\\delta\_\{a\}\+\\delta\_\{b\}\), so every comparison based on intensities—in particular the level\-2 MMD of mean embeddings—vanishes, and Proposition[B\.3](https://arxiv.org/html/2610.20743#A2.Thmtheorem3)is uninformative\. Fixf∈C⁡\(Ω\)f\\in C\(\\Omega\)\. Underℙ\\mathbb\{P\}the projected law is the point mass ats:=12​\(f​\(a\)\+f​\(b\)\)s:=\\frac\{1\}\{2\}\(f\(a\)\+f\(b\)\); underℚ\\mathbb\{Q\}it is12​δf⁡\(a\)\+12​δf⁡\(b\)\\frac\{1\}\{2\}\\delta\_\{f\(a\)\}\+\\frac\{1\}\{2\}\\delta\_\{f\(b\)\}\. The only coupling of a Dirac with another law is the product, andf⁡\(a\)−s=s−f⁡\(b\)=12​\(f⁡\(a\)−f⁡\(b\)\)f\(a\)\-s=s\-f\(b\)=\\frac\{1\}\{2\}\(f\(a\)\-f\(b\)\), so

Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)=12​\|f⁡\(a\)−s\|p\+12​\|f⁡\(b\)−s\|p=2−p​\|f⁡\(a\)−f⁡\(b\)\|p\.\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)=\\tfrac\{1\}\{2\}\|f\(a\)\-s\|^\{p\}\+\\tfrac\{1\}\{2\}\|f\(b\)\-s\|^\{p\}=2^\{\-p\}\\,\|f\(a\)\-f\(b\)\|^\{p\}\.Sincef⁡\(a\)−f⁡\(b\)∼𝒩⁡\(0,dκ​\(a,b\)2\)f\(a\)\-f\(b\)\\sim\\mathcal\{N\}\\big\(0,d\_\{\\kappa\}\(a,b\)^\{2\}\\big\), integrating inffgives

VSWpp​\(ℙ,ℚ\)=2−p​𝔼σ​\|f⁡\(a\)−f⁡\(b\)\|p=2−p​cp​dκ​\(a,b\)p\>0,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=2^\{\-p\}\\,\\mathbb\{E\}\_\{\\sigma\}\|f\(a\)\-f\(b\)\|^\{p\}=2^\{\-p\}c\_\{p\}\\,d\_\{\\kappa\}\(a,b\)^\{p\}\>0,the strict positivity becausedκ​\(a,b\)\>0d\_\{\\kappa\}\(a,b\)\>0wheneverκ\\kappais strictly positive definite anda≠ba\\neq b\. ThusVSW\\mathrm\{VSW\}sees the outer \(transport\) structure that intensity comparisons cannot; every slice withf⁡\(a\)≠f⁡\(b\)f\(a\)\\neq f\(b\)separates the two meta\-measures\.

###### Corollary B\.6\(Universal characteristic kernels on meta\-measures\)\.

For everyγ\>0\\gamma\>0andβ∈\(0,2\]\\beta\\in\(0,2\], the kernel

Kγ,β​\(ℙ,ℚ\)=exp⁡\(−γ​VSW2​\(ℙ,ℚ\)β\)K\_\{\\gamma,\\beta\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\exp\\big\(\-\\gamma\\,\\mathrm\{VSW\}\_\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)^\{\\beta\}\\big\)is positive definite on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. Moreover, it is continuous and*universal*on the compact metric space\(𝒫⁡\(𝒫⁡\(Ω\)\),weak\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{weak\}\), hence characteristic\. Consequently, the MMD induced byKγ,βK\_\{\\gamma,\\beta\}is a metric on the space of Borel probability measures over𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\.

###### Proof of Corollary[B\.6](https://arxiv.org/html/2610.20743#A2.Thmtheorem6)\.

*Positive definiteness\.*By Theorem[3\.6](https://arxiv.org/html/2610.20743#S3.Thmtheorem6), there is an isometric embeddingT:\(𝒫⁡\(𝒫⁡\(Ω\)\),VSW2\)→HT:\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{2\}\)\\to Hinto a Hilbert spaceHHwithVSW2​\(ℙ,ℚ\)=‖T⁡\(ℙ\)−T⁡\(ℚ\)‖H\\mathrm\{VSW\}\_\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\\|T\(\\mathbb\{P\}\)\-T\(\\mathbb\{Q\}\)\\\|\_\{H\}\. HenceVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}is conditionally negative definite \(CND\)\. For everyβ∈\(0,2\]\\beta\\in\(0,2\], we haveβ/2∈\(0,1\]\\beta/2\\in\(0,1\], sor↦rβ/2r\\mapsto r^\{\\beta/2\}is a Bernstein function on\[0,∞\)\[0,\\infty\)\. Applying it to the nonnegative CND functionVSW22\\mathrm\{VSW\}\_\{2\}^\{2\}shows that\(VSW22\)β/2=VSW2β\(\\mathrm\{VSW\}\_\{2\}^\{2\}\)^\{\\beta/2\}=\\mathrm\{VSW\}\_\{2\}^\{\\beta\}is CND\. Schoenberg’s theorem then yields thatKγ,β​\(ℙ,ℚ\)=exp⁡\(−γ​VSW2​\(ℙ,ℚ\)β\)K\_\{\\gamma,\\beta\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\exp\\big\(\-\\gamma\\,\\mathrm\{VSW\}\_\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)^\{\\beta\}\\big\)is positive definite for allγ\>0\\gamma\>0andβ∈\(0,2\]\\beta\\in\(0,2\]\.

*Continuity\.*By Proposition[3\.5](https://arxiv.org/html/2610.20743#S3.Thmtheorem5),VSW2\\mathrm\{VSW\}\_\{2\}induces the weak topology on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), hence\(ℙ,ℚ\)↦VSW2​\(ℙ,ℚ\)\(\\mathbb\{P\},\\mathbb\{Q\}\)\\mapsto\\mathrm\{VSW\}\_\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)is continuous for the weak topology\. Composing witht↦exp⁡\(−γ​tβ\)t\\mapsto\\exp\(\-\\gamma t^\{\\beta\}\)gives continuity ofKγ,βK\_\{\\gamma,\\beta\}\.

*Universality and characteristicness\.*The space\(𝒫⁡\(𝒫⁡\(Ω\)\),weak\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\text\{weak\}\)is compact\. Forβ=2\\beta=2,Kγ,2​\(ℙ,ℚ\)=exp⁡\(−γ​‖T⁡\(ℙ\)−T⁡\(ℚ\)‖H2\)K\_\{\\gamma,2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\exp\(\-\\gamma\\\|T\(\\mathbb\{P\}\)\-T\(\\mathbb\{Q\}\)\\\|\_\{H\}^\{2\}\)is the restriction of a Gaussian kernel on the Hilbert spaceHHto the compact setT⁡\(𝒫⁡\(𝒫⁡\(Ω\)\)\)T\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\)\. Such a Gaussian kernel is universal on compact subsets of a separable Hilbert space \(e\.g\.[Christmann and Steinwart, 2010](https://arxiv.org/html/2610.20743#bib.bib13)\); thereforeKγ,2K\_\{\\gamma,2\}is universal on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\.

Forβ∈\(0,2\)\\beta\\in\(0,2\), the functionψ⁡\(t\):=exp⁡\(−γ​tβ/2\)\\psi\(t\):=\\exp\(\-\\gamma t^\{\\beta/2\}\)is completely monotone on\[0,∞\)\[0,\\infty\), so by Bernstein’s theorem there exists a probability measureνβ\\nu\_\{\\beta\}on\[0,∞\)\[0,\\infty\)such thatψ⁡\(t\)=∫0∞e−s​t​d​νβ​\(s\)\\psi\(t\)=\\int\_\{0\}^\{\\infty\}e^\{\-st\}\\,\\mathrm\{d\}\\nu\_\{\\beta\}\(s\)andνβ​\(\(0,∞\)\)\>0\\nu\_\{\\beta\}\(\(0,\\infty\)\)\>0\. Substitutingt=VSW22​\(ℙ,ℚ\)t=\\mathrm\{VSW\}\_\{2\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)gives

Kγ,β​\(ℙ,ℚ\)=∫0∞exp⁡\(−s​VSW22​\(ℙ,ℚ\)\)​d​νβ​\(s\)\.K\_\{\\gamma,\\beta\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\int\_\{0\}^\{\\infty\}\\exp\\big\(\-s\\,\\mathrm\{VSW\}\_\{2\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\big\)\\,\\mathrm\{d\}\\nu\_\{\\beta\}\(s\)\.If a finite signed Borel measureη\\etaon𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)satisfies∬Kγ,β​𝑑η⊗𝑑η=0\\iint K\_\{\\gamma,\\beta\}\\,\\mathrm\{d\}\\eta\\otimes\\,\\mathrm\{d\}\\eta=0, Fubini applies because the kernels are bounded andη\\etahas finite total variation\. The nonnegative Gaussian\-kernel quadratic forms therefore vanish forνβ\\nu\_\{\\beta\}\-a\.e\.ss, hence for somes\>0s\>0\. Universality ofexp⁡\(−s​VSW22\)\\exp\(\-s\\,\\mathrm\{VSW\}\_\{2\}^\{2\}\)forcesη=0\\eta=0by the equivalence between universality and integral strict positive definiteness on compact spaces\([Sriperumbudur et al\., 2011](https://arxiv.org/html/2610.20743#bib.bib40), Prop\. 4\)\. The same equivalence proves universality ofKγ,βK\_\{\\gamma,\\beta\}; universal continuous kernels on compact metric spaces are characteristic\. ∎

## Appendix CProofs of the Statements in Section[3](https://arxiv.org/html/2610.20743#S3)

###### Proof of Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\.

\(i\) Let\(fn,μn\)→\(f,μ\)\(f\_\{n\},\\mu\_\{n\}\)\\to\(f,\\mu\)inC⁡\(Ω\)×𝒫⁡\(Ω\)C\(\\Omega\)\\times\\mathcal\{P\}\(\\Omega\)\. In particular:

- •fn→ff\_\{n\}\\to fin\(C⁡\(Ω\),‖⋅‖∞\)\(C\(\\Omega\),\\left\\lVert\\cdot\\right\\rVert\_\{\\infty\}\), i\.e\.‖fn−f‖∞=supx∈Ω\|fn​\(x\)−f⁡\(x\)\|→0\\left\\lVert f\_\{n\}\-f\\right\\rVert\_\{\\infty\}=\\sup\_\{x\\in\\Omega\}\|f\_\{n\}\(x\)\-f\(x\)\|\\to 0, which is uniform convergence onΩ\\Omega;
- •μn→μ\\mu\_\{n\}\\to\\muin𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\), i\.e\.μn⇀μ\\mu\_\{n\}\\rightharpoonup\\muweakly:∫Ωg​d​μn→∫Ωg​𝑑μ\\int\_\{\\Omega\}g\\,\\mathrm\{d\}\\mu\_\{n\}\\to\\int\_\{\\Omega\}g\\,\\mathrm\{d\}\\mufor everyg∈C⁡\(Ω\)g\\in C\(\\Omega\)\.

Then

\|⟨fn,μn⟩−⟨f,μ⟩\|\\displaystyle\\left\|\\langle f\_\{n\},\\mu\_\{n\}\\rangle\-\\langle f,\\mu\\rangle\\right\|≤\|⟨fn−f,μn⟩\|\+\|⟨f,μn⟩−⟨f,μ⟩\|≤‖fn−f‖∞\+\|⟨f,μn⟩−⟨f,μ⟩\|\.\\displaystyle\\leq\\left\|\\langle f\_\{n\}\-f,\\mu\_\{n\}\\rangle\\right\|\+\\left\|\\langle f,\\mu\_\{n\}\\rangle\-\\langle f,\\mu\\rangle\\right\|\\leq\\left\\lVert f\_\{n\}\-f\\right\\rVert\_\{\\infty\}\+\\left\|\\langle f,\\mu\_\{n\}\\rangle\-\\langle f,\\mu\\rangle\\right\|\.The first term tends to zero by uniform convergence, while the second tends to zero by weak convergence, since the fixed functionf∈C⁡\(Ω\)f\\in C\(\\Omega\)is bounded and continuous\. HencePPis jointly continuous\.

\(ii\) Choose fixedf,g∈C⁡\(Ω\)f,g\\in C\(\\Omega\), the deterministic mapμ↦\(Pf​\(μ\),Pg​\(μ\)\)\\mu\\mapsto\(P\_\{f\}\(\\mu\),P\_\{g\}\(\\mu\)\)pushesℙ\\mathbb\{P\}to a coupling between\(Pf\)\#​ℙ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}and\(Pg\)\#​ℙ\(P\_\{g\}\)\_\{\\\#\}\\mathbb\{P\}\. We have:

𝔼μ∼ℙ​\|Pf​\(μ\)−Pg​\(μ\)\|p\\displaystyle\\mathbb\{E\}\_\{\\mu\\sim\\mathbb\{P\}\}\|P\_\{f\}\(\\mu\)\-P\_\{g\}\(\\mu\)\|^\{p\}=𝔼μ∼ℙ​\|∫\(f−g\)​𝑑μ\|p\\displaystyle=\\mathbb\{E\}\_\{\\mu\\sim\\mathbb\{P\}\}\|\\int\(f\-g\)d\\mu\|^\{p\}≤𝔼μ∼ℙ​∫\|f−g\|p​𝑑μ\\displaystyle\\leq\\mathbb\{E\}\_\{\\mu\\sim\\mathbb\{P\}\}\\int\|f\-g\|^\{p\}d\\mu≤supx∈Ω\|f⁡\(x\)−g⁡\(x\)\|p\\displaystyle\\leq\\sup\_\{x\\in\\Omega\}\|f\(x\)\-g\(x\)\|^\{p\}=‖f−g‖∞p;\\displaystyle=\\left\\lVert f\-g\\right\\rVert\_\{\\infty\}^\{p\};henceWp​\(\(Pf\)\#​ℙ,\(Pg\)\#​ℙ\)≤‖f−g‖∞\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{g\}\)\_\{\\\#\}\\mathbb\{P\}\\big\)\\leq\\left\\lVert f\-g\\right\\rVert\_\{\\infty\}, and similarly forℚ\\mathbb\{Q\}\. The triangle inequality forWp\\mathrm\{W\}\_\{p\}on𝒫p​\(ℝ\)\\mathcal\{P\}\_\{p\}\(\\mathbb\{R\}\)then gives\|Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)−Wp​\(\(Pg\)\#​ℙ,\(Pg\)\#​ℚ\)\|≤2​‖f−g‖∞\\big\|\\mathrm\{W\}\_\{p\}\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\)\-\\mathrm\{W\}\_\{p\}\(\(P\_\{g\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{g\}\)\_\{\\\#\}\\mathbb\{Q\}\)\\big\|\\leq 2\\left\\lVert f\-g\\right\\rVert\_\{\\infty\}\. Thusf↦Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)f\\mapsto\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)is22\-Lipschitz\.

*Finiteness\.*Fixf∈C⁡\(Ω\)f\\in C\(\\Omega\)\. As noted after Definition[3\.1](https://arxiv.org/html/2610.20743#S3.Thmtheorem1), both\(Pf\)\#​ℙ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}and\(Pf\)\#​ℚ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}are supported in the intervalIf:=\[−‖f‖∞,‖f‖∞\]I\_\{f\}:=\[\-\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert f\\right\\rVert\_\{\\infty\}\], of length\|If\|=2​‖f‖∞\|I\_\{f\}\|=2\\left\\lVert f\\right\\rVert\_\{\\infty\}\. Consequently every couplingπ∈Π⁡\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)\\pi\\in\\Pi\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)is supported inIf×IfI\_\{f\}\\times I\_\{f\}, on which\|x−y\|≤2​‖f‖∞\|x\-y\|\\leq 2\\left\\lVert f\\right\\rVert\_\{\\infty\}, so

Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)=infπ∫ℝ×ℝ\|x−y\|p​𝑑π​\(x,y\)≤\(2​‖f‖∞\)p\.\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)=\\inf\_\{\\pi\}\\int\_\{\\mathbb\{R\}\\times\\mathbb\{R\}\}\|x\-y\|^\{p\}\\,\\mathrm\{d\}\\pi\(x,y\)\\;\\leq\\;\\big\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\\big\)^\{p\}\.Integrating this bound againstσ\\sigmaand using Lemma[A\.1](https://arxiv.org/html/2610.20743#A1.Thmtheorem1)\(i\),

VSWpp​\(ℙ,ℚ\)=∫C⁡\(Ω\)Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)​𝑑σ​\(f\)≤2p​∫C⁡\(Ω\)‖f‖∞p​𝑑σ​\(f\)=2p​Λp<∞\.\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\int\_\{C\(\\Omega\)\}\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\\,\\mathrm\{d\}\\sigma\(f\)\\;\\leq\\;2^\{p\}\\int\_\{C\(\\Omega\)\}\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{p\}\\,\\mathrm\{d\}\\sigma\(f\)\\;=\\;2^\{p\}\\Lambda\_\{p\}\\;<\\;\\infty\.The integral in \([4](https://arxiv.org/html/2610.20743#S3.E4)\) is therefore well defined\. ∎

###### Proof of Theorem[3\.3](https://arxiv.org/html/2610.20743#S3.Thmtheorem3)\.

We first show thatVSWp\\mathrm\{VSW\}\_\{p\}is a pseudometric on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\): it is finite, symmetric, vanishes on the diagonal, and satisfies the triangle inequality\.

*Finiteness*was established in Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(ii\):VSWpp​\(ℙ,ℚ\)≤2p​Λp<∞\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq 2^\{p\}\\Lambda\_\{p\}<\\inftyfor allℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\.

*Symmetry*is inherited slicewise:Wp\\mathrm\{W\}\_\{p\}is symmetric on𝒫⁡\(ℝ\)\\mathcal\{P\}\(\\mathbb\{R\}\)\.

*Vanishing on the diagonal*: ifℙ=ℚ\\mathbb\{P\}=\\mathbb\{Q\}then\(Pf\)\#​ℙ=\(Pf\)\#​ℚ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}=\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}for everyf∈C⁡\(Ω\)f\\in C\(\\Omega\)\.

*Triangle inequality\.*Letℙ,ℚ,ℝ′∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\},\\mathbb\{R\}^\{\\prime\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)and writehℙ​ℚ​\(f\):=Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)h\_\{\\mathbb\{P\}\\mathbb\{Q\}\}\(f\):=\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\), so thatVSWp​\(ℙ,ℚ\)=‖hℙ​ℚ‖Lp​\(σ\)\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\left\\lVert h\_\{\\mathbb\{P\}\\mathbb\{Q\}\}\\right\\rVert\_\{L^\{p\}\(\\sigma\)\}by \([4](https://arxiv.org/html/2610.20743#S3.E4)\)\. For each fixedff, the triangle inequality forWp\\mathrm\{W\}\_\{p\}on𝒫⁡\(ℝ\)\\mathcal\{P\}\(\\mathbb\{R\}\)gives the pointwise bound

hℙ​ℚ​\(f\)≤hℙ​ℝ′​\(f\)\+hℝ′​ℚ​\(f\)\.h\_\{\\mathbb\{P\}\\mathbb\{Q\}\}\(f\)\\;\\leq\\;h\_\{\\mathbb\{P\}\\mathbb\{R\}^\{\\prime\}\}\(f\)\+h\_\{\\mathbb\{R\}^\{\\prime\}\\mathbb\{Q\}\}\(f\)\.TakingLp​\(σ\)L^\{p\}\(\\sigma\)norms and applying Minkowski’s inequality gives the result\.

The only property left to verify is identity of indiscernibles\. AssumeVSWp​\(ℙ,ℚ\)=0\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=0\.

Define

A:=\{f∈C⁡\(Ω\):\(Pf\)\#​ℙ=\(Pf\)\#​ℚ\}=\{f∈C⁡\(Ω\):Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)=0\}\.A:=\\big\\\{f\\in C\(\\Omega\):\\ \(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}=\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\\\}=\\big\\\{f\\in C\(\\Omega\):\\ \\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)=0\\big\\\}\.The two descriptions agree because both projected laws are compactly supported, hence lie in𝒫p​\(ℝ\)\\mathcal\{P\}\_\{p\}\(\\mathbb\{R\}\), the subspace of𝒫⁡\(ℝ\)\\mathcal\{P\}\(\\mathbb\{R\}\)of measures with finitepp\-moment, on whichWp\\mathrm\{W\}\_\{p\}is a metric\.

*Step 0 \(σ⁡\(A\)=1\\sigma\(A\)=1\)\.*By assumption and \([4](https://arxiv.org/html/2610.20743#S3.E4)\),

0=VSWpp​\(ℙ,ℚ\)=∫C⁡\(Ω\)Wpp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)​𝑑σ​\(f\)\.0=\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\int\_\{C\(\\Omega\)\}\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\\,\\mathrm\{d\}\\sigma\(f\)\.The integrand is nonnegative and Borel \(Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(ii\)\), so it vanishesσ\\sigma\-a\.e\., i\.e\.σ⁡\(A\)=1\\sigma\(A\)=1\.

*Step 1 \(A=C⁡\(Ω\)A=C\(\\Omega\)\)\.*The maph⁡\(f\):=Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)h\(f\):=\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)is continuous by Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(ii\), henceA=h−1​\(\{0\}\)A=h^\{\-1\}\(\\\{0\\\}\)is closed\. Sinceσ⁡\(A\)=1\\sigma\(A\)=1, closedness impliessupp​σ⊆A\\mathrm\{supp\}\\sigma\\subseteq A\. By Lemma[A\.1](https://arxiv.org/html/2610.20743#A1.Thmtheorem1)\(ii\) and universality \(Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1)\),supp​σ=C⁡\(Ω\)\\mathrm\{supp\}\\sigma=C\(\\Omega\), henceA=C⁡\(Ω\)A=C\(\\Omega\)\. Therefore

\(Pf\)\#​ℙ=\(Pf\)\#​ℚfor every​f∈C⁡\(Ω\)\.\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}=\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\qquad\\text\{for every \}f\\in C\(\\Omega\)\.\(9\)
*Step 2 \(finite\-dimensional distributions agree\)\.*Fixk∈ℕk\\in\\mathbb\{N\}andf1,…,fk∈C⁡\(Ω\)f\_\{1\},\\dots,f\_\{k\}\\in C\(\\Omega\)\. ConsiderX:𝒫⁡\(Ω\)→ℝkX:\\mathcal\{P\}\(\\Omega\)\\to\\mathbb\{R\}^\{k\},X⁡\(μ\)=\(Pf1​\(μ\),…,Pfk​\(μ\)\)X\(\\mu\)=\(P\_\{f\_\{1\}\}\(\\mu\),\\dots,P\_\{f\_\{k\}\}\(\\mu\)\)\. For anyt∈ℝkt\\in\\mathbb\{R\}^\{k\}, Lemma[2\.1](https://arxiv.org/html/2610.20743#S2.Thmtheorem1)gives

⟨t,X⁡\(μ\)⟩=∑j=1ktj​Pfj​\(μ\)=Pgt​\(μ\),gt:=∑j=1ktj​fj∈C⁡\(Ω\)\.\\langle t,X\(\\mu\)\\rangle=\\sum\_\{j=1\}^\{k\}t\_\{j\}P\_\{f\_\{j\}\}\(\\mu\)=P\_\{g\_\{t\}\}\(\\mu\),\\qquad g\_\{t\}:=\\sum\_\{j=1\}^\{k\}t\_\{j\}f\_\{j\}\\in C\(\\Omega\)\.Applying \([9](https://arxiv.org/html/2610.20743#A3.E9)\) togtg\_\{t\}shows that⟨t,X⟩\\langle t,X\\ranglehas the same law underμ∼ℙ\\mu\\sim\\mathbb\{P\}and underμ∼ℚ\\mu\\sim\\mathbb\{Q\}\. By the Cramér–Wold theorem, the laws ofXXunderℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}coincide, i\.e\.

\(Pf1,…,Pfk\)\#​ℙ=\(Pf1,…,Pfk\)\#​ℚin​𝒫​\(ℝk\)\.\\big\(P\_\{f\_\{1\}\},\\dots,P\_\{f\_\{k\}\}\\big\)\_\{\\\#\}\\mathbb\{P\}=\\big\(P\_\{f\_\{1\}\},\\dots,P\_\{f\_\{k\}\}\\big\)\_\{\\\#\}\\mathbb\{Q\}\\qquad\\text\{in \}\\mathcal\{P\}\(\\mathbb\{R\}^\{k\}\)\.\(10\)
*Step 3 \(ℙ=ℚ\\mathbb\{P\}=\\mathbb\{Q\}\)\.*Let

𝒞=\{\{μ∈𝒫\(Ω\):\(Pf1\(μ\),…,Pfk\(μ\)\)∈B\}:k∈ℕ,f1,…,fk∈C\(Ω\),B∈ℬ\(ℝk\)\}\.\\mathcal\{C\}=\\Big\\\{\\\{\\mu\\in\\mathcal\{P\}\(\\Omega\):\(P\_\{f\_\{1\}\}\(\\mu\),\\dots,P\_\{f\_\{k\}\}\(\\mu\)\)\\in B\\\}:\\ k\\in\\mathbb\{N\},\\ f\_\{1\},\\dots,f\_\{k\}\\in C\(\\Omega\),\\ B\\in\\mathcal\{B\}\(\\mathbb\{R\}^\{k\}\)\\Big\\\}\.Then𝒞\\mathcal\{C\}is aπ\\pi\-system \(closed under finite intersections\)\. Moreoverσ⁡\(𝒞\)=ℬ⁡\(𝒫⁡\(Ω\)\)\\sigma\(\\mathcal\{C\}\)=\\mathcal\{B\}\(\\mathcal\{P\}\(\\Omega\)\)because the weak Borelσ\\sigma\-algebra is generated by the mapsPfP\_\{f\}\(Section[2](https://arxiv.org/html/2610.20743#S2)\)\.

Defineℒ:=\{C∈ℬ⁡\(𝒫⁡\(Ω\)\):ℙ⁡\(C\)=ℚ⁡\(C\)\}\\mathcal\{L\}:=\\\{C\\in\\mathcal\{B\}\(\\mathcal\{P\}\(\\Omega\)\):\\mathbb\{P\}\(C\)=\\mathbb\{Q\}\(C\)\\\}\. This is aλ\\lambda\-system\. By \([10](https://arxiv.org/html/2610.20743#A3.E10)\), we have𝒞⊆ℒ\\mathcal\{C\}\\subseteq\\mathcal\{L\}\. Dynkin’sπ\\pi–λ\\lambdatheorem yieldsσ⁡\(𝒞\)⊆ℒ\\sigma\(\\mathcal\{C\}\)\\subseteq\\mathcal\{L\}, henceℬ⁡\(𝒫⁡\(Ω\)\)⊆ℒ\\mathcal\{B\}\(\\mathcal\{P\}\(\\Omega\)\)\\subseteq\\mathcal\{L\}and thereforeℙ=ℚ\\mathbb\{P\}=\\mathbb\{Q\}\. ∎

###### Proof of Proposition[3\.5](https://arxiv.org/html/2610.20743#S3.Thmtheorem5)\.

*Weak convergence impliesVSWp→0\\mathrm\{VSW\}\_\{p\}\\to 0\.*Letℙm⇀ℙ\\mathbb\{P\}\_\{m\}\\rightharpoonup\\mathbb\{P\}in the weak topology on𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), and fixf∈C⁡\(Ω\)f\\in C\(\\Omega\)\. SincePf:𝒫⁡\(Ω\)→ℝP\_\{f\}:\\mathcal\{P\}\(\\Omega\)\\to\\mathbb\{R\}is continuous \(Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(i\)\), for everyg∈Cb​\(ℝ\)g\\in C\_\{b\}\(\\mathbb\{R\}\)the compositiong∘Pfg\\circ P\_\{f\}belongs toC⁡\(𝒫⁡\(Ω\)\)C\(\\mathcal\{P\}\(\\Omega\)\); hence

∫ℝg​d​\(\(Pf\)\#​ℙm\)=∫𝒫⁡\(Ω\)g∘Pf​d​ℙm⟶∫𝒫⁡\(Ω\)g∘Pf​𝑑ℙ=∫ℝg​d​\(\(Pf\)\#​ℙ\),\\int\_\{\\mathbb\{R\}\}g\\,\\mathrm\{d\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\_\{m\}\\big\)=\\int\_\{\\mathcal\{P\}\(\\Omega\)\}g\\circ P\_\{f\}\\,\\mathrm\{d\}\\mathbb\{P\}\_\{m\}\\;\\longrightarrow\\;\\int\_\{\\mathcal\{P\}\(\\Omega\)\}g\\circ P\_\{f\}\\,\\mathrm\{d\}\\mathbb\{P\}=\\int\_\{\\mathbb\{R\}\}g\\,\\mathrm\{d\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\\big\),i\.e\.\(Pf\)\#​ℙm⇀\(Pf\)\#​ℙ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\_\{m\}\\rightharpoonup\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}in𝒫⁡\(ℝ\)\\mathcal\{P\}\(\\mathbb\{R\}\)\. Moreover\|Pf\|≤‖f‖∞\|P\_\{f\}\|\\leq\\left\\lVert f\\right\\rVert\_\{\\infty\}on𝒫⁡\(Ω\)\\mathcal\{P\}\(\\Omega\), so all these laws are supported in the fixed compact intervalIf=\[−‖f‖∞,‖f‖∞\]I\_\{f\}=\[\-\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert f\\right\\rVert\_\{\\infty\}\]\. On probability measures supported in a common compact interval,Wp\\mathrm\{W\}\_\{p\}metrizes weak convergence, so

Wpp​\(\(Pf\)\#​ℙm,\(Pf\)\#​ℙ\)⟶0for each fixed​f∈C⁡\(Ω\)\.\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\_\{m\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\\big\)\\;\\longrightarrow\\;0\\qquad\\text\{for each fixed \}f\\in C\(\\Omega\)\.The integrand is dominated by\(2​‖f‖∞\)p\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{p\}, which lies inL1​\(σ\)L^\{1\}\(\\sigma\)by Lemma[A\.1](https://arxiv.org/html/2610.20743#A1.Thmtheorem1)\(i\)\. Dominated convergence therefore yieldsVSWp​\(ℙm,ℙ\)→0\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\}\_\{m\},\\mathbb\{P\}\)\\to 0\.

*VSWp→0\\mathrm\{VSW\}\_\{p\}\\to 0implies weak convergence\.*Consider the identity map

ι:\(𝒫⁡\(𝒫⁡\(Ω\)\),weak\)→\(𝒫⁡\(𝒫⁡\(Ω\)\),VSWp\),\\iota:\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\text\{weak\}\)\\to\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{p\}\),which is a bijection\. The previous paragraph shows it is continuous\. Since\(𝒫⁡\(𝒫⁡\(Ω\)\),weak\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\text\{weak\}\)is compact and\(𝒫⁡\(𝒫⁡\(Ω\)\),VSWp\)\(\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\),\\mathrm\{VSW\}\_\{p\}\)is Hausdorff \(being metric\), a continuous bijection from a compact space onto a Hausdorff space is a homeomorphism\. HenceVSWp\\mathrm\{VSW\}\_\{p\}induces the weak topology\. ∎

###### Proof of Theorem[3\.6](https://arxiv.org/html/2610.20743#S3.Thmtheorem6)\.

*Measurability\.*The evaluation map\(f,μ\)↦Pf​\(μ\)\(f,\\mu\)\\mapsto P\_\{f\}\(\\mu\)is jointly continuous \(Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(i\)\)\. Hence, for fixedℙ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\), the function

G⁡\(f,s\):=ℙ⁡\{μ∈𝒫⁡\(Ω\):Pf​\(μ\)≤s\}G\(f,s\):=\\mathbb\{P\}\\\{\\mu\\in\\mathcal\{P\}\(\\Omega\):P\_\{f\}\(\\mu\)\\leq s\\\}is jointly Borel in\(f,s\)\(f,s\): the set\{\(f,s,μ\):Pf​\(μ\)−s≤0\}\\\{\(f,s,\\mu\):P\_\{f\}\(\\mu\)\-s\\leq 0\\\}is closed inC⁡\(Ω\)×ℝ×𝒫⁡\(Ω\)C\(\\Omega\)\\times\\mathbb\{R\}\\times\\mathcal\{P\}\(\\Omega\), so its indicator is jointly Borel, and Tonelli \(integrating outμ\\muagainstℙ\\mathbb\{P\}\) yields joint measurability ofGG\. Define the quantile

q⁡\(f,u\):=inf\{s∈ℚrat:G⁡\(f,s\)≥u\}\.q\(f,u\):=\\inf\\\{s\\in\\mathbb\{Q\}\_\{\\mathrm\{rat\}\}:G\(f,s\)\\geq u\\\}\.Since the infimum runs over a countable set and each subgraph\{\(f,u\):u≤G⁡\(f,s\)\}\\\{\(f,u\):u\\leq G\(f,s\)\\\}is Borel,qqis jointly Borel in\(f,u\)\(f,u\)\.

*Integrability\.*Since\(Pf\)\#​ℙ\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}is supported in\[−‖f‖∞,‖f‖∞\]\[\-\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert f\\right\\rVert\_\{\\infty\}\], we have\|q⁡\(f,u\)\|≤‖f‖∞\|q\(f,u\)\|\\leq\\left\\lVert f\\right\\rVert\_\{\\infty\}for all\(f,u\)\(f,u\)\. By Lemma[A\.1](https://arxiv.org/html/2610.20743#A1.Thmtheorem1)\(i\),𝔼σ​‖f‖∞2<∞\\mathbb\{E\}\_\{\\sigma\}\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{2\}<\\infty, soT⁡\(ℙ\)∈L2​\(σ⊗λ\)T\(\\mathbb\{P\}\)\\in L^\{2\}\(\\sigma\\otimes\\lambda\)\.

*Isometry\.*Forp=2p=2, the one\-dimensional OT formula \([2](https://arxiv.org/html/2610.20743#S2.E2)\) gives, for eachff,

W22​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)=∫01\|q\(Pf\)\#​ℙ​\(u\)−q\(Pf\)\#​ℚ​\(u\)\|2​𝑑u\.\\mathrm\{W\}\_\{2\}^\{2\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)=\\int\_\{0\}^\{1\}\\big\|q\_\{\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\}\(u\)\-q\_\{\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\}\(u\)\\big\|^\{2\}\\,\\mathrm\{d\}u\.Integrating inffand applying Tonelli yields

VSW22​\(ℙ,ℚ\)=∫C⁡\(Ω\)∫01\|T⁡\(ℙ\)​\(f,u\)−T⁡\(ℚ\)​\(f,u\)\|2​𝑑u​𝑑σ​\(f\)=‖T⁡\(ℙ\)−T⁡\(ℚ\)‖L2​\(σ⊗λ\)2\.\\mathrm\{VSW\}\_\{2\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=\\int\_\{C\(\\Omega\)\}\\\!\\int\_\{0\}^\{1\}\\big\|T\(\\mathbb\{P\}\)\(f,u\)\-T\(\\mathbb\{Q\}\)\(f,u\)\\big\|^\{2\}\\,\\mathrm\{d\}u\\,\\mathrm\{d\}\\sigma\(f\)=\\left\\lVert T\(\\mathbb\{P\}\)\-T\(\\mathbb\{Q\}\)\\right\\rVert\_\{L^\{2\}\(\\sigma\\otimes\\lambda\)\}^\{2\}\.This proves the claimed isometric embedding\. ∎

###### Proof of Theorem[3\.7](https://arxiv.org/html/2610.20743#S3.Thmtheorem7)\.

By the triangle inequality \(Theorem[3\.3](https://arxiv.org/html/2610.20743#S3.Thmtheorem3)\) and\(a\+b\)p≤2p−1​\(ap\+bp\)\(a\+b\)^\{p\}\\leq 2^\{p\-1\}\(a^\{p\}\+b^\{p\}\),

𝔼​VSWpp​\(ℙ,ℙ~\)≤2p−1​\(𝔼​VSWpp​\(ℙ,ℙ^N\)\+𝔼​VSWpp​\(ℙ^N,ℙ~\)\)\.\\mathbb\{E\}\\,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\widetilde\{\\mathbb\{P\}\}\)\\leq 2^\{p\-1\}\\Big\(\\mathbb\{E\}\\,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\widehat\{\\mathbb\{P\}\}\_\{N\}\)\+\\mathbb\{E\}\\,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\widehat\{\\mathbb\{P\}\}\_\{N\},\\widetilde\{\\mathbb\{P\}\}\)\\Big\)\.*Outer term\.*Fixffand letνf:=\(Pf\)\#​ℙ∈𝒫⁡\(If\)\\nu\_\{f\}:=\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\}\\in\\mathcal\{P\}\(I\_\{f\}\),If=\[−‖f‖∞,‖f‖∞\]I\_\{f\}=\[\-\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert f\\right\\rVert\_\{\\infty\}\]\. Then\(Pf\)\#​ℙ^N\(P\_\{f\}\)\_\{\\\#\}\\widehat\{\\mathbb\{P\}\}\_\{N\}is exactly the empirical measure of theNNi\.i\.d\. scalarsPf​\(μi\)∼νfP\_\{f\}\(\\mu\_\{i\}\)\\sim\\nu\_\{f\}\. UsingWpp≤\|If\|p−1​W1\\mathrm\{W\}\_\{p\}^\{p\}\\leq\|I\_\{f\}\|^\{p\-1\}\\mathrm\{W\}\_\{1\}onIfI\_\{f\}and \([3](https://arxiv.org/html/2610.20743#S2.E3)\),

𝔼​Wpp​\(\(Pf\)\#​ℙ^N,νf\)≤\(2​‖f‖∞\)p−1⋅2​‖f‖∞2​N=\(2​‖f‖∞\)p2​N\.\\mathbb\{E\}\\,\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\widehat\{\\mathbb\{P\}\}\_\{N\},\\nu\_\{f\}\\big\)\\leq\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{p\-1\}\\cdot\\frac\{2\\left\\lVert f\\right\\rVert\_\{\\infty\}\}\{2\\sqrt\{N\}\}=\\frac\{\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{p\}\}\{2\\sqrt\{N\}\}\.Integrating inff\(Tonelli; joint measurability as before\) gives

𝔼VSWpp\(ℙ,ℙ^N\)≤12⋅2pΛpN−1/2\.\\mathbb\{E\}\\,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\mathbb\{P\},\\widehat\{\\mathbb\{P\}\}\_\{N\}\)\\leq\\tfrac\{1\}\{2\}\\cdot 2^\{p\}\\Lambda\_\{p\}N^\{\-1/2\}\.
*Inner term\.*Pairμi\\mu\_\{i\}withμ^i\\widehat\{\\mu\}\_\{i\}: the coupling of\(Pf\)\#​ℙ^N\(P\_\{f\}\)\_\{\\\#\}\\widehat\{\\mathbb\{P\}\}\_\{N\}and\(Pf\)\#​ℙ~\(P\_\{f\}\)\_\{\\\#\}\\widetilde\{\\mathbb\{P\}\}that matches indexiito indexiiyields

Wpp​\(\(Pf\)\#​ℙ^N,\(Pf\)\#​ℙ~\)≤1N​∑i=1N\|Pf​\(μi\)−Pf​\(μ^i\)\|p\.\\mathrm\{W\}\_\{p\}^\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\widehat\{\\mathbb\{P\}\}\_\{N\},\(P\_\{f\}\)\_\{\\\#\}\\widetilde\{\\mathbb\{P\}\}\\big\)\\leq\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\big\|P\_\{f\}\(\\mu\_\{i\}\)\-P\_\{f\}\(\\widehat\{\\mu\}\_\{i\}\)\\big\|^\{p\}\.Conditionally onμi\\mu\_\{i\},Pf​\(μ^i\)−Pf​\(μi\)=1n​∑j\(f⁡\(xi​j\)−𝔼​f​\(xi​j\)\)P\_\{f\}\(\\widehat\{\\mu\}\_\{i\}\)\-P\_\{f\}\(\\mu\_\{i\}\)=\\frac\{1\}\{n\}\\sum\_\{j\}\\big\(f\(x\_\{ij\}\)\-\\mathbb\{E\}f\(x\_\{ij\}\)\\big\)is a centered average of i\.i\.d\. variables bounded by2​‖f‖∞2\\left\\lVert f\\right\\rVert\_\{\\infty\}\. Forp∈\[1,2\]p\\in\[1,2\], Jensen and the variance bound give𝔼\|⋅\|p≤\(\(2​‖f‖∞\)24​n\)p/2≤\(2‖f‖∞\)pn−p/2\\mathbb\{E\}\\big\|\\cdot\\big\|^\{p\}\\leq\\big\(\\tfrac\{\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{2\}\}\{4n\}\\big\)^\{p/2\}\\leq\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{p\}n^\{\-p/2\}; forp\>2p\>2, the Marcinkiewicz–Zygmund inequality gives𝔼\|⋅\|p≤Cp\(2‖f‖∞\)pn−p/2\\mathbb\{E\}\|\\cdot\|^\{p\}\\leq C\_\{p\}\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{p\}n^\{\-p/2\}\. Taking expectations, then theσ\\sigma\-integral \(Tonelli\), yields

𝔼VSWpp\(ℙ^N,ℙ~\)≤Cp2pΛpn−p/2\.\\mathbb\{E\}\\,\\mathrm\{VSW\}\_\{p\}^\{p\}\(\\widehat\{\\mathbb\{P\}\}\_\{N\},\\widetilde\{\\mathbb\{P\}\}\)\\leq C\_\{p\}\\,2^\{p\}\\Lambda\_\{p\}\\,n^\{\-p/2\}\.Combining the outer and inner bounds proves the theorem\. The distance rates in Remark[3\.8](https://arxiv.org/html/2610.20743#S3.Thmtheorem8)follow by Jensen \(𝔼​VSWp≤\(𝔼​VSWpp\)1/p\\mathbb\{E\}\\mathrm\{VSW\}\_\{p\}\\leq\(\\mathbb\{E\}\\mathrm\{VSW\}\_\{p\}^\{p\}\)^\{1/p\}\) and subadditivity ofx↦x1/px\\mapsto x^\{1/p\}\. ∎

###### Proof of Proposition[3\.9](https://arxiv.org/html/2610.20743#S3.Thmtheorem9)\.

The summands are i\.i\.d\. with meanVSWpp\\mathrm\{VSW\}\_\{p\}^\{p\}and second moment at most𝔼σ​\(2​‖f‖∞\)2​p=22​p​Λ2​p\\mathbb\{E\}\_\{\\sigma\}\(2\\left\\lVert f\\right\\rVert\_\{\\infty\}\)^\{2p\}=2^\{2p\}\\Lambda\_\{2p\}; apply𝔼​\|X−𝔼​X\|≤Var⁡X\\mathbb\{E\}\|X\-\\mathbb\{E\}X\|\\leq\\sqrt\{\\operatorname\{Var\}X\}to the empirical mean\. \(Cf\. the analogous bound for sliced divergences in[Nadjahi et al\., 2020](https://arxiv.org/html/2610.20743#bib.bib30)\.\) ∎

#### The shared finite\-feature implementation\.

Condition on the sampled Fourier basis, letσR\\sigma\_\{R\}be the conditional law offRf\_\{R\}, and defineϕR​\(x\)=2/R​\(cos⁡\(⟨ωr,x⟩\+br\)\)r=1R\\phi\_\{R\}\(x\)=\\sqrt\{2/R\}\\big\(\\cos\(\\langle\\omega\_\{r\},x\\rangle\+b\_\{r\}\)\\big\)\_\{r=1\}^\{R\}andzR​\(μ\)=∫ϕR​\(x\)​𝑑μ​\(x\)z\_\{R\}\(\\mu\)=\\int\\phi\_\{R\}\(x\)\\,\\mathrm\{d\}\\mu\(x\)\. ThenPfR​\(μ\)=a⊤​zR​\(μ\)P\_\{f\_\{R\}\}\(\\mu\)=a^\{\\top\}z\_\{R\}\(\\mu\)fora∼𝒩⁡\(0,IR\)a\\sim\\mathcal\{N\}\(0,I\_\{R\}\)\. WritingSWp\\mathrm\{SW\}\_\{p\}for classical slicing with uniform unit directions, Gaussian radial symmetry gives the exact identity

VSWp\(σR\)​\(ℙ,ℚ\)p=𝔼​‖a‖p​SWpp​\(\(zR\)\#​ℙ,\(zR\)\#​ℚ\)\.\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\_\{R\}\)\}\(\\mathbb\{P\},\\mathbb\{Q\}\)^\{p\}=\\mathbb\{E\}\\left\\lVert a\\right\\rVert^\{p\}\\,\\mathrm\{SW\}\_\{p\}^\{p\}\\big\(\(z\_\{R\}\)\_\{\\\#\}\\mathbb\{P\},\(z\_\{R\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\.The distance itself has factor\(𝔼​‖a‖p\)1/p\(\\mathbb\{E\}\\left\\lVert a\\right\\rVert^\{p\}\)^\{1/p\}, equal toR\\sqrt\{R\}forp=2p=2\. The finite\-feature map can send distinct measures to the same vector, so this is generally a pseudometric on the original meta\-measures\. Conditional on the basis, Proposition[3\.9](https://arxiv.org/html/2610.20743#S3.Thmtheorem9)’s variance argument applies withσR\\sigma\_\{R\}and its sup\-norm moments in place ofσ\\sigma\. IncreasingLLcontrols this conditional Monte Carlo error, while no quantitative feature\-error bound is provided here \(Remark[C\.4](https://arxiv.org/html/2610.20743#A3.Thmtheorem4)\)\. For a direct finite\-feature reference check, Proposition[B\.1](https://arxiv.org/html/2610.20743#A2.Thmtheorem1)givesVSW2​\(δμ,δν\)=MMDκ​\(μ,ν\)\\mathrm\{VSW\}\_\{2\}\(\\delta\_\{\\mu\},\\delta\_\{\\nu\}\)=\\mathrm\{MMD\}\_\{\\kappa\}\(\\mu,\\nu\), whereas the conditional finite\-feature value is‖zR​\(μ\)−zR​\(ν\)‖2\\left\\lVert z\_\{R\}\(\\mu\)\-z\_\{R\}\(\\nu\)\\right\\rVert\_\{2\}\.

#### Gradient Flow

###### Lemma C\.5\(No ties, almost surely\)\.

Supposeκ\\kappais characteristic and the cloudsμ^1,…,μ^N\\widehat\{\\mu\}\_\{1\},\\dots,\\widehat\{\\mu\}\_\{N\}are pairwise distinct as measures \(and similarly for the targets\)\. Then forσ\\sigma\-a\.e\.ff, the projected scalarss1​\(f\),…,sN​\(f\)s\_\{1\}\(f\),\\dots,s\_\{N\}\(f\)are pairwise distinct \(and similarlyt1,…,tNt\_\{1\},\\dots,t\_\{N\}\), so the sorting permutationπf\\pi\_\{f\}matchings\(i\)s\_\{\(i\)\}tot\(i\)t\_\{\(i\)\}is locally constant inXXaround any configuration with distinct clouds\.

###### Proof\.

Fori≠i′i\\neq i^\{\\prime\},si​\(f\)−si′​\(f\)=⟨f,μ^i−μ^i′⟩s\_\{i\}\(f\)\-s\_\{i^\{\\prime\}\}\(f\)=\\langle f,\\widehat\{\\mu\}\_\{i\}\-\\widehat\{\\mu\}\_\{i^\{\\prime\}\}\\rangleis a centered Gaussian with varianceMMDκ2​\(μ^i,μ^i′\)\\mathrm\{MMD\}\_\{\\kappa\}^\{2\}\(\\widehat\{\\mu\}\_\{i\},\\widehat\{\\mu\}\_\{i^\{\\prime\}\}\), which is strictly positive becauseκ\\kappais characteristic and the clouds are distinct; a nondegenerate Gaussian hits00with probability zero\. A union over the finitely many pairs concludes\. ∎

Algorithm 2VSW\\mathrm\{VSW\}flow for populations of point clouds \(one step\)1:particles

X=\(xi​j\)X=\(x\_\{ij\}\), targets

\{yk​l\}\\\{y\_\{kl\}\\\}, kernel

κ\\kappa, slices

LL, features

RR, step

η\\eta
2:sample fresh slices

f1,…,fLf\_\{1\},\\dots,f\_\{L\}via \([5](https://arxiv.org/html/2610.20743#S3.E5)\)

3:for

l=1,…,Ll=1,\\dots,Ldo

4:compute

si\(l\),tk\(l\)s\_\{i\}^\{\(l\)\},t\_\{k\}^\{\(l\)\}; sort both; obtain matching

πl\\pi\_\{l\}
5:endfor

6:

gi​j←2N​n​L∑l\(si\(l\)−tπl​\(i\)\(l\)\)∇fl\(xi​j\)g\_\{ij\}\\leftarrow\\frac\{2\}\{NnL\}\\sum\_\{l\}\\big\(s\_\{i\}^\{\(l\)\}\-t^\{\(l\)\}\_\{\\pi\_\{l\}\(i\)\}\\big\)\\nabla f\_\{l\}\(x\_\{ij\}\)
7:

xi​j←xi​j−η​gi​jx\_\{ij\}\\leftarrow x\_\{ij\}\-\\eta\\,g\_\{ij\}⊳\\trianglerightor passggto Adam / reparametrized generator

Finally, we prove Proposition[3\.10](https://arxiv.org/html/2610.20743#S3.Thmtheorem10)which holds under Assumptions[1](https://arxiv.org/html/2610.20743#Thmassumption1)–[2](https://arxiv.org/html/2610.20743#Thmassumption2)and the hypotheses of Lemma[C\.5](https://arxiv.org/html/2610.20743#A3.Thmtheorem5)\.

###### Proof of Proposition[3\.10](https://arxiv.org/html/2610.20743#S3.Thmtheorem10)\.

Forf∈C1f\\in C^\{1\}write

Ff​\(X\):=minπ∈SN⁡1N​∑i=1N\(si​\(f\)−tπ⁡\(i\)​\(f\)\)2,F\_\{f\}\(X\):=\\min\_\{\\pi\\in S\_\{N\}\}\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\big\(s\_\{i\}\(f\)\-t\_\{\\pi\(i\)\}\(f\)\\big\)^\{2\},so thatℱ=𝔼σ​\[Ff\]\\mathcal\{F\}=\\mathbb\{E\}\_\{\\sigma\}\[F\_\{f\}\]by \([2](https://arxiv.org/html/2610.20743#S2.E2)\) \(the increasing rearrangement realizes the minimum\)\. Fix a coordinatexi​jx\_\{ij\}and letΩ′⊃Ω\\Omega^\{\\prime\}\\supset\\Omegabe a fixed compact neighborhood containing all configurations under consideration\. In this proof,‖f‖∞\\left\\lVert f\\right\\rVert\_\{\\infty\}denotessupx∈Ω′\|f⁡\(x\)\|\\sup\_\{x\\in\\Omega^\{\\prime\}\}\|f\(x\)\|\. We assume that the regularity and moment conditions of Assumption[2](https://arxiv.org/html/2610.20743#Thmassumption2)hold onΩ′\\Omega^\{\\prime\}; then the field∇f\\nabla fis defined and Borell–TIS applies onΩ′\\Omega^\{\\prime\}\(for the Gaussian RBF,ffextends to all ofℝd\\mathbb\{R\}^\{d\}\)\. The interchange of one partial derivative with𝔼σ\\mathbb\{E\}\_\{\\sigma\}proceeds in three steps\.

*Step 1 \(pointwise differentiability forσ\\sigma\-a\.e\.ff\)\.*LetAAbe the full\-measure set of Lemma[C\.5](https://arxiv.org/html/2610.20743#A3.Thmtheorem5)\. Forf∈Af\\in Athe projected scalarss1​\(f\),…,sN​\(f\)s\_\{1\}\(f\),\\dots,s\_\{N\}\(f\)\(resp\.t1​\(f\),…,tN​\(f\)t\_\{1\}\(f\),\\dots,t\_\{N\}\(f\)\) are pairwise distinct, so the minimizing permutationπf\\pi\_\{f\}is unique and remains optimal on a ballB⁡\(X,rf\)B\(X,r\_\{f\}\)whose radiusrf\>0r\_\{f\}\>0*depends onff*, by continuity ofX↦si​\(f\)X\\mapsto s\_\{i\}\(f\)\. On this ballFfF\_\{f\}coincides with the smooth function1N​∑i\(si−tπf​\(i\)\)2\\frac\{1\}\{N\}\\sum\_\{i\}\(s\_\{i\}\-t\_\{\\pi\_\{f\}\(i\)\}\)^\{2\}, whence

∂xi​jFf\(X\)=2N​n\(si\(f\)−tπf​\(i\)\(f\)\)∇f\(xi​j\),∂si/∂xi​j=1n∇f\(xi​j\)\.\\partial\_\{x\_\{ij\}\}F\_\{f\}\(X\)=\\tfrac\{2\}\{Nn\}\\big\(s\_\{i\}\(f\)\-t\_\{\\pi\_\{f\}\(i\)\}\(f\)\\big\)\\nabla f\(x\_\{ij\}\),\\qquad\\partial s\_\{i\}/\\partial x\_\{ij\}=\\tfrac\{1\}\{n\}\\nabla f\(x\_\{ij\}\)\.
*Step 2 \(uniform integrable envelope for the difference quotients\)\.*For everyf∈C1f\\in C^\{1\}\(not onlyf∈Af\\in A\) and every fixedπ∈SN\\pi\\in S\_\{N\}, the mapX↦1N​∑i\(si−tπ⁡\(i\)\)2X\\mapsto\\frac\{1\}\{N\}\\sum\_\{i\}\(s\_\{i\}\-t\_\{\\pi\(i\)\}\)^\{2\}is differentiable with partial gradient inxi​jx\_\{ij\}bounded in norm by

E⁡\(f\):=4N​n​‖f‖∞​supΩ′‖∇f‖,E\(f\):=\\tfrac\{4\}\{Nn\}\\,\\left\\lVert f\\right\\rVert\_\{\\infty\}\\,\\sup\_\{\\Omega^\{\\prime\}\}\\left\\lVert\\nabla f\\right\\rVert,since\|si−tπ⁡\(i\)\|≤2​‖f‖∞\|s\_\{i\}\-t\_\{\\pi\(i\)\}\|\\leq 2\\left\\lVert f\\right\\rVert\_\{\\infty\}\. A pointwise minimum of finitely many functions that areMM\-Lipschitz along a segment is itselfMM\-Lipschitz along that segment; hence every difference quotient ofFfF\_\{f\}in the coordinatexi​jx\_\{ij\}is bounded byE⁡\(f\)E\(f\),*uniformly in the step size*\. MoreoverE∈L1​\(σ\)E\\in L^\{1\}\(\\sigma\): by Cauchy–Schwarz,

𝔼σ​\[‖f‖∞​supΩ′‖∇f‖\]≤\(𝔼σ​‖f‖∞2\)1/2​\(𝔼σ​supΩ′‖∇f‖2\)1/2<∞\\mathbb\{E\}\_\{\\sigma\}\\big\[\\left\\lVert f\\right\\rVert\_\{\\infty\}\\,\\sup\_\{\\Omega^\{\\prime\}\}\\left\\lVert\\nabla f\\right\\rVert\\big\]\\leq\\big\(\\mathbb\{E\}\_\{\\sigma\}\\left\\lVert f\\right\\rVert\_\{\\infty\}^\{2\}\\big\)^\{1/2\}\\big\(\\mathbb\{E\}\_\{\\sigma\}\\sup\_\{\\Omega^\{\\prime\}\}\\left\\lVert\\nabla f\\right\\rVert^\{2\}\\big\)^\{1/2\}<\\inftyby Lemma[A\.1](https://arxiv.org/html/2610.20743#A1.Thmtheorem1)\(i\) and Assumption[2](https://arxiv.org/html/2610.20743#Thmassumption2)\.

*Step 3 \(dominated convergence\)\.*Ash→0h\\to 0, the difference quotientsh−1​\(Ff​\(X\+h​e\)−Ff​\(X\)\)h^\{\-1\}\\big\(F\_\{f\}\(X\+he\)\-F\_\{f\}\(X\)\\big\)converge forσ\\sigma\-a\.e\.ff\(Step 1\) and are dominated byE∈L1​\(σ\)E\\in L^\{1\}\(\\sigma\)\(Step 2\); dominated convergence yields the partial derivative ofℱ\\mathcal\{F\}and formula \([7](https://arxiv.org/html/2610.20743#S3.E7)\)\. Finally, the right\-hand side of \([7](https://arxiv.org/html/2610.20743#S3.E7)\) is continuous at each configurationXXwith pairwise distinct clouds\. Indeed, ifXr→XX\_\{r\}\\to X, then for the full\-measure set of slices having no ties atXX, continuity of the projected scalars impliesπf​\(Xr\)=πf​\(X\)\\pi\_\{f\}\(X\_\{r\}\)=\\pi\_\{f\}\(X\)for all sufficiently largerr\. Hence the integrand in \([7](https://arxiv.org/html/2610.20743#S3.E7)\) converges pointwise, and the same integrable envelopeEEallows another application of dominated convergence\. Thus, all partial derivatives ofℱ\\mathcal\{F\}exist and are continuous nearXX, andℱ\\mathcal\{F\}is differentiable atXX\. ∎

## Appendix DStatistical Properties

###### Corollary D\.1\(Error of the fully empirical estimator\)\.

Letℙ,ℚ∈𝒫⁡\(𝒫⁡\(Ω\)\)\\mathbb\{P\},\\mathbb\{Q\}\\in\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)be observed through the two\-stage scheme with parameters\(N,n\)\(N,n\)and\(M,m\)\(M,m\)respectively, and letVSW^p,L\\widehat\{\\mathrm\{VSW\}\}\_\{p,L\}denote theLL\-slice Monte Carlo estimator of Proposition[3\.9](https://arxiv.org/html/2610.20743#S3.Thmtheorem9)\. Then

𝔼​\|VSW^p,L​\(ℙ~N,n,ℚ~M,m\)−VSWp​\(ℙ,ℚ\)\|\\displaystyle\\mathbb\{E\}\\Big\|\\widehat\{\\mathrm\{VSW\}\}\_\{p,L\}\\big\(\\widetilde\{\\mathbb\{P\}\}\_\{N,n\},\\widetilde\{\\mathbb\{Q\}\}\_\{M,m\}\\big\)\-\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\Big\|≤C\(N−1/\(2p\)\+n−1/2\+M−1/\(2p\)\+m−1/2\+L−1/\(2p\)\),\\displaystyle\\leq C\\Big\(N^\{\-1/\(2p\)\}\+n^\{\-1/2\}\+M^\{\-1/\(2p\)\}\+m^\{\-1/2\}\+L^\{\-1/\(2p\)\}\\Big\),withC=C⁡\(p,Λ2​p\)C=C\\big\(p,\\Lambda\_\{2p\}\\big\)\.

###### Proof\.

By the triangle inequality forVSWp\\mathrm\{VSW\}\_\{p\}\(Theorem[3\.3](https://arxiv.org/html/2610.20743#S3.Thmtheorem3)\),\|VSWp​\(ℙ~N,n,ℚ~M,m\)−VSWp​\(ℙ,ℚ\)\|≤VSWp​\(ℙ,ℙ~N,n\)\+VSWp​\(ℚ,ℚ~M,m\)\\big\|\\mathrm\{VSW\}\_\{p\}\(\\widetilde\{\\mathbb\{P\}\}\_\{N,n\},\\widetilde\{\\mathbb\{Q\}\}\_\{M,m\}\)\-\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\big\|\\leq\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{P\},\\widetilde\{\\mathbb\{P\}\}\_\{N,n\}\)\+\\mathrm\{VSW\}\_\{p\}\(\\mathbb\{Q\},\\widetilde\{\\mathbb\{Q\}\}\_\{M,m\}\), and Theorem[3\.7](https://arxiv.org/html/2610.20743#S3.Thmtheorem7)bounds the expectation of each term after takingpp\-th roots and applying Jensen\. For the slicing error, Proposition[3\.9](https://arxiv.org/html/2610.20743#S3.Thmtheorem9)gives𝔼​\|VSW^p,Lp−VSWpp\|≤2p​\(Λ2​p/L\)1/2\\mathbb\{E\}\|\\widehat\{\\mathrm\{VSW\}\}\_\{p,L\}^\{p\}\-\\mathrm\{VSW\}\_\{p\}^\{p\}\|\\leq 2^\{p\}\(\\Lambda\_\{2p\}/L\)^\{1/2\}at the empirical meta\-measures, and\|a1/p−b1/p\|≤\|a−b\|1/p\|a^\{1/p\}\-b^\{1/p\}\|\\leq\|a\-b\|^\{1/p\}fora,b≥0a,b\\geq 0converts this into a bound on the estimator itself\. Combining the three contributions gives the claim\. ∎

###### Proposition D\.2\(Stability in the slicing measure\)\.

Letσ,σ′\\sigma,\\sigma^\{\\prime\}be two slicing measures onC⁡\(Ω\)C\(\\Omega\)with finite sup\-norm moments of order2​\(p−1\)∨22\(p\-1\)\\vee 2, and letVSWp\(σ\),VSWp\(σ′\)\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\)\},\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma^\{\\prime\}\)\}denote the corresponding distances\. Then for any couplingΞ\\Xiof\(σ,σ′\)\(\\sigma,\\sigma^\{\\prime\}\),

\|VSWp\(σ\)​\(ℙ,ℚ\)p−VSWp\(σ′\)​\(ℙ,ℚ\)p\|≤2​p​𝔼\(f,g\)∼Ξ​\[\(2​max⁡\(‖f‖∞,‖g‖∞\)\)p−1​‖f−g‖∞\]\.\\big\|\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\)\}\(\\mathbb\{P\},\\mathbb\{Q\}\)^\{p\}\-\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma^\{\\prime\}\)\}\(\\mathbb\{P\},\\mathbb\{Q\}\)^\{p\}\\big\|\\;\\leq\\;2p\\;\\mathbb\{E\}\_\{\(f,g\)\\sim\\Xi\}\\Big\[\\big\(2\\max\(\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert g\\right\\rVert\_\{\\infty\}\)\\big\)^\{p\-1\}\\,\\left\\lVert f\-g\\right\\rVert\_\{\\infty\}\\Big\]\.In particular, if a sequence of finite\-rank approximationsfR∼σRf\_\{R\}\\sim\\sigma\_\{R\}can be coupled tof∼σf\\sim\\sigmaso that𝔼​‖fR−f‖∞2→0\\mathbb\{E\}\\left\\lVert f\_\{R\}\-f\\right\\rVert\_\{\\infty\}^\{2\}\\to 0andsupR𝔼​‖fR‖∞2​\(p−1\)<∞\\sup\_\{R\}\\mathbb\{E\}\\left\\lVert f\_\{R\}\\right\\rVert\_\{\\infty\}^\{2\(p\-1\)\}<\\infty\(both automatic fromL2L^\{2\}convergence whenp≤2p\\leq 2\), thenVSWp\(σR\)→VSWp\(σ\)\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\_\{R\}\)\}\\to\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\)\}uniformly on𝒫⁡\(𝒫⁡\(Ω\)\)×𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\\times\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\. Lemma[D\.3](https://arxiv.org/html/2610.20743#A4.Thmtheorem3)verifies this hypothesis for truncated Karhunen–Loève expansions\.

###### Proof\.

Writeh⁡\(f\):=Wp​\(\(Pf\)\#​ℙ,\(Pf\)\#​ℚ\)∈\[0,2​‖f‖∞\]h\(f\):=\\mathrm\{W\}\_\{p\}\\big\(\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{P\},\(P\_\{f\}\)\_\{\\\#\}\\mathbb\{Q\}\\big\)\\in\[0,2\\left\\lVert f\\right\\rVert\_\{\\infty\}\]\. By Lemma[3\.2](https://arxiv.org/html/2610.20743#S3.Thmtheorem2)\(ii\),\|h⁡\(f\)−h⁡\(g\)\|≤2​‖f−g‖∞\|h\(f\)\-h\(g\)\|\\leq 2\\left\\lVert f\-g\\right\\rVert\_\{\\infty\}, and\|h​\(f\)p−h​\(g\)p\|≤p​max⁡\(h⁡\(f\),h⁡\(g\)\)p−1​\|h⁡\(f\)−h⁡\(g\)\|\|h\(f\)^\{p\}\-h\(g\)^\{p\}\|\\leq p\\max\(h\(f\),h\(g\)\)^\{p\-1\}\|h\(f\)\-h\(g\)\|\. Take expectations underΞ\\Xiand usemax⁡\(h⁡\(f\),h⁡\(g\)\)≤2​max⁡\(‖f‖∞,‖g‖∞\)\\max\(h\(f\),h\(g\)\)\\leq 2\\max\(\\left\\lVert f\\right\\rVert\_\{\\infty\},\\left\\lVert g\\right\\rVert\_\{\\infty\}\); the marginals reproduce the two sliced distances\. The uniformity claim follows by Cauchy–Schwarz on the right\-hand side\. ∎

###### Lemma D\.3\(KL truncations satisfy the coupling hypothesis\)\.

Letκ\\kappasatisfy Assumption[1](https://arxiv.org/html/2610.20743#Thmassumption1), let\(λk,φk\)k≥1\(\\lambda\_\{k\},\\varphi\_\{k\}\)\_\{k\\geq 1\}be the Mercer eigenpairs ofκ\\kappaonL2​\(Ω,ι\)L^\{2\}\(\\Omega,\\iota\)for a fully supported finite Borel measureι\\iotaonΩ\\Omega, and set

fR=∑k=1Rλk​ξk​φk,ξk∼iid𝒩⁡\(0,1\)\.f\_\{R\}\\;=\\;\\sum\_\{k=1\}^\{R\}\\sqrt\{\\lambda\_\{k\}\}\\,\\xi\_\{k\}\\,\\varphi\_\{k\},\\qquad\\xi\_\{k\}\\stackrel\{\{\\scriptstyle\\text\{iid\}\}\}\{\{\\sim\}\}\\mathcal\{N\}\(0,1\)\.Then\{λk​φk\}k≥1\\\{\\sqrt\{\\lambda\_\{k\}\}\\varphi\_\{k\}\\\}\_\{k\\geq 1\}is an orthonormal basis of the Cameron–Martin spaceℋκ\\mathcal\{H\}\_\{\\kappa\}ofσ\\sigma, the series converges almost surely in\(C⁡\(Ω\),‖⋅‖∞\)\(C\(\\Omega\),\\left\\lVert\\cdot\\right\\rVert\_\{\\infty\}\)with limitffdistributed asσ\\sigma, and𝔼​‖f−fR‖∞2→0\\mathbb\{E\}\\left\\lVert f\-f\_\{R\}\\right\\rVert\_\{\\infty\}^\{2\}\\to 0\. ConsequentlyVSWp\(σR\)→VSWp\(σ\)\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\_\{R\}\)\}\\to\\mathrm\{VSW\}\_\{p\}^\{\(\\sigma\)\}uniformly on𝒫⁡\(𝒫⁡\(Ω\)\)×𝒫⁡\(𝒫⁡\(Ω\)\)\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)\\times\\mathcal\{P\}\(\\mathcal\{P\}\(\\Omega\)\)by Proposition[D\.2](https://arxiv.org/html/2610.20743#A4.Thmtheorem2)\.

###### Proof\.

By Mercer’s theorem on the compact spaceΩ\\Omega\(withκ\\kappacontinuous andι\\iotafully supported\), the eigenfunctionsφk\\varphi\_\{k\}are continuous and\{λk​φk\}\\\{\\sqrt\{\\lambda\_\{k\}\}\\varphi\_\{k\}\\\}is an orthonormal basis ofℋκ\\mathcal\{H\}\_\{\\kappa\}, which coincides with the Cameron–Martin space of the Gaussian measureσ\\sigmaonC⁡\(Ω\)C\(\\Omega\)\([Bogachev, 1998](https://arxiv.org/html/2610.20743#bib.bib6), Ch\. 2–3\)\. For any orthonormal basis\{ek\}\\\{e\_\{k\}\\\}of the Cameron–Martin space of a centered Radon Gaussian measure on a separable Banach space, the series∑kξk​ek\\sum\_\{k\}\\xi\_\{k\}e\_\{k\}with i\.i\.d\. standard Gaussian coefficients converges almost surely in the Banach norm, with limit distributed as the measure\([Bogachev, 1998](https://arxiv.org/html/2610.20743#bib.bib6), Thm\. 3\.5\.1\); this is an Itô–Nisio\-type theorem\. Finally, almost surely convergent Gaussian series in a separable Banach space converge inLrL^\{r\}for everyr<∞r<\\infty\([Ledoux and Talagrand, 1991](https://arxiv.org/html/2610.20743#bib.bib26), Ch\. 3\), so𝔼​‖f−fR‖∞2→0\\mathbb\{E\}\\left\\lVert f\-f\_\{R\}\\right\\rVert\_\{\\infty\}^\{2\}\\to 0; the couplingΞ=law⁡\(fR,f\)\\Xi=\\mathrm\{law\}\(f\_\{R\},f\)in Proposition[D\.2](https://arxiv.org/html/2610.20743#A4.Thmtheorem2)together with Cauchy–Schwarz gives the uniform convergence of the sliced distances\. ∎

## Appendix EAblation

Ablation onRR\.The number of random Fourier featuresRRis specific to VSW\. We therefore sweepR∈\{8,16,…,1024\}R\\in\\\{8,16,\\dots,1024\\\}in the dataset\-distance setting of Figure[2](https://arxiv.org/html/2610.20743#S4.F2)on FashionMNIST, at the two largest matched budgetsL=S=103L=S=10^\{3\}andL=S=104L=S=10^\{4\}\(Table[3](https://arxiv.org/html/2610.20743#A5.T3)\), with the baseline configurations of Appendix[F](https://arxiv.org/html/2610.20743#A6)\. Each entry is the mean±\\pmstandard deviation over55independent draws of the features and slices, for VSW and both baselines\. Runtimes follow the protocol of Figure[3](https://arxiv.org/html/2610.20743#S4.F3)and are given on the3030pairs at their own sizes, which is the cost of the correlation beside them, and at fixed sizes10410^\{4\}and6×1046\\times 10^\{4\}\. LargerRRgenerally improves correlation at increased runtime, with limited gains beyondR=512R=512\. For slicing budgetL=S=1000L=S=1000, all testedR≥16R\\geq 16give higher mean Pearson and Spearman correlations than DSW and s\-OTDD;R=8R=8is below DSW\. AtL=S=104L=S=10^\{4\},R=128R=128is competitive with DSW \(Pearson0\.8468±0\.00800\.8468\\pm 0\.0080versus0\.8396±0\.01250\.8396\\pm 0\.0125\), with overlapping variability across random draws\. We useR=128R=128for its balance of correlation and runtime\.

Table 3:Ablation on the number of random Fourier featuresRRon FashionMNIST, at the two matched slicing budgetsL=S=103L=S=10^\{3\}\(top\) andL=S=104L=S=10^\{4\}\(bottom\)\. Each entry is the mean±\\pmstandard deviation over55independent draws of the features and slices\. Times are per distance evaluation: on the3030pairs at their own sizes, at a fixed size of10410^\{4\}, and on the full training set of6×1046\\times 10^\{4\}\.
## Appendix FImplementation Details

Below, we provide experimental details for all of the reported experiments in the paper\. All experiments are conducted on an AMD EPYC 7702P CPU, or NVIDIA RTX A5000 GPUs\. Unless stated otherwise, VSW uses the Gaussian RBF kernelκℓ\(x,y\)=exp\(−∥x−y∥2/\(2ℓ2\)\)\\kappa\_\{\\ell\}\(x,y\)=\\exp\\\!\\big\(\-\\\|x\-y\\\|^\{2\}/\(2\\ell^\{2\}\)\\big\), whose spectral measure in \([5](https://arxiv.org/html/2610.20743#S3.E5)\) isτκℓ=𝒩⁡\(0,ℓ−2​Id\)\\tau\_\{\\kappa\_\{\\ell\}\}=\\mathcal\{N\}\(0,\\ell^\{\-2\}I\_\{d\}\)\. The bandwidthℓ\>0\\ell\>0is set by the median heuristic\([Gretton et al\., 2012](https://arxiv.org/html/2610.20743#bib.bib20)\)\. We pool the points of all inner measures of the two meta\-measures being compared, draw20002000of them uniformly at random, and setℓ\\ellto the median of their pairwise Euclidean distances\. The bandwidth is computed once, before the slices are drawn\.

Statistical Properties\.Each meta\-measure is a mixture of33Gaussian inner measures with uniform weights\. The meta\-measure isℙ=13​∑k=13δ𝒩⁡\(μk,Id\)\\mathbb\{P\}=\\frac\{1\}\{3\}\\sum\_\{k=1\}^\{3\}\\delta\_\{\\mathcal\{N\}\(\\mu\_\{k\},I\_\{d\}\)\}, a uniform distribution over three Gaussian inner measures whose meansμk\\mu\_\{k\}are equally spaced on a circle of radius44in the first two coordinates\. To sample an empirical meta\-measure, we drawNNlabelsk1,…,kNk\_\{1\},\\dots,k\_\{N\}independently and uniformly from\{1,2,3\}\\\{1,2,3\\\}, and observe each inner measure𝒩⁡\(μki,Id\)\\mathcal\{N\}\(\\mu\_\{k\_\{i\}\},I\_\{d\}\)throughnni\.i\.d\. points\. Each empirical meta\-measure hasNNinner measures observed throughnnpoints each\. VSW usesR=256R=256, a median\-heuristic bandwidth, and every curve in Figure[1](https://arxiv.org/html/2610.20743#S3.F1)is the mean over200200runs\. For the Monte Carlo rate, we compare two independent samples withN=n=1000N=n=1000and report\|VSW^2,L−VSW^2,Lmax\|\|\\widehat\{\\mathrm\{VSW\}\}\_\{2,L\}\-\\widehat\{\\mathrm\{VSW\}\}\_\{2,L\_\{\\max\}\}\|forL≤104L\\leq 10^\{4\}, where the referenceLmax=105L\_\{\\max\}=10^\{5\}uses the same slices\. For the outer and inner rates, both meta\-measures are drawn from the same distribution, and we useL=104L=10^\{4\}\. The outer rate variesNNwithn=1000n=1000\. The inner rate variesnnwithN=1000N=1000, and both meta\-measures share the sameNNcomponent labels, so that only the inner sampling error remains\.

Optimal Transport Dataset Distance\.Following[Nguyen et al\. \(2025\)](https://arxiv.org/html/2610.20743#bib.bib34), we draw3030random pairs of disjoint sub\-datasets from the training sets of MNIST, FashionMNIST, and CIFAR\-10\. For each pair, we sample a size uniformly from\{5000,6000,…,10000\}\\\{5000,6000,\\dots,10000\\\}and take the first and the last that many indices of a random permutation of the training set\. Each sub\-dataset is a meta\-measure whose inner measures are its class\-conditional distributions of flattened images\. As a reference, we compute the exact OTDD\([Alvarez\-Melis and Fusi, 2020](https://arxiv.org/html/2610.20743#bib.bib2)\)with its default implementation\. Following[Piening and Beinert \(2026\)](https://arxiv.org/html/2610.20743#bib.bib35), we set the weight of its feature cost to zero, so that OTDD reduces to a Wasserstein\-over\-Wasserstein distance between the class\-conditional distributions\. s\-OTDD\([Nguyen et al\., 2025](https://arxiv.org/html/2610.20743#bib.bib34)\)and DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\)are used with their default hyper\-parameters\. We use a vectorized implementation of DSW that returns the same values as the released code under identical random draws, while being optimized so that the runtime comparison remains fair\. VSW usesR=128R=128, with a bandwidth set once per pair of sub\-datasets by the median heuristic\.

The correlations in Figure[2](https://arxiv.org/html/2610.20743#S4.F2)pool all3030pairs across the sampled sizes\. They are not conditioned on sample size, so shared sample\-size dependence can contribute to the reported agreement\. This protocol introduces no controlled class\-proportion or domain shifts\.

For Figure[3](https://arxiv.org/html/2610.20743#S4.F3), we time one distance evaluation between two sub\-datasets whose size ranges from10001000to the full training set in steps of50005000, forL=S∈\{103,5×103,104\}L=S\\in\\\{10^\{3\},5\\times 10^\{3\},10^\{4\}\\\}\. All methods run on CPU\. We report the mean and standard deviation over1010random draws of the sub\-datasets\.

Comparing Distributions of Point Clouds\.We follow the protocol and released code of[Piening and Beinert \(2026, Sec\. 5\.3\)](https://arxiv.org/html/2610.20743#bib.bib35)\. The reference meta\-measure containsN=10N=10training shapes of one class, and the target containsMMtest shapes of the same class\. We vary one factor at a time: \(i\) the number of target shapesM∈\{1,…,19\}M\\in\\\{1,\\dots,19\\\}\(class*bed*,n=m=50n=m=50\); \(ii\) the standard deviationσnoise∈\[0,0\.5\]\\sigma\_\{\\mathrm\{noise\}\}\\in\[0,0\.5\]of Gaussian noise added to the target points \(class*sofa*,M=10M=10,n=m=50n=m=50\); and \(iii\) the target resolutionm∈\{50,100,…,600\}m\\in\\\{50,100,\\dots,600\\\}\(class*chair*,M=10M=10,n=500n=500\)\. Each setting is repeated over55random draws of shapes\. VSW usesL=2000L=2000slices withR=128R=128, and bandwidth is set by the median heuristic\. For OT\-NNA\([Yang et al\., 2019](https://arxiv.org/html/2610.20743#bib.bib47)\), WoW, and DSW, we use the implementations and hyperparameters of[Piening and Beinert \(2026\)](https://arxiv.org/html/2610.20743#bib.bib35)\. Our only change is a vectorized DSW implementation, which ensures optimal implementation of DSW and fair runtime comparison\. Under identical random draws it returns the same values as the released code, so it reduces the runtime of DSW without changing its values\. In Figure[4](https://arxiv.org/html/2610.20743#S4.F4), each curve is divided by its mean over the sweep, and shaded regions show±1\\pm 1standard deviation over the repetitions\.

For Table[1](https://arxiv.org/html/2610.20743#S4.T1), every method compares the full sets of662662ShapeNet test chairs\([Chang et al\., 2015](https://arxiv.org/html/2610.20743#bib.bib11)\)and662662PVD samples\([Zhou et al\., 2021](https://arxiv.org/html/2610.20743#bib.bib49)\), with20482048points per cloud\. All methods use the settings above and run on one NVIDIA RTX A5000 GPU\. For VSW and DSW, we report the mean±\\pmstandard deviation over66timed batches, while OT\-NNA and WoW are timed once\.

Comparing Image Distributions via Patch Distributions\.We follow the protocol and released code of[Piening and Beinert \(2026, Sec\. 5\.4\)](https://arxiv.org/html/2610.20743#bib.bib35)\. Each meta\-measure is a batch of3232grayscale64×6464\\times 64Perlin\-noise images \(scale100100, pixel values rescaled to\[0,1\]\[0,1\]\), and each image is represented by the uniform empirical measure of its32493249overlapping8×88\\times 8patches inℝ64\\mathbb\{R\}^\{64\}\. In the persistence sweep, the reference batch has persistence0\.50\.5\(lacunarity2\.52\.5,55octaves\) and the target persistence varies over\{0\.1,0\.25,0\.5,0\.75,1\}\\\{0\.1,0\.25,0\.5,0\.75,1\\\}\. In the lacunarity sweep, the reference has lacunarity1\.51\.5\(persistence11,66octaves\) and the target lacunarity varies over\{1,1\.25,1\.5,1\.75,2\}\\\{1,1\.25,1\.5,1\.75,2\\\}\. Reference and target batches are generated from different random seeds, each sweep is repeated55times, and all methods are evaluated on the same pairs of batches\. VSW usesL=2000L=2000slices withR=128R=128random Fourier features of a Gaussian RBF kernel, whose bandwidth is set by the median heuristic on the pooled patches\. WoW and DSW are implemented as in the point\-cloud experiment above\. Following the released code, DSW uses100×100100\\times 100slices, a5050\-point quantile grid, and length scale0\.10\.1, and we again use a vectorized implementation that returns the same values as the released code\. In Figure[5](https://arxiv.org/html/2610.20743#S4.F5), each curve is divided by its maximum within a run before averaging, which preserves the depth of its minimum at the reference parameter\. Shaded regions show±1\\pm 1standard deviation over the55runs\. The runtimes in Figure[5](https://arxiv.org/html/2610.20743#S4.F5)are for one distance evaluation between two batches\. For VSW and DSW, each evaluation is repeated55times, and the mean is averaged over all5050pairs of batches\. All runtimes are measured on the CPU\.

Gradient Flow on Natural Images\.The target meta\-measureℚ\\mathbb\{Q\}consists ofN=5N=5ImageNet classes\([Deng et al\., 2009](https://arxiv.org/html/2610.20743#bib.bib14)\)\(goose, koala, Persian cat, monarch, and macaw\), each represented by the uniform empirical measure of its first2020training images\. Each image is resized and centre\-cropped to64×6464\\times 64, scaled to\[0,1\]\[0,1\], and flattened, so thatΩ⊂ℝ12288\\Omega\\subset\\mathbb\{R\}^\{12288\}\. The source meta\-measure has the same shape, withN=5N=5clouds ofn=20n=20particles drawn i\.i\.d\. from𝒩⁡\(0,I\)\\mathcal\{N\}\(0,I\)\. We run the particle scheme of Section[3\.4](https://arxiv.org/html/2610.20743#S3.SS4)for110,000110\{,\}000steps\. At every step, we draw fresh slices \(L=8000L=8000slices,R=4096R=4096random Fourier features of a Gaussian RBF kernel\) and a Monte Carlo estimate of the gradient in Eq\. \([7](https://arxiv.org/html/2610.20743#S3.E7)\) by automatic differentiation\. Since each particle carries mass1/\(N​n\)1/\(Nn\), we multiply the gradient byN​nNnso that the step size does not depend on the number of particles\. The particles are updated by gradient descent with heavy\-ball momentum0\.90\.9, and the momentum is reset when the bandwidth changes\. The bandwidth follows a two\-phase schedule\. The large bandwidth first moves each cloud towards a target cloud, and the smaller one then resolves individual images\. The full run takes about66minutes on one GPU\.

In Figure[6](https://arxiv.org/html/2610.20743#S4.F6)\(left\), each row follows one particle of one class, and the last column shows its nearest ground\-truth image of the matched class in Euclidean distance\. For the right panel, we clip all snapshots to\[0,1\]\[0,1\], embed them with CLIP ViT\-L/14\([Radford et al\., 2021](https://arxiv.org/html/2610.20743#bib.bib36)\)using its standard preprocessing, andℓ2\\ell\_\{2\}\-normalize the embeddings\. We then project them with a 2D PCA fitted on the100100ground\-truth embeddings only, so that all snapshots share the same axes\. Each line traces one particle through the snapshots at steps00,11k,22k,33k,55k,88k,1515k,2525k, and110110k\.

Parametric Generative Modeling\.DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\)uses100×100100\\times 100slices and a5050\-point quantile grid, with its GP length scale tuned per dataset \(0\.030\.03and0\.0030\.003\)\. MMDSW\([Bonet et al\., 2025b](https://arxiv.org/html/2610.20743#bib.bib9)\)is the MMD with the sliced\-Wasserstein Riesz kernel−SW2r\-\\mathrm\{SW\}\_\{2\}^\{r\}, with128128projections andr=1r=1andr=0\.5r=0\.5, respectively\. For DSW\([Piening and Beinert, 2026](https://arxiv.org/html/2610.20743#bib.bib35)\), we again use a vectorized implementation that returns the same values as the reference one\. We report the point\-levelW22\\mathrm\{W\}\_\{2\}^\{2\}between40964096generated points, one from each of40964096generated clouds, and40964096points from the target, computed with POT\([Flamary et al\., 2021](https://arxiv.org/html/2610.20743#bib.bib18)\)and averaged over22draws\. The ground\-truth level isW22\\mathrm\{W\}\_\{2\}^\{2\}between two independent target samples of the same size\. Curves show the mean and standard deviation over33seeds, and the scatter plots in Figure[7](https://arxiv.org/html/2610.20743#S4.F7)show seed11\. The wall\-clock is measured on GPU\.

Shape Classification\.We follow the protocol of[Piening and Beinert \(2026, Sec\. 5\.1\)](https://arxiv.org/html/2610.20743#bib.bib35)and use their released implementations of all baselines\. Each shape is a metric\-measure space given by its pairwise distance matrix, which we map to a meta\-measure onΩ⊂ℝ\\Omega\\subset\\mathbb\{R\}: the outer measure is uniform over the points, and the inner measure of pointiiis the empirical distribution of its distances to all points, i\.e\., rowiiof the distance matrix\. For every method, we compute the full matrix of distances between shapes and report the mean and standard deviation of the33\-NN accuracy over10001000random stratified25%/75%25\\%/75\\%train/test splits\. FAUST\-500 and FAUST\-1000 use the100100MPI\-FAUST training registrations\([Bogo et al\., 2014](https://arxiv.org/html/2610.20743#bib.bib7)\)\. For 2D Shapes\([Beier et al\., 2022](https://arxiv.org/html/2610.20743#bib.bib3)\)and Animals\([Sumner and Popović, 2004](https://arxiv.org/html/2610.20743#bib.bib42)\), we use the preprocessed data released by[Beier et al\. \(2022\)](https://arxiv.org/html/2610.20743#bib.bib3)\. For MNIST\-2000, we follow the construction of[Piening and Beinert \(2026\)](https://arxiv.org/html/2610.20743#bib.bib35): for each digit00–44, we build2020shapes by sampling20002000points from the mean training image proportionally to pixel intensity, adding Gaussian noise with standard deviation33pixels, and applying a random rotation, with Euclidean distances\.

VSW usesL=100L=100andR=16R=16\. SQW and STLB use100100projections and a1010\-point quantile grid \(length scale0\.010\.01for SQW\), and TLB, AE, and GW are used as in the released code, without a feature term\. To compare the methods fairly, VSW, SQW, and STLB each draw a single random basis for the whole distance matrix, so that every shape is projected once rather than once per pair\. AE caches its per\-shape energy self\-terms, which changes its distances by less than0\.2%0\.2\\%\. TLB and GW are deterministic and admit no such amortization\. Runtimes in Table[2](https://arxiv.org/html/2610.20743#S4.T2)are the mean time per distance on CPU\.