Vertical Sliced Wasserstein Distances for Meta-Measures

Elaheh Akbari; Ahmadreza Moradipari; Kyungtae Han; Hamed Pirsiavash; Yikun Bai; Rocio Diaz Martin; Soheil Kolouri

cs.LGOct 14, 2026

Vertical Sliced Wasserstein Distances for Meta-Measures

Authors: Elaheh Akbari, Ahmadreza Moradipari, Kyungtae Han, Hamed Pirsiavash, Yikun Bai, Rocio Diaz Martin, Soheil Kolouri

Organizations: College of Connected Computing, Vanderbilt University · Toyota Motor North America, InfoTech Labs · Department of Computer Science, University of California, Davis · Department of Mathematics, Purdue University ·

Abstract

Many modern learning problems require comparing probability distributions whose individual observations are themselves probability measures, naturally giving rise to meta-measures in $\mathcal{P}(\mathcal{P}(Ω))$ . However, optimal transport distances on this space can be prohibitively expensive to evaluate. We introduce the vertical sliced Wasserstein ( $\mathrm{VSW}$ ) distance, a scalable metric on $\mathcal{P}(\mathcal{P}(Ω))$ . Rather than slicing along the transport geometry of $\mathcal{P}(Ω)$ , $\mathrm{VSW}$ projects each inner measure using continuous test functions and compares the resulting one-dimensional distributions with one-dimensional Wasserstein distance. We prove that $\mathrm{VSW}$ is a valid metric on $\mathcal{P}(\mathcal{P}(Ω))$ , metrizes weak convergence, and admits dimension-independent statistical convergence guarantees. Empirically, $\mathrm{VSW}$ provides high accuracy and efficiency compared with existing distances for meta-measures. We further derive a practical particle gradient for the empirical $\mathrm{VSW}_2^2$ objective and demonstrate its use for gradient-based transport of meta-measures. $\mathrm{VSW}$ provides a simple and scalable alternative for comparing and optimizing distributions of distributions.

Figures & tables

Figure 1: Statistical complexity analysis for empirical meta-measures

X,Y

. Setup in Appendix F . The outer law has three atoms, so the

N^{-1/2}

reference for

\mathrm{VSW}_{2}^{2}

corresponds to the atomic-outer rate (Remark C.2 ). Left: distance error against a

10^{5}

-slice reference. Right:

X,Y

share component labels to isolate inner sampling error.

Figure 2: Pearson and Spearman correlations with exact OTDD versus the number of slices, computed over

30

pairs per dataset. Both sub-datasets in a pair are drawn from the same training set and have equal size (

5{,}000

–

10{,}000

samples); sizes vary across pairs.

Figure 3: Runtime (s) vs the dataset size (each of 2 sets). VSW remains fastest among all methods.

	OT-NNA	WoW	DSW	VSW (Ours)
runtime (s)	$6077.65^{\dagger}$	$3094.22^{\dagger}$	$0.0709\pm 0.00017$	$\mathbf{0.0125\pm 0.00003}$

Table 1: Wall-clock time of a single distance evaluation between the 662 ground-truth ShapeNet chairs and 662 generated samples (

2048

points 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.

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) .

Figure 7: Parametric generation on Checkerboard and 8-Gaussians datasets.

Distance	2D Shapes		Animals		FAUST-500		FAUST-1000		MNIST-2000
Distance	Acc. (%)	Time (ms)	Acc. (%)	Time (ms)	Acc. (%)	Time (ms)	Acc. (%)	Time (ms)	Acc. (%)	Time (ms)
VSW	$97.7\pm 2.6$	$0.35$	$99.0\pm 1.3$	$0.34$	$39.0\pm 5.5$	$2.80$	$41.2\pm 6.0$	$3.76$	$96.5\pm 3.4$	$7.48$
SQW	$99.4\pm 1.2$	$0.93$	$99.3\pm 1.0$	$0.95$	$37.5\pm 5.5$	$6.51$	$40.8\pm 6.0$	$10.15$	$86.0\pm 4.3$	$18.46$
TLB	$100.0\pm 0.2$	$0.91$	$100.0\pm 0.0$	$0.97$	$36.7\pm 5.6$	$40.96$	$40.2\pm 6.0$	$163.06$	$95.5\pm 3.3$	$653.84$
STLB	$98.5\pm 2.0$	$0.99$	$99.0\pm 1.3$	$1.00$	$35.2\pm 5.5$	$10.01$	$38.4\pm 5.9$	$10.65$	$82.2\pm 3.8$	$18.36$
AE	$99.7\pm 0.8$	$0.13$	$97.9\pm 1.8$	$0.13$	$37.5\pm 5.7$	$2.63$	$42.0\pm 5.9$	$4.46$	$93.2\pm 4.0$	$19.79$

Table 2: Shape classification with KNN (Acc.,

\uparrow

) and runtime per distance computation (Time).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Matched slicing budget $L=S=10^{3}$
Method	Correlation with OTDD ( $\uparrow$ )		Time per evaluation (ms, $\downarrow$ )
Method	Pearson	Spearman	$5$ – $10$ k	$10^{4}$	$6\times 10^{4}$
VSW, $R=8$	$0.6030\pm 0.0548$	$0.6015\pm 0.0695$	$3.4$	$3.2$	$12.8$
VSW, $R=16$	$0.7931\pm 0.0362$	$0.7750\pm 0.0519$	$3.6$	$3.8$	$17.7$
VSW, $R=32$	$0.7698\pm 0.0393$	$0.7623\pm 0.0653$	$4.6$	$4.8$	$25.3$
VSW, $R=64$	$0.8044\pm 0.0164$	$0.8174\pm 0.0309$	$5.6$	$6.5$	$36.4$

Appendix

Table 3: Ablation on the number of random Fourier features

R

on FashionMNIST, at the two matched slicing budgets

L=S=10^{3}

(top) and

L=S=10^{4}

(bottom). Each entry is the mean

\pm

standard deviation over

5

independent draws of the features and slices. Times are per distance evaluation: on the

30

pairs at their own sizes, at a fixed size of

10^{4}

, and on the full training set of

6\times 10^{4}

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Wasserstein distance compares probability distributions by the minimum cost of transporting mass between them.

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