Double-Sliced Wasserstein Metric

• Double-Sliced Wasserstein is a metric that compares probability meta-measures using two sequential slicing operations, preserving the topology of the original Wasserstein-over-Wasserstein distance.
• It combines Euclidean projections and quantile-space slicing to achieve computational efficiency and numerical robustness in high-dimensional data analysis.
• Empirical evaluations demonstrate that DSW provides comparable discriminative power to WoW while accelerating computation and reducing sensitivity to unstable high-order moment estimation.

The Double-Sliced Wasserstein (DSW) metric is a recent development in the study of optimal transport on spaces of probability measures, specifically designed as a computationally efficient and statistically robust surrogate for the Wasserstein-over-Wasserstein (WoW) distance between meta-measures. The DSW metric achieves speed and stability by combining traditional Euclidean slicing with an inner slicing in quantile function space, avoiding reliance on high-order moments or unstable operations. DSW is topologically equivalent to WoW on empirical meta-measures and empirically offers substantial speedups with comparable discriminative power for applications in dataset similarity, point-cloud analysis, and perceptual evaluation of images and shapes (Piening et al., 26 Sep 2025).

1. Meta-Measure Spaces and the Wasserstein-Over-Wasserstein Problem

Let $\mathcal{X}$ be a Polish space and $P_2(\mathcal{X})$ the set of Borel probability measures with finite second moment, equipped with the 2-Wasserstein distance,

$$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$

A meta-measure is defined as $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$, that is, a probability law over probability measures on $\mathcal{X}$. The Wasserstein-over-Wasserstein (WoW) metric lifts the $W_2$ distance to the meta-measure space: $$\mathrm{WoW}(\alpha, \beta) = \left[ \inf_{\Pi \in \Gamma(\alpha,\beta)} \int_{P_2(\mathcal{X}) \times P_2(\mathcal{X})} W_2^2(\mu,\nu)\, d\Pi(\mu,\nu) \right]^{1/2},$$ which is computationally prohibitive for large collections of distributions, especially in high-dimensions due to quadratic scaling in the number of inner measures.

2. Quantile Isometry and Functional Slicing

For measures on $\mathbb{R}$, the 1D 2-Wasserstein metric admits an isometry to $L^2([0,1])$, mapping a measure $\mu$ to its quantile function $P_2(\mathcal{X})$0: $P_2(\mathcal{X})$1 This isometry underpins the functional optimal transport approach used in DSW. Sliced-Wasserstein distances on general Banach spaces $P_2(\mathcal{X})$2 make use of projections $P_2(\mathcal{X})$3 for $P_2(\mathcal{X})$4, and for a probability measure $P_2(\mathcal{X})$5 on $P_2(\mathcal{X})$6,

$P_2(\mathcal{X})$7

This construction, under appropriate support conditions on $P_2(\mathcal{X})$8, yields a true metric on $P_2(\mathcal{X})$9.

In the specific setting of meta-measures on $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$0, the quantile map $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$1 pushes $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$2 to a law $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$3 on $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$4, yielding a “sliced-quantile WoW” (SQW) metric,

$$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$5

3. Construction and Mathematical Formulation of Double-Sliced Wasserstein

The Double-Sliced Wasserstein metric is constructed through consecutive application of two slicing steps:

1. Euclidean Slicing: For each $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$6, project every inner measure $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$7 onto $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$8 via $$W_2(\mu,\nu) = \left(\inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal{X}^2} d^2(x,x')\,d\pi(x,x')\right)^{1/2}.$$9, inducing a pushed-forward measure $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$0.
2. Quantile-Space Slicing: For fixed $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$1, one obtains two 1D meta-measures $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$2. Using a Gaussian process prior $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$3 on $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$4 (e.g., with an RBF kernel), the SQW distance between the meta-measures is

$\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$5

1. Aggregation: Integrate the inner SQW metric over $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$6 to obtain the Double-Sliced Wasserstein: $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$7

The full expansion writes: $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$8

For computation, inner integrals are estimated using Monte Carlo samples $\alpha\in P_2\bigl(P_2(\mathcal{X})\bigr)$9 and $\mathcal{X}$0 Gaussian process paths.

4. Topological Properties and Equivalence with WoW

Let empirical meta-measures $\mathcal{X}$1 be composed of $\mathcal{X}$2 inner empirical measures, each with $\mathcal{X}$3 support points. The DSW metric is topologically equivalent to the WoW metric: $\mathcal{X}$4 for any positive Gaussian $\mathcal{X}$5 (Piening et al., 26 Sep 2025). The argument combines a discrete Cramér–Wold theorem at each slice $\mathcal{X}$6 and the quantile-space isometry. This ensures that DSW is a true metric and it preserves the geometry induced by WoW on the space of empirical meta-measures.

5. Computational Complexity and Numerical Stability

Metric	Complexity per evaluation	Stability Considerations
WoW	$\mathcal{X}$7	Requires all pairwise inner 2-Wasserstein computations; slow for large $\mathcal{X}$8 and $\mathcal{X}$9; sensitive to moment estimation
DSW	$W_2$0	Only $W_2$1 projections needed; relies on quantile functions (no high-order moments); numerically robust

• For full WoW on $W_2$2 meta-points of size $W_2$3, cost is dominated by an $W_2$4 matrix of pairwise 2-Wasserstein computations, each in $W_2$5 (entropic case).
• For DSW, sampling $W_2$6 directions and computing 1D quantile-transport per meta-measure, total complexity is $W_2$7; usually $W_2$8 or constant.
• DSW avoids the unstable high-order moments used in s-OTDD, maintaining stability even with non-Gaussian or heavy-tailed meta-distributions.

6. Empirical Results and Applications

Experimental evaluations in (Piening et al., 26 Sep 2025) demonstrate that DSW achieves strong performance in several tasks:

• Shape classification via local distance distributions (mm-spaces): DSW matches accuracy of Gromov–Wasserstein and sliced GW (STLB), with an order of magnitude faster runtime.
• Dataset similarity (OTDD surrogate): On MNIST, Fashion-MNIST, and CIFAR-10 splits, DSW correlates with exact OTDD (Pearson $W_2$9), outperforming s-OTDD in stability and speed.
• Point-cloud evaluation: For batches of 3D shapes modeled as meta-measures in $$\mathrm{WoW}(\alpha, \beta) = \left[ \inf_{\Pi \in \Gamma(\alpha,\beta)} \int_{P_2(\mathcal{X}) \times P_2(\mathcal{X})} W_2^2(\mu,\nu)\, d\Pi(\mu,\nu) \right]^{1/2},$$0, DSW matches OT-NNA and WoW in sensitivity but is 10–20$$\mathrm{WoW}(\alpha, \beta) = \left[ \inf_{\Pi \in \Gamma(\alpha,\beta)} \int_{P_2(\mathcal{X}) \times P_2(\mathcal{X})} W_2^2(\mu,\nu)\, d\Pi(\mu,\nu) \right]^{1/2},$$1 faster, providing similar robustness to mode collapse and sampling noise.
• Image perceptual distance: Image batches are represented as meta-measures on patch distributions. DSW defines a perceptual metric sensitive to qualitative similarity, aligns with standard fiducial metrics (e.g., Kernel Inception Distance), and is 40$$\mathrm{WoW}(\alpha, \beta) = \left[ \inf_{\Pi \in \Gamma(\alpha,\beta)} \int_{P_2(\mathcal{X}) \times P_2(\mathcal{X})} W_2^2(\mu,\nu)\, d\Pi(\mu,\nu) \right]^{1/2},$$2 faster than full WoW.

These results indicate that DSW yields operationally efficient metrics for meta-measure comparison without the compromises of parametric forms or unstable statistical estimators.

7. Significance and Prospects

Double-Sliced Wasserstein provides a tractable, mathematically principled metric for meta-level optimal transport problems, preserving the topology and discriminative power of WoW while mitigating prohibitive computational demands. Its combination of classical slicing and functional quantile-space slicing leverages both geometry and statistical properties of optimal transport. The approach is widely applicable to large-scale shape analysis, dataset comparison, and the evaluation of structured or hierarchical data distributions.

Pending open questions include: optimizing DSW kernel choices for task-adaptiveness, theoretical dual formulations for functional slicing, and extensions beyond empirical meta-measures to infinite or continuous families. A plausible implication is that DSW could serve as a foundation for scalable learning frameworks in high-level data spaces where conventional OT remains intractable, especially in large-dimensional and nonparametric distributional regimes (Piening et al., 26 Sep 2025).