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 paper 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 $Q_\mu(s)= \inf\{x:\mu(-\infty,x]\geq s\}$: $$W_2(\mu,\nu;\mathbb{R}) = \left[ \int_0^1 |Q_\mu(s)-Q_\nu(s)|^2 ds \right]^{1/2} = \|Q_\mu-Q_\nu\|_{L^2([0,1])}.$$ This isometry underpins the functional optimal transport approach used in DSW. Sliced-Wasserstein distances on general Banach spaces $U$ make use of projections $\pi_v(x)=\langle v, x\rangle$ for $v\in U^*$, and for a probability measure $\xi$ on $U^*$,

$$SW(\mu,\nu; \xi) = \left[ \int_{v \in U^*} W_2^2(\pi_{v\#}\mu, \pi_{v\#}\nu; \mathbb{R})\, d\xi(v) \right]^{1/2}.$$

This construction, under appropriate support conditions on $\xi$, yields a true metric on $P_2(U)$.

In the specific setting of meta-measures on $P_2(\mathbb{R})$, the quantile map $q:\mu\to Q_\mu$ pushes $\alpha$ to a law $q_\#\alpha$ on $L^2([0,1])$, yielding a “sliced-quantile WoW” (SQW) metric,

$$SQW(\alpha, \beta; \xi) = SW(q_\#\alpha, q_\#\beta; \xi).$$

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 $\theta \in S^{d-1}$, project every inner measure $\mu \in P_2(\mathbb{R}^d)$ onto $\mathbb{R}$ via $\pi_\theta(x) = \langle \theta, x \rangle$, inducing a pushed-forward measure $\pi_{\theta\#}\mu$.
2. Quantile-Space Slicing: For fixed $\theta$, one obtains two 1D meta-measures $_{\theta\#}\alpha,\,_{\theta\#}\beta \in P_2(P_2(\mathbb{R}))$. Using a Gaussian process prior $\xi$ on $L^2([0,1])$ (e.g., with an RBF kernel), the SQW distance between the meta-measures is

$SW(_{\theta\#}\alpha,\,_{\theta\#}\beta; \xi) = \left[ \int_{v \in L^2([0,1])} W_2^2\left(\pi_{v\#}q_\#(_{\theta\#}\alpha),\, \pi_{v\#}q_\#(_{\theta\#}\beta) \right) d\xi(v) \right]^{1/2}.$

1. Aggregation: Integrate the inner SQW metric over $S^{d-1}$ to obtain the Double-Sliced Wasserstein: $DSW(\alpha, \beta; \xi) = \left[ \int_{S^{d-1}} SW^2(_{\theta\#}\alpha,\,_{\theta\#}\beta; \xi) dS^{d-1}(\theta) \right]^{1/2}.$

The full expansion writes: $$DSW(\alpha, \beta) = \left\{ \int_{S^{d-1}} \int_{v \in L^2([0,1])} \int_{u \in [0,1]} \left\langle Q_{P_\theta\#\alpha}(u) - Q_{P_\theta\#\beta}(u), v(u) \right\rangle^2 du\, d\xi(v)\, dS^{d-1}(\theta) \right\}^{1/2}.$$

For computation, inner integrals are estimated using Monte Carlo samples $\theta_s \in S^{d-1}$ and $v_{s,t}$ Gaussian process paths.

4. Topological Properties and Equivalence with WoW

Let empirical meta-measures $$\alpha_n, \beta_n \in P^N(P^{\tilde n}(\mathbb{R}^d))$$ be composed of $N$ inner empirical measures, each with $\tilde n$ support points. The DSW metric is topologically equivalent to the WoW metric: $$DSW(\alpha_n, \beta_n; \xi) \to 0 \quad\Longleftrightarrow\quad \mathrm{WoW}(\alpha_n, \beta_n) \to 0$$ for any positive Gaussian $\xi$ (Piening et al., 26 Sep 2025). The argument combines a discrete Cramér–Wold theorem at each slice $\theta$ 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	$\mathcal{O}(N^2 n^2\log n)$	Requires all pairwise inner 2-Wasserstein computations; slow for large $N$ and $n$; sensitive to moment estimation
DSW	$\mathcal{O}(S N n\log n)$	Only $S \ll N^2$ projections needed; relies on quantile functions (no high-order moments); numerically robust

• For full WoW on $N$ meta-points of size $n$, cost is dominated by an $N\times N$ matrix of pairwise 2-Wasserstein computations, each in $O(n^2\log n)$ (entropic case).
• For DSW, sampling $S$ directions and computing 1D quantile-transport per meta-measure, total complexity is $O(S N n \log n)$; usually $S = O(N)$ 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 $> 0.9$), outperforming s-OTDD in stability and speed.
• Point-cloud evaluation: For batches of 3D shapes modeled as meta-measures in $P(P(\mathbb{R}^3))$, DSW matches OT-NNA and WoW in sensitivity but is 10–20$\times$ 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$\times$ 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).