Quantile Isometry & Functional Slicing

• Quantile isometry and functional slicing are mathematical frameworks that embed probability measures into L2 spaces via quantile functions for efficient Wasserstein computations.
• They underpin the double-sliced Wasserstein distance, dramatically reducing computational complexity while ensuring robust metric evaluation in diverse applications.
• By extending classical sliced-Wasserstein approaches with functional projections, these methods enable stable, scalable optimal transport analysis in high-dimensional and meta-measure settings.

Quantile isometry and functional slicing constitute the core mathematical innovations underpinning scalable optimal transport (OT) for probability measures and, in particular, for meta-measures (measures over measures). These concepts connect the structure of 1D Wasserstein spaces to $L^2(0,1)$ function spaces through the geometry of quantile functions, enabling efficient computation of metrics such as the Wasserstein over Wasserstein (WoW) and its double-sliced surrogates. This article delineates the formal underpinnings, algorithmic realizations, and empirical characteristics of these methods, as developed in the context of the double-sliced Wasserstein (DSW) distance (Piening et al., 26 Sep 2025).

1. Quantile Isometry in One Dimension

The 2-Wasserstein distance $W_2$ on the real line between probability measures $\mu, \nu \in P_2(\mathbb{R})$—those with finite second moment—admits a canonical formulation: $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$ where $\Gamma(\mu, \nu)$ is the set of all couplings of $\mu$ and $\nu$. For such measures, the quantile or inverse–CDF function, $Q_\mu : (0, 1) \rightarrow \mathbb{R}$, assigns to $s$ the smallest $x$ with $W_2$0. $W_2$1 is in $W_2$2.

The critical isometry theorem holds: $W_2$3 Thus, the mapping $W_2$4 given by $W_2$5 is an isometric embedding for $W_2$6. The uniqueness and monotonicity of optimal couplings in one dimension (monotone rearrangement) ensure that this embedding exactly captures transport geometry.

The practical implication is that computation of $W_2$7 in 1D reduces to the $W_2$8 norm between quantiles, allowing OT on $W_2$9 to be computed via quantile sorting—a key enabler for scalable OT algorithms.

2. Functional Slicing in Banach Spaces

Extending to general separable Banach spaces $\mu, \nu \in P_2(\mathbb{R})$0, with $\mu, \nu \in P_2(\mathbb{R})$1 the continuous dual, slicing is formalized as follows. For $\mu, \nu \in P_2(\mathbb{R})$2, the linear projection $\mu, \nu \in P_2(\mathbb{R})$3 defines the push-forward $\mu, \nu \in P_2(\mathbb{R})$4 for any probability measure $\mu, \nu \in P_2(\mathbb{R})$5 on $\mu, \nu \in P_2(\mathbb{R})$6. For a probability measure $\mu, \nu \in P_2(\mathbb{R})$7 on $\mu, \nu \in P_2(\mathbb{R})$8, the $\mu, \nu \in P_2(\mathbb{R})$9-sliced Wasserstein distance is

$$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$0

If $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$1 is not contained in a proper subspace of $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$2, this defines a metric on $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$3. Along each direction $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$4, the measure is reduced to a 1D problem computable by quantiles; thus, slicing facilitates scalable, projection-based OT on high- or infinite-dimensional spaces.

This framework generalizes the classical sliced-Wasserstein approach (uniform integration over $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$5 in $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$6) and provides a template for slicing in functional spaces using probability distributions (e.g., Gaussian processes) over $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$7.

3. Double-Sliced Wasserstein Distance for Meta-Measures

Comparing meta-measures $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$8 (probability measures over measures) necessitates more structure. The double-sliced Wasserstein (DSW) metric operates by:

1. Outer (Euclidean) slice: For $$W_2(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathbb{R}^2} (x - y)^2 \, d\gamma(x, y) \right)^{1/2},$$9, the induced 1D meta-measure $\Gamma(\mu, \nu)$0 pushes each atomic measure in $\Gamma(\mu, \nu)$1 to its 1D projection along $\Gamma(\mu, \nu)$2.
2. Inner (functional) slice: For $\Gamma(\mu, \nu)$3 and a 1D meta-measure $\Gamma(\mu, \nu)$4, the map $\Gamma(\mu, \nu)$5 with $\Gamma(\mu, \nu)$6 pushes via projections against random $\Gamma(\mu, \nu)$7 directions.

The DSW metric is

$\Gamma(\mu, \nu)$8

where $\Gamma(\mu, \nu)$9 is uniform measure on the sphere, and $\mu$0 is a probability law (e.g., Gaussian process prior) on $\mu$1. DSW thus combines spatial (Euclidean) projection with functional (quantile space) slicing.

For discrete meta-measures supported on finitely many atomic measures, DSW minimization is equivalent to minimizing the original WoW metric: $\mu$2 This demonstrates fidelity of DSW as a surrogate for WoW when applied to practical data representations.

4. Algorithmic Structure and Computational Complexity

The computation of DSW proceeds as follows:

1. Sample $\mu$3 directions $\mu$4 on $\mu$5.
2. For each $\mu$6, calculate the induced 1D meta-measures $\mu$7 and $\mu$8.
3. For each $\mu$9, sample $\nu$0 directions $\nu$1 in $\nu$2.
4. For each $\nu$3, project each 1D atomic measure $\nu$4 to the scalar $\nu$5.
5. Compute $\nu$6 between the resulting empirical distributions via quantile sorting.
6. Aggregate via a two-level Monte Carlo average.

The computational cost is $\nu$7, where $\nu$8 is the maximum support of the atomic measures. In practice, $\nu$9 (with $Q_\mu : (0, 1) \rightarrow \mathbb{R}$0 the number of meta-measure atoms), leading to considerable computational savings over the naive $Q_\mu : (0, 1) \rightarrow \mathbb{R}$1 and $Q_\mu : (0, 1) \rightarrow \mathbb{R}$2 required for direct WoW on discrete meta-measures. The method avoids calculation of high-order moments and high-dimensional LPs, relying solely on quantile evaluation and random projections.

5. Empirical Properties and Practical Significance

Numerical experiments demonstrate:

• For shape classification based on local distance distributions, DSW matches WoW in KNN accuracy while reducing computation time by orders of magnitude on large meshes.
• In OTDD (Optimal Transport Dataset Distance) settings for batches of images, DSW correlates with the ground-truth OTDD (Pearson/Spearman $Q_\mu : (0, 1) \rightarrow \mathbb{R}$3 with $Q_\mu : (0, 1) \rightarrow \mathbb{R}$4 projections), outperforming moment-based approaches that are unstable for high-order moments.
• In generative point-cloud testing, DSW captures distributional phenomena (mode collapse, sensitivity to noise) similarly to OT-NNA, but with linear rather than cubic or quadratic cost in batch or point resolution.
• For image patch distribution matching, DSW aligns with both Euclidean Wasserstein and KID metrics and maintains robustness under various image transformations and sampling artifacts.

These results establish DSW as a scalable, reliable surrogate for high-dimensional and meta-measure OT, without requiring parametric assumptions or higher-order statistics. The adoption of quantile-based isometry ensures numerical stability even with limited projection samples and irregular supports.

6. Connections to Related Methods and Theoretical Implications

Quantile isometry and functional slicing generalize classical sliced-Wasserstein techniques (Bonnotte et al. 2015) and intersect with recent measures for datasets and distributions, including s-OTDD (Nguyen et al. 2025) and point-cloud OT (Piening & Beinert 2025). Functional slicing (integration over infinite-dimensional $Q_\mu : (0, 1) \rightarrow \mathbb{R}$5 directions) parallels developments in random projections on Hilbert spaces (Han 2023).

A plausible implication is that the isometric embedding of probability measures into $Q_\mu : (0, 1) \rightarrow \mathbb{R}$6 quantile space enables further dimensionality reduction and kernelization strategies for OT beyond Euclidean settings. The restriction to $Q_\mu : (0, 1) \rightarrow \mathbb{R}$7 (quadratic cost) is crucial, as only for $Q_\mu : (0, 1) \rightarrow \mathbb{R}$8 does the quantile isometry hold; analogous constructions for $Q_\mu : (0, 1) \rightarrow \mathbb{R}$9 would not generally preserve metric structure in $s$0.

Furthermore, the avoidance of high-order moments circumvents numerical instability endemic to earlier sliced meta-OT methods—particularly for empirical measures with heavy tails or irregular support—while still providing separation and stability guarantees for discrete cases.

Table: Sliced Wasserstein vs. Double-Sliced (Functional) Wasserstein

Metric	Slicing Domain	Direction Distribution
Sliced Wasserstein	$s$1 in $s$2	Uniform (Haar)
DSW	$s$3	Sphere $s$4 GP on $s$5

This tabulation emphasizes the concatenated, two-level projection structure distinguishing DSW from traditional methods.

Overall, quantile isometry and functional slicing establish a theoretically sound and computationally efficient framework for optimal transport comparisons of both data and meta-data distributions, with direct impact on scalable learning and distributional geometry in high dimensions (Piening et al., 26 Sep 2025).