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 $\mu((-\infty, x]) \geq s$. $Q_\mu$ is in $L^2(0,1)$.

The critical isometry theorem holds: $$W_2(\mu, \nu) = \| Q_\mu - Q_\nu \|_{L^2(0,1)} = \left( \int_0^1 |Q_\mu(s) - Q_\nu(s)|^2 ds \right)^{1/2}.$$ Thus, the mapping $q: P_2(\mathbb{R}) \to L^2(0,1)$ given by $\mu \mapsto Q_\mu$ is an isometric embedding for $W_2$. 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 $W_2$ in 1D reduces to the $L^2$ norm between quantiles, allowing OT on $\mathbb{R}$ 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 $(E, \|\cdot\|)$, with $E^*$ the continuous dual, slicing is formalized as follows. For $v \in E^*$, the linear projection $\pi_v(x) = \langle v, x \rangle$ defines the push-forward $\pi_{v\sharp}\mu \in P_2(\mathbb{R})$ for any probability measure $\mu$ on $E$. For a probability measure $\xi$ on $E^*$, the $\xi$-sliced Wasserstein distance is

$$SW(\mu, \nu; \xi) = \left( \int_{E^*} W_2^2(\pi_{v\sharp}\mu, \pi_{v\sharp}\nu) d\xi(v) \right)^{1/2}.$$

If $\operatorname{supp} \xi$ is not contained in a proper subspace of $E^*$, this defines a metric on $P_2(E)$. Along each direction $v$, 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 $S^{d-1}$ in $\mathbb{R}^d$) and provides a template for slicing in functional spaces using probability distributions (e.g., Gaussian processes) over $E^*$.

3. Double-Sliced Wasserstein Distance for Meta-Measures

Comparing meta-measures $\alpha, \beta \in P_2(P_2(\mathbb{R}^d))$ (probability measures over measures) necessitates more structure. The double-sliced Wasserstein (DSW) metric operates by:

1. Outer (Euclidean) slice: For $u \in S^{d-1}$, the induced 1D meta-measure $$\alpha_u = \pi_{u\sharp}\alpha \in P_2(P_2(\mathbb{R}))$$ pushes each atomic measure in $\alpha$ to its 1D projection along $u$.
2. Inner (functional) slice: For $v \in L^2(0,1)$ and a 1D meta-measure $\gamma$, the map $$\pi_{v\sharp}(\gamma) = \operatorname{Law}(\langle v, Q_\mu \rangle)$$ with $\mu \sim \gamma$ pushes via projections against random $L^2$ directions.

The DSW metric is

$$\mathrm{DSW}(\alpha, \beta) = \left( \int_{u \in S^{d-1}} \left( \int_{v \in L^2(0,1)} W_2^2(\pi_{v\sharp} \alpha_u, \pi_{v\sharp} \beta_u) d\rho(v) \right) d\sigma(u) \right)^{1/2},$$

where $\sigma$ is uniform measure on the sphere, and $\rho$ is a probability law (e.g., Gaussian process prior) on $L^2(0,1)$. 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: $$\mathrm{DSW}(\alpha, \beta) \rightarrow 0 \iff W_2(\alpha, \beta; P_2(\mathbb{R}^d)) \rightarrow 0.$$ 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 $N_u$ directions $u_i \sim \sigma$ on $S^{d-1}$.
2. For each $u_i$, calculate the induced 1D meta-measures $\alpha_{u_i}$ and $\beta_{u_i}$.
3. For each $u_i$, sample $N_v$ directions $v_{i,j} \sim \rho$ in $L^2(0,1)$.
4. For each $v_{i,j}$, project each 1D atomic measure $\mu$ to the scalar $\langle v_{i,j}, Q_\mu \rangle$.
5. Compute $W_2$ between the resulting empirical distributions via quantile sorting.
6. Aggregate via a two-level Monte Carlo average.

The computational cost is $O(N_u N_v n \log n)$, where $n$ is the maximum support of the atomic measures. In practice, $N_u N_v \ll N^2$ (with $N$ the number of meta-measure atoms), leading to considerable computational savings over the naive $O(N^3 \log N)$ and $O(N^2 n \log n)$ 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 $>0.9$ with $10^4$ 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 $L^2$ directions) parallels developments in random projections on Hilbert spaces (Han 2023).

A plausible implication is that the isometric embedding of probability measures into $L^2$ quantile space enables further dimensionality reduction and kernelization strategies for OT beyond Euclidean settings. The restriction to $p=2$ (quadratic cost) is crucial, as only for $W_2$ does the quantile isometry hold; analogous constructions for $p \neq 2$ would not generally preserve metric structure in $L^p$.

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	$S^{d-1}$ in $\mathbb{R}^d$	Uniform (Haar)
DSW	$S^{d-1} \times L^2(0,1)$	Sphere $\times$ GP on $L^2$

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).