Double-Sliced Wasserstein (DSW) Overview

• Double-Sliced Wasserstein (DSW) is a two-stage transport metric that compares meta-measures by applying an outer Euclidean slicing followed by an inner functional slicing via quantile embeddings.
• The approach computes one-dimensional Wasserstein distances after sequential projections, bypassing costly high-dimensional optimal transport computations while retaining a strong connection to Wasserstein over Wasserstein minimization.
• Empirical studies demonstrate that DSW approximates WoW with improved runtime in applications such as shape classification, image patch analysis, and point cloud comparison.

Searching arXiv for the specified DSW and closely related sliced-Wasserstein papers. Double-Sliced Wasserstein (DSW) is a hierarchical sliced optimal transport construction for comparing meta-measures, namely probability measures whose atoms are themselves probability measures. In the formulation introduced in "Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport" (Piening et al., 26 Sep 2025), DSW is defined on $P_2(P_2(\mathbb{R}^d))$ by applying two successive reductions: an outer Euclidean slicing that maps measures in $P_2(\mathbb{R}^d)$ to one-dimensional measures in $P_2(\mathbb{R})$, and an inner functional slicing that exploits the isometric embedding of the one-dimensional Wasserstein space into $L_2([0,1])$ via quantile functions. The resulting metric is designed as a practical surrogate for Wasserstein over Wasserstein (WoW), with the stated aim of avoiding the unstable higher-order moment machinery used by earlier sliced WoW approaches while retaining a strong theoretical connection to WoW minimization on discretized meta-measures (Piening et al., 26 Sep 2025).

1. Object of study and mathematical setting

The primary domain of DSW is the Wasserstein space of Wasserstein spaces: $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$ Such an object is a probability measure over probability measures on $\mathbb{R}^d$. This setting is used to compare distributions over images, shapes, point clouds, and class-conditional distributions in datasets (Piening et al., 26 Sep 2025).

The natural comparison principle in this space is the Wasserstein distance on $P_2(\mathbb{R}^d)$, written as

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$

which the paper refers to as Wasserstein over Wasserstein (WoW) (Piening et al., 26 Sep 2025). For a complete separable metric space $(\mathcal{X},d)$, the paper defines

$$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$

This outer use of Wasserstein is conceptually direct but computationally costly when the atoms of the meta-measures are themselves empirical measures (Piening et al., 26 Sep 2025).

If two empirical meta-measures are supported on $P_2(\mathbb{R}^d)$0 empirical measures, each having $P_2(\mathbb{R}^d)$1 support points in $P_2(\mathbb{R}^d)$2, then the approximate computation of the pairwise ground cost matrix for WoW requires

$P_2(\mathbb{R}^d)$3

operations before solving the outer transport problem (Piening et al., 26 Sep 2025). This computational bottleneck motivates sliced surrogates.

A possible source of ambiguity is that the acronym DSW also appears in earlier work to denote distributional Sliced-Wasserstein, meaning a sliced Wasserstein distance with a learned distribution over projection directions rather than a literal two-stage slicing procedure. "Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances" (Ohana et al., 2022) explicitly uses DSW in that sense and defines

$P_2(\mathbb{R}^d)$4

with $P_2(\mathbb{R}^d)$5 chosen by optimization under a diversity constraint. That paper also states unambiguously that it does not study a second nested slicing construction (Ohana et al., 2022). In contemporary usage, therefore, "Double-Sliced Wasserstein" refers specifically to the two-level meta-measure construction of (Piening et al., 26 Sep 2025), whereas "distributional Sliced-Wasserstein" refers to learned slice distributions in the sense of (Ohana et al., 2022).

2. Foundational ingredients

The first ingredient is the standard sliced Wasserstein construction in Euclidean space. For $P_2(\mathbb{R}^d)$6, the classical sliced Wasserstein distance is defined by

$P_2(\mathbb{R}^d)$7

where $P_2(\mathbb{R}^d)$8 and $P_2(\mathbb{R}^d)$9 denotes the uniform probability measure on the sphere (Piening et al., 26 Sep 2025). This replaces a high-dimensional transport problem by an average of one-dimensional transport problems.

The second ingredient is the exact one-dimensional Wasserstein–quantile formula. For $P_2(\mathbb{R})$0, the quantile function is

$P_2(\mathbb{R})$1

and for $P_2(\mathbb{R})$2,

$P_2(\mathbb{R})$3

equivalently

$P_2(\mathbb{R})$4

(Piening et al., 26 Sep 2025). This implies that the map

$P_2(\mathbb{R})$5

is an isometric embedding (Piening et al., 26 Sep 2025). The inner stage of DSW is built entirely on this isometry.

The third ingredient is a general sliced Wasserstein framework on Banach spaces. Let $P_2(\mathbb{R})$6 be a separable Banach space with dual $P_2(\mathbb{R})$7. Slicing is performed with continuous linear functionals

$P_2(\mathbb{R})$8

and an arbitrary slicing measure

$P_2(\mathbb{R})$9

The paper defines

$L_2([0,1])$0

(Piening et al., 26 Sep 2025). The motivation is that there is no uniform probability distribution on the unit sphere of an infinite-dimensional Banach space, so the role of “random direction” is taken by a random functional sampled from $L_2([0,1])$1.

The paper proves that $L_2([0,1])$2 is always a well-defined pseudo-metric on $L_2([0,1])$3, and becomes a metric if

$L_2([0,1])$4

(Piening et al., 26 Sep 2025). This Banach-space framework is the formal mechanism that allows the second slicing stage in $L_2([0,1])$5.

3. Construction of Double-Sliced Wasserstein

The outer slicing stage maps each inner measure $L_2([0,1])$6 to a one-dimensional projected measure. For $L_2([0,1])$7, the paper defines the meta-projection

$L_2([0,1])$8

A meta-measure $L_2([0,1])$9 is therefore pushed forward to a one-dimensional meta-measure

$\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$0

If one computes WoW after this first slicing and averages over $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$1, one obtains the intermediate sliced WoW

$\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$2

(Piening et al., 26 Sep 2025).

The inner slicing stage operates on $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$3. Because the quantile map $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$4 embeds $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$5 isometrically into $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$6, it pushes a one-dimensional meta-measure $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$7 to

$\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$8

Using the Banach-space sliced framework with $\boldsymbol{\mu} \in P_2(P_2(\mathbb{R}^d)).$9, the paper defines the sliced quantile WoW (SQW) distance: $\mathbb{R}^d$0 (Piening et al., 26 Sep 2025).

Combining the two stages yields the main definition: $\mathbb{R}^d$1 Unfolding $\mathbb{R}^d$2, the definition becomes

$\mathbb{R}^d$3

(Piening et al., 26 Sep 2025).

This definition gives a literal interpretation of “double slicing”:

• Outer slicing: $\mathbb{R}^d$4 via $\mathbb{R}^d$5.
• Inner slicing: $\mathbb{R}^d$6 via $\mathbb{R}^d$7, after the quantile embedding.

The distribution over outer projections is the uniform distribution on $\mathbb{R}^d$8, whereas the distribution over inner projections is $\mathbb{R}^d$9, chosen in practice as a Gaussian measure corresponding to a Gaussian process $P_2(\mathbb{R}^d)$0 with RBF kernel

$P_2(\mathbb{R}^d)$1

(Piening et al., 26 Sep 2025). Because $P_2(\mathbb{R}^d)$2 is universal, the induced Gaussian measure has full support, which ensures positivity and metricity of the inner sliced distance (Piening et al., 26 Sep 2025).

4. Relation to WoW and metric properties

The principal theoretical claim is not that DSW is pointwise equal to WoW. Rather, DSW is presented as a surrogate metric with a strong minimization equivalence to WoW on discretized meta-measures (Piening et al., 26 Sep 2025).

The central theorem states:

For positive $P_2(\mathbb{R}^d)$3, $P_2(\mathbb{R}^d)$4 defines a metric on $P_2(\mathbb{R}^d)$5. Moreover, for $P_2(\mathbb{R}^d)$6 with compact $P_2(\mathbb{R}^d)$7 and positive Gaussian $P_2(\mathbb{R}^d)$8, it holds $P_2(\mathbb{R}^d)$9 (Piening et al., 26 Sep 2025).

Accordingly, DSW minimization is said to be equivalent to WoW minimization for discretized meta-measures. The exact statement concerns zero-convergence behavior rather than equality of objective values (Piening et al., 26 Sep 2025). A plausible implication is that, in the empirical fixed-cardinality compact-support regime of the theorem, DSW can be substituted for WoW without altering the asymptotic location of minimizers.

A key comparison inequality in the appendix is

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$0

with

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$1

(Piening et al., 26 Sep 2025). This shows that DSW is controlled by WoW via the intermediate once-sliced object.

Metricity of the inner Banach-space sliced distance follows from Theorem 1 of (Piening et al., 26 Sep 2025), which establishes non-negativity, symmetry, and the triangle inequality, with definiteness under the support condition on $$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$2. For DSW on empirical meta-measures, definiteness is proved using a Cramér–Wold-type result due to Cuesta-Albertos, Fraiman, and Ransford: if two measures satisfy the Carleman condition and agree on projections over a set of directions of positive spherical measure, then they are equal. Since empirical measures satisfy the Carleman condition, this yields definiteness of DSW in that regime (Piening et al., 26 Sep 2025).

The appendix also establishes a general Banach-space stability bound

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$3

together with a Lipschitz dependence on the projection parameter: $$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$4 (Piening et al., 26 Sep 2025). These results support measurability and Monte Carlo approximation.

The scope of the strongest theoretical guarantees is limited. The most explicit definiteness and convergence-equivalence results are stated for empirical or discretized meta-measures and, more generally, for classes satisfying the Carleman condition (Piening et al., 26 Sep 2025). The paper remarks that broader compact-support extensions are plausible but are not fully developed.

5. Computation and approximation

For empirical meta-measures of the form

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$5

and similarly for $$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$6, the computational pipeline follows the two slicing levels (Piening et al., 26 Sep 2025).

For each sampled outer direction $$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$7, one first computes the projected one-dimensional empirical measures

$$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$8

Their quantile functions are then obtained by sorting projected support points. Next, one samples an inner projection $$W(\boldsymbol{\mu},\boldsymbol{\nu};P_2(\mathbb{R}^d)),$$9 from the Gaussian process corresponding to $(\mathcal{X},d)$0 and computes the functional inner product

$(\mathcal{X},d)$1

This pushes the meta-measure to a scalar empirical measure over its atoms.

In practice, the inner product is approximated by quadrature: $(\mathcal{X},d)$2 so that

$(\mathcal{X},d)$3

(Piening et al., 26 Sep 2025).

With sampled pairs $(\mathcal{X},d)$4, $(\mathcal{X},d)$5, the estimator is

$(\mathcal{X},d)$6

All remaining transport calculations are one-dimensional (Piening et al., 26 Sep 2025).

For general Banach slicing, the paper proves the Monte Carlo rate

$(\mathcal{X},d)$7

which implies the standard $(\mathcal{X},d)$8 decay in the number of projection samples (Piening et al., 26 Sep 2025).

The claimed computational advantage over WoW comes from replacing many high-dimensional pairwise Wasserstein calculations by repeated operations of four kinds: one-dimensional projection of support points, sorting to obtain quantiles, scalar quadrature against Gaussian functions, and one-dimensional Wasserstein computation (Piening et al., 26 Sep 2025). The implementation further amortizes cost by reusing quantile computations across many inner projections for the same outer direction.

6. Relation to adjacent sliced-Wasserstein constructions

The most immediate conceptual relative of DSW is sliced WoW itself, namely $(\mathcal{X},d)$9, which performs only the outer slicing step. DSW refines that approach by using the quantile isometry to slice again in $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$0, thereby replacing WoW on $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$1 by a functional sliced surrogate (Piening et al., 26 Sep 2025). The paper argues that this avoids the moment truncation used by earlier sliced WoW methods.

A second related line is adaptive sliced-Wasserstein, where the distribution over projection directions is learned rather than fixed. In (Ohana et al., 2022), sliced Wasserstein with a slice distribution $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$2 is written as

$$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$3

That paper treats standard SW as the special case $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$4, max-SW as the case in which $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$5 is a Dirac mass, and distributional SW as the case in which $$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$6 is optimized over a family of distributions (Ohana et al., 2022). It explicitly states that its framework is

$$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$7

rather than a nested double-slicing architecture (Ohana et al., 2022).

This distinction is important because the same acronym DSW may otherwise mask two unrelated ideas:

Term	Core construction	Source
Double-Sliced Wasserstein	Two nested slicing levels for meta-measures	(Piening et al., 26 Sep 2025)
Distributional Sliced-Wasserstein	One slicing level with a learned distribution over projections	(Ohana et al., 2022)

The PAC-Bayesian theory in (Ohana et al., 2022) is nevertheless relevant by analogy. That paper interprets adaptive sliced Wasserstein as an average empirical risk and proves a generalization bound of the form

$$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$8

(Ohana et al., 2022). This suggests that if one wished to analyze learned outer or inner projection laws in DSW, a PAC-Bayesian treatment of projection distributions could be a plausible extension, although such a theory is not provided in (Piening et al., 26 Sep 2025).

A third neighboring construction is the sliced multi-marginal Wasserstein distance of (Cohen et al., 2021), which also reduces a complex OT problem to averaged one-dimensional subproblems. There the objective is multi-marginal rather than pairwise or meta-measure-based. The paper defines

$$W(\mu, \nu; \mathcal X) \coloneqq \inf_{\gamma \in\Gamma(\mu, \nu)} \biggl( \int_{\mathcal X \times \mathcal X} d^2(x_1, x_2) \, d \gamma(x_1, x_2) \biggr)^{\frac{1}{2}}.$$9

for $P_2(\mathbb{R}^d)$00 measures and proves that it is a generalized metric, induces the same topology as standard Wasserstein, admits a dimension-free sample complexity, and is tightly connected with barycentric averaging under the sliced-Wasserstein metric (Cohen et al., 2021). The conceptual overlap with DSW is the reliance on exact one-dimensional OT structure, projection-based relaxation, and Monte Carlo approximation, but (Cohen et al., 2021) involves only one slicing layer and no Wasserstein-over-Wasserstein geometry.

7. Empirical behavior, scope, and limitations

The experiments in (Piening et al., 26 Sep 2025) are intended to show that DSW is a practical substitute for WoW across several application domains.

For shape classification via local distance distributions, shapes are represented as meta-measures in $P_2(\mathbb{R}^d)$01. Using datasets 2D Shapes, Animals, FAUST-500, and FAUST-1000 with settings $P_2(\mathbb{R}^d)$02, $P_2(\mathbb{R}^d)$03, $P_2(\mathbb{R}^d)$04, the paper reports that SQW yields comparable KNN classification accuracy to TLB, STLB, AE, and GW, with significant runtime advantage over GW on larger datasets (Piening et al., 26 Sep 2025). For FAUST-1000, the reported values are: Ours $P_2(\mathbb{R}^d)$05 accuracy, $P_2(\mathbb{R}^d)$06 ms; TLB $P_2(\mathbb{R}^d)$07, $P_2(\mathbb{R}^d)$08 ms; STLB $P_2(\mathbb{R}^d)$09, $P_2(\mathbb{R}^d)$10 ms; AE $P_2(\mathbb{R}^d)$11, $P_2(\mathbb{R}^d)$12 ms; GW $P_2(\mathbb{R}^d)$13, $P_2(\mathbb{R}^d)$14 ms (Piening et al., 26 Sep 2025).

For OTDD comparison on image datasets, using MNIST, FashionMNIST, and CIFAR-10, DSW and s-OTDD are compared against OTDD over 100 random dataset splits, with DSW settings $P_2(\mathbb{R}^d)$15, $P_2(\mathbb{R}^d)$16, $P_2(\mathbb{R}^d)$17. The reported result is that DSW correlates strongly with OTDD and can serve as a drop-in replacement for the expensive OTDD/WoW computation (Piening et al., 26 Sep 2025).

For comparing distributions of point clouds, with each point cloud treated as an empirical measure in $P_2(\mathbb{R}^d)$18 and batches treated as empirical meta-measures in $P_2(\mathbb{R}^d)$19, the paper compares OT-NNA, WoW, and DSW while varying the number of target shapes, Gaussian noise level, and point cloud resolution. It reports that WoW and DSW both detect mode collapse, plateau sensibly as the number of target shapes increases, react robustly to noise and point resolution, and that DSW closely tracks WoW’s qualitative behavior (Piening et al., 26 Sep 2025). For $P_2(\mathbb{R}^d)$20 and $P_2(\mathbb{R}^d)$21, the runtime example is approximately $P_2(\mathbb{R}^d)$22 s for DSW with $P_2(\mathbb{R}^d)$23, $P_2(\mathbb{R}^d)$24, $P_2(\mathbb{R}^d)$25, compared with about $P_2(\mathbb{R}^d)$26 s for WoW and about $P_2(\mathbb{R}^d)$27 s for OT-NNA (Piening et al., 26 Sep 2025).

For comparing image distributions via patch distributions, images are represented as empirical distributions of vectorized $P_2(\mathbb{R}^d)$28 patches: $P_2(\mathbb{R}^d)$29 and batches become meta-measures in $P_2(\mathbb{R}^d)$30. On synthetic $P_2(\mathbb{R}^d)$31 Perlin-noise textures with batch size $P_2(\mathbb{R}^d)$32, patch size $P_2(\mathbb{R}^d)$33, and DSW settings $P_2(\mathbb{R}^d)$34, $P_2(\mathbb{R}^d)$35, $P_2(\mathbb{R}^d)$36, the paper reports that patch-based DSW is minimized at the true reference parameters, is more sensitive to parameter changes than Euclidean Wasserstein-on-images, behaves similarly to patch-based WoW, and reduces runtime from about $P_2(\mathbb{R}^d)$37 s for patch-based WoW to about $P_2(\mathbb{R}^d)$38 s for DSW (Piening et al., 26 Sep 2025).

The limitations identified in (Piening et al., 26 Sep 2025) are specific and consequential. First, the strongest equivalence to WoW is proved for discretized empirical meta-measures with fixed support cardinalities and compact support. Second, the inner slicing measure $P_2(\mathbb{R}^d)$39 is implemented through a Gaussian process, introducing hyperparameters such as the bandwidth $P_2(\mathbb{R}^d)$40; supplementary experiments show that very small $P_2(\mathbb{R}^d)$41 can make estimates sensitive to discretization. Third, DSW remains approximate in practice because it relies on Monte Carlo sampling over projections and quadrature on $P_2(\mathbb{R}^d)$42. Fourth, the present formulation targets $P_2(\mathbb{R}^d)$43; extension to labelled settings such as $P_2(\mathbb{R}^d)$44 is mentioned as future work. Finally, the conclusion notes that additional topological properties remain to be analyzed (Piening et al., 26 Sep 2025).

Taken together, these results characterize Double-Sliced Wasserstein as a two-stage projection-based OT surrogate for meta-measures: it preserves the outer philosophy of sliced Wasserstein, replaces the inner one-dimensional Wasserstein geometry by its exact quantile-space linearization, and thereby provides a metric on empirical meta-measures with convergence-equivalent minimization behavior relative to WoW in the discretized regime studied in (Piening et al., 26 Sep 2025).