Sliced Wasserstein Distance Estimation

• Sliced Wasserstein Distance Estimation is a method that projects high-dimensional distributions onto 1D spaces to efficiently compute optimal transport distances.
• It leverages integral transforms like the Radon transform and employs sorting algorithms to reduce complexity in comparing empirical measures.
• The estimator’s performance is influenced by sample size and projection count, achieving optimal convergence rates under standard regularity conditions.

The sliced Wasserstein distance (SWD) is a computationally tractable variant of the Wasserstein (optimal transport) distance that leverages one-dimensional projections to compare high-dimensional probability measures. SWD is widely studied for its statistical properties, empirical performance, and central role in large-scale generative modeling, distributional regression, and high-dimensional data analysis. Estimating the SWD from finite samples—termed "sliced Wasserstein distance estimation"—is essential for both theoretical inference and practical implementation.

1. Definition and Slicing Transforms

Given probability measures $P, Q$ on $\mathbb{R}^p$ with finite $p$-th moments, for each unit vector $\theta \in S^{p-1}$ (the unit sphere), define the associated one-dimensional marginals: $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$ where $\langle \theta, x \rangle$ denotes projection and $(\cdot)_\#$ the pushforward. The $L^p$-sliced Wasserstein distance is

$$\mathrm{SW}_p(P, Q) = \left( \int_{S^{p-1}} W_p^p( P_\theta, Q_\theta ) \, d\theta \right)^{1/p},$$

where $W_p(\cdot, \cdot)$ is the standard 1D Wasserstein distance and $\mathbb{R}^p$0 is the uniform Haar measure (Chen et al., 2023).

More generally, slicing may be interpreted via integral transforms, such as the Radon transform,

$\mathbb{R}^p$1

for $\mathbb{R}^p$2 a function or density and $\mathbb{R}^p$3 the surface measure (Chen et al., 2023, Nguyen et al., 2022). Injectivity and invertibility of the chosen transform are essential for establishing metric properties.

2. Sample-Based Estimation and Algorithms

In practical settings, one observes i.i.d. samples $\mathbb{R}^p$4, $\mathbb{R}^p$5 and constructs empirical measures $\mathbb{R}^p$6. For each projection direction $\mathbb{R}^p$7, one computes the projected samples, sorts them, and evaluates the 1D $\mathbb{R}^p$8. The empirical estimator with $\mathbb{R}^p$9 directions is

$p$0

Directions $p$1 are typically selected i.i.d. uniformly on $p$2, but deterministic quadrature (e.g., spherical designs) is also possible (Chen et al., 2023). The sorting cost for each projection is $p$3, so the total complexity is $p$4 for moderate sample sizes.

Algorithmic Summary:

Step	Description	Complexity
Project/sort	For each $p$5, compute and sort	$p$6
1D OT	Compute $p$7 via quantile/empirical CDF	$p$8
Average	Aggregate over all directions ($p$9)	$\theta \in S^{p-1}$0

(Chen et al., 2023, Nguyen et al., 2022, Nietert et al., 2022)

3. Theoretical Guarantees

Consistency and Rates

Under standard regularity conditions (compact support, bounded density, finite moments), the empirical estimator satisfies

$\theta \in S^{p-1}$1

as $\theta \in S^{p-1}$2 (Chen et al., 2023). When sample size and the number of directions are of the same order, the error decreases at the optimal parametric rate.

Uniform Concentration

With suitable bracketing or VC-type entropy control over the class of projections, uniform concentration inequalities apply, and for bounded one-dimensional Wasserstein terms, Hoeffding-type deviation bounds yield

$\theta \in S^{p-1}$3

for some constant $\theta \in S^{p-1}$4 (Chen et al., 2023).

4. Comparison to Alternative Paradigms

Slice-Average Regression (SAW):

Performs minimization of the full sliced Wasserstein (i.e., over the space of probability measures on $\theta \in S^{p-1}$5), which is non-convex globally but reflects the full geometry of the underlying distributions.

Slice-Wise Regression (SWW):

Performs independent 1D regression per direction (Fréchet mean), then reconstructs the multivariate distribution via inversion of the slice transform. While each 1D problem is convex and parallelizable, the aggregation step may be unstable or high-variance in large $\theta \in S^{p-1}$6.

Each paradigm's statistical and computational tradeoffs depend on the properties of the slicing/inversion schemes and the geometry of the regression functional (Chen et al., 2023).

5. Practical Considerations in SWD Estimation

• Number of Directions ($\theta \in S^{p-1}$7): Empirically, $\theta \in S^{p-1}$8 suffices in moderate dimensions ($\theta \in S^{p-1}$9). Cost grows linearly with $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$0. Adaptive approaches (e.g., cross-validation) can guide selection (Chen et al., 2023).
• Sampling of Directions: Random uniform sampling is unbiased and easy to implement; deterministic designs can yield lower variance/quadrature error.
• Computational Complexity: For common settings ($$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$1), estimation is feasible on standard hardware in seconds.
• Bias and Variance: Finite $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$2 yields $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$3 bias; finite $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$4 yields $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$5 variance. The total RMSE is of the form $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$6 for appropriate constants.
• Empirical Performance: Simulations confirm predicted rates and demonstrate suitability for data analysis and regression tasks with distributional responses (Chen et al., 2023).

6. Extensions: Slicing Transforms and Stability

The definition of slicing is extendable: any transform $$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$7 that satisfies injectivity, stability (Lipschitz), invertibility, and admits regularized inverses can, in principle, serve as the basis for a generalized SWD. The Radon transform is a canonical choice due to its known properties in tomography and harmonic analysis (Chen et al., 2023, Nguyen et al., 2022). For stable inference and regression, the existence of a continuous regularized inverse is necessary.

7. Applications and Limitations

SWD estimation is widely deployed in statistical modeling with distributional responses, such as Fréchet regression, and in generative modeling for high-dimensional data where full-dimensional OT is computationally prohibitive (Nguyen et al., 2022, Nietert et al., 2022). Its favorable statistical properties, fast computational algorithms, and adaptability to different slicing paradigms make SWD estimation a central tool for modern data-driven studies of probability distributions.

Limitations include the non-convexity of the global regression objective when using the full SWD (SAW), instability of the inversion step when employing slice-wise regression (SWW) in high dimensions, and the linear scaling of computation with the number of projections.

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Summary Table: Estimator Error Sources

Error Source	Rate	Main Control Parameter
Finite sample size	$$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$8	$$P_\theta = \langle \theta, \cdot \rangle_\# P, \quad Q_\theta = \langle \theta, \cdot \rangle_\# Q,$$9, $\langle \theta, x \rangle$0
Number of slices	$\langle \theta, x \rangle$1	$\langle \theta, x \rangle$2

These rates are minimax optimal under classical regularity conditions for optimal transport processes (Chen et al., 2023).