Minimum Sliced Wasserstein Estimation

• Minimum sliced Wasserstein estimation is a framework that minimizes the averaged one-dimensional Wasserstein distances to compare high-dimensional probability measures.
• It employs methods like random and orthogonal projections, learnable slicing, and control variates to achieve computational tractability and reduced variance.
• The approach is underpinned by strong statistical guarantees including consistency, CLTs, and minimax rates, ensuring reliable inference and robust optimization.

Minimum Sliced Wasserstein Estimation is the methodology and theory surrounding statistical inference, generative modeling, and parameter estimation procedures that minimize the sliced Wasserstein (SW) distance between probability distributions. The SW distance is constructed by averaging the Wasserstein distances between the one-dimensional projections ("slices") of two measures, and has become central in addressing computational and statistical challenges of optimal transport (OT) in high-dimensional spaces. Minimum SW estimation provides computational tractability, robust optimization landscapes, and strong theoretical guarantees, making it a foundational component in a range of modern machine learning and statistical applications.

1. Sliced Wasserstein Distance: Structure and Rationale

The sliced Wasserstein distance (SWD) of order $p$ between two high-dimensional probability measures $\mu, \nu$ on $\mathbb{R}^d$ is defined as

$$\mathrm{SW}_p^p(\mu, \nu) = \int_{S^{d-1}} W_p^p(P_\theta\#\mu, ~P_\theta\#\nu) \, d\sigma(\theta),$$

where $P_\theta(x) = \langle \theta, x \rangle$, $W_p$ is the standard one-dimensional $p$-Wasserstein distance, and $\sigma$ is the uniform measure on the unit sphere $S^{d-1}$ (Wu et al., 2017, Kolouri et al., 2017). The key computational advantage is that $W_p$ is tractable in 1D (via sorting and CDF inversion), while the aggregation over directions yields a metric on high-dimensional distributions.

Averaging over one-dimensional projections avoids the curse of dimensionality that affects high-dimensional OT solvers, permits fast O($n\log n$) implementations for empirical distributions, and supports closed-form solutions for many subproblems. This makes the SW distance a structurally attractive object for minimum distance estimation: comparing model and data distributions by minimizing SWD is fundamentally lower-complexity and often numerically more stable than minimizing the true $d$-Wasserstein distance.

2. Algorithms and Approximation Schemes

The classic approach to SW estimation relies on Monte Carlo integration: random or quasi-random projection directions $\{\theta_\ell\}$ are sampled from $S^{d-1}$, and the empirical average over $L$ directions approximates the full integral

$$\widehat{\mathrm{SW}}_p^p(\mu, \nu; L) = \frac{1}{L} \sum_{\ell=1}^L W_p^p(P_{\theta_\ell}\#\mu, P_{\theta_\ell}\#\nu).$$

Selecting and structuring these projections is central to practical minimum SW estimation:

• Random Projections: Purely random Monte Carlo directions ensure unbiasedness but may incur high variance in finite $L$ regimes.
• Orthogonal Coupling: Sampling projections as orthogonal directions (for instance, via Haar measure on the orthogonal group) reduces estimator variance and approximates stratified sampling over $S^{d-1}$ by increasing coverage diversity (Rowland et al., 2019).
• Learnable/Parameterized Projections: Instead of random directions, a small set of learned orthogonal projection matrices (optimized end-to-end, typically on the Stiefel manifold) can dramatically reduce the number of projections required for accurate estimation. In generative models, deep architectures use differentiable SWD "blocks" with parameterized orthogonal projections, yielding efficient, highly informative distance approximations with as few as 128 learned projection directions (Wu et al., 2017).
• Control Variates: Variance reduction via control variates (using Gaussian-matched projections) enables more accurate SW estimation without increased computational load (Nguyen et al., 2023).
• Hierarchical and Bottleneck Projections: Hierarchical Sliced Wasserstein (HSW) employs a two-stage projection architecture (a small $k$-dimensional bottleneck, followed by linear mixing into $L$ directions) to further trade off computational complexity and estimation fidelity in high-dimensional or mini-batch settings (Nguyen et al., 2022).

Recent advances have introduced random-path projection schemes—sampling directions as normalized differences $X-Y$ between samples $X \sim p$, $Y \sim \nu$, potentially regularized by a location-scale distribution—to preferentially explore "discriminative" directions and provide more informative minimum base directions for the discrepancy (Nguyen et al., 29 Jan 2024).

3. Statistical and Optimization Theory

Minimum SW estimation is statistically well-behaved, with theoretical results paralleling those for classical Wasserstein minimum distance estimators:

• Consistency: Under mild regularity, convergence in SW distance implies weak convergence of distributions (Nadjahi et al., 2019). Minimum SW estimators (minimizing SWD over model parameters $\theta$) are consistent estimators of the population minimizer.
• Central Limit Theorems (CLT): Recent work rigorously establishes that, for $p > 1$, the empirical SWD (or plug-in minimum SWD estimator) is asymptotically normal, and—importantly—can be centered at the population cost under suitable regularity, enabling valid frequentist inference and hypothesis testing in parametric and nonparametric settings (Rodríguez-Vítores et al., 24 Mar 2025, Nadjahi et al., 2019, Xi et al., 2022).
• Minimax Rates: The minimax convergence rates for minimum SW estimation match (or improve upon) those for full Wasserstein estimation under weak moment assumptions, achieving $O(n^{-1/2})$ parametric rates under regularity of projections or slightly slower rates otherwise (Singh et al., 2018, Manole et al., 2019). These rates imply that minimum SW estimation is dimension-free in the sample complexity.
• Optimization Landscape: The SW energy (as a function of model parameters or point supports) is piecewise quadratic, smooth within polytopal "cells" defined by permutations of projected points, and globally locally Lipschitz. Both block coordinate descent and stochastic gradient descent methods provably converge to (Clarke) critical points, even when using a finite number of projections (Tanguy et al., 2023, Kolouri et al., 2017).

The use of "minimum distance estimation" with SWD can also mitigate multi-modality and adverse local minima, as commonly encountered in likelihood or KL-divergence-based maximum likelihood estimation frameworks.

4. Applications in Generative Models and Beyond

Minimum SW estimation is widely applied in generative modeling and parameter inference across multiple domains:

Application Type	Approach	SW Role
Autoencoders (SWAE)	Primal SWD blocks	Push encoder output to match prior without extra regularizer (Wu et al., 2017)
Generative Adversarial Networks (SWGAN)	Dual SWD blocks	Discriminator loss via efficient 1D SWD critics
Gaussian Mixture Models	SW means estimation	Minimize SWD between GMM and empirical data
Flow Matching / Diffusion Models	Sliced OT-computed couplings	Enable scalable high-quality transport plans
Robust Estimation	Partial OT/Sliced robust SWD	Dual formulation for minimax-optimal estimation under contamination (Nietert et al., 2023)
Multi-task Learning	Sliced multi-marginal OT	Shared structure/regularization over multiple tasks
Likelihood-Free Inference	SWD-based confidence intervals	Nonparametric uncertainty quantification

Experiments demonstrate that even a small number of learnable projections yields statistically and visually superior generative models (quantified via FID score), outperforming classical models and random-projection-based SW models (Wu et al., 2017, Nguyen et al., 2022). In clustering, e.g., SW-GMM outperforms EM-GMM in both convergence robustness and resulting sample purity (Kolouri et al., 2017). Gradient flows leveraging SWD demonstrate improved stability and faster convergence when using control variates or discriminative projection strategies (Nguyen et al., 2023, Nguyen et al., 29 Jan 2024).

5. Geometry, Regularity, and Metric Properties

The geometry and analytic structure of the SW metric space underpins its effectiveness in minimum distance estimation:

• Comparison with Negative Sobolev Norms: On "nice" measures (absolutely continuous with bounded densities), the SW metric is equivalent to the homogeneous negative Sobolev norm $\dot H^{-(d+1)/2}$, providing RKHS-like local structure and explaining parametric rates (Park et al., 2023). At discrete approximations, SW and classical Wasserstein metrics are comparable up to dimension-dependent constants.
• Gradient Flows and Tangent Structure: Although SW space is not a length space (does not guarantee geodesics), its length-space completion has well-behaved geodesics and well-defined tangent spaces, allowing for higher-order (negative Sobolev) gradient flows and variational schemes.
• Metricity: Variants such as the Hierarchical SW, min-SWGG, and expected sliced transport (EST) distance are all proven to be (quasi-)metrics under appropriate conditions, satisfying symmetry, triangle inequality, and identity of indiscernibles (Mahey et al., 2023, Liu et al., 16 Oct 2024, Nguyen et al., 2022). Metricity persists under generalizations, including importance-weighted and random-path based sliced distances.

These geometric properties ensure that minimum SW estimation provides rigorous, robust measures of discrepancy, supporting both theoretical inference and practical comparisons in high dimensions.

6. Variants, Extensions, and Future Directions

Recent advances have expanded the scope of minimum SW estimation:

• Random-Path and Discriminative Slicing: Construction of slicing distributions based on normalized sample differences (random-path directions) enhances discrimination and accelerates convergence in optimization and gradient flows (Nguyen et al., 29 Jan 2024).
• Transport Plan Construction: Methods such as min-SWGG and expected sliced transport (EST) plans "lift" optimal 1D matchings back to high dimensions, yielding computationally tractable, explicit transport plans and providing proxies or upper bounds to the true Wasserstein distance (Mahey et al., 2023, Liu et al., 16 Oct 2024). EST with Gibbs-weighted averaging interpolates between classic SW and min-SWGG.
• Generalized and Differentiable Slicing: The min-SWGG framework has been reformulated as a bilevel optimization problem, and extended to differentiable approximations enabling gradient-based tuning of slices, and further to non-linear projections (e.g., neural parameterizations and manifold-valued data) (Chapel et al., 28 May 2025).
• Variance-Reduced Estimation: Incorporation of control variates using Gaussian-matched approximations provides statistically optimal variance properties with matched computational cost, boosting reliability in both estimation and downstream model training (Nguyen et al., 2023).
• Hierarchical and Multi-marginal Slicing: Hierarchical and multi-marginal SW bring further computational advantages and are foundational for scalable structure-sharing in multi-task learning (Nguyen et al., 2022, Cohen et al., 2021).

Active research areas include minimizing the required number of slices in high dimension, robustly estimating SW under adversarial contamination, further tightening the link between geometric structure and statistical efficiency, and harnessing learned slicing or generalized projections for data on non-Euclidean manifolds or spaces with complex dependencies (Nietert et al., 2023, Chapel et al., 28 May 2025).

7. Statistical Inference and Confidence Intervals

Recent work has placed minimum SW estimation on firm inferential ground:

• Finite-sample and Asymptotic Inference: Construction of confidence intervals for SWD is possible via quantile confidence bands on projected measures, yielding finite-sample and minimax-optimal intervals (adaptive to regularity), as well as CLTs for both parameter estimation and hypothesis testing (Manole et al., 2019, Rodríguez-Vítores et al., 24 Mar 2025, Xi et al., 2022).
• Bootstrap Deficiencies: Standard bootstrap schemes may under-cover in low-smoothness regimes; finite-sample analytic intervals provide robust coverage (Manole et al., 2019).
• Monte Carlo and Variance Estimation: The variance arising from both empirical (sample) error and Monte Carlo integration error in slicing can be explicitly estimated, and their trade-off governs the confidence interval length and power of tests (Rodríguez-Vítores et al., 24 Mar 2025).

This inferential machinery enables rigorous application of minimum SW estimation to both parameter learning and model selection in high-dimensional data science applications.

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In summary, minimum sliced Wasserstein estimation combines the computational tractability of one-dimensional OT, modern algorithmic advances in projection selection and smoothing, robust statistical theory, rigorous metric geometry, and versatile modeling applications. As variants and extensions (including random-path, hierarchical, and transport-plan-forming approaches) are developed, minimum SW estimation is positioned as a core methodology for scalable, robust, and theoretically principled quantification and learning in high-dimensional probability and generative modeling.