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Quasi-Monte Carlo for 3D Sliced Wasserstein

Khai Nguyen, Nicola Bariletto, Nhat Ho

TL;DR

The paper tackles the computational bottleneck of the Sliced Wasserstein distance by introducing Quasi-Sliced Wasserstein (QSW) with low-discrepancy sphere point sets and Randomized Quasi-Sliced Wasserstein (RQSW) for unbiased optimization. It develops several sphere-sampling mappings (Gaussian-based, equal-area, generalized spiral, maximizing distance, minimizing Coulomb energy), proves asymptotic convergence of QSW and unbiasedness of RQSW, and demonstrates superior performance over standard Monte Carlo SW across 3D tasks such as point-cloud comparison/interpolation, image style transfer, and deep point-cloud autoencoding. The results show that QSW/RQSW yield better approximation accuracy and more stable gradients, with RCQSW and related randomized variants recommended for practical use. This work broadens the applicability of SW in 3D data contexts and lays groundwork for extending QSW/RQSW to higher dimensions and alternative SW forms.

Abstract

Monte Carlo (MC) integration has been employed as the standard approximation method for the Sliced Wasserstein (SW) distance, whose analytical expression involves an intractable expectation. However, MC integration is not optimal in terms of absolute approximation error. To provide a better class of empirical SW, we propose quasi-sliced Wasserstein (QSW) approximations that rely on Quasi-Monte Carlo (QMC) methods. For a comprehensive investigation of QMC for SW, we focus on the 3D setting, specifically computing the SW between probability measures in three dimensions. In greater detail, we empirically evaluate various methods to construct QMC point sets on the 3D unit-hypersphere, including the Gaussian-based and equal area mappings, generalized spiral points, and optimizing discrepancy energies. Furthermore, to obtain an unbiased estimator for stochastic optimization, we extend QSW to Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness in the discussed point sets. Theoretically, we prove the asymptotic convergence of QSW and the unbiasedness of RQSW. Finally, we conduct experiments on various 3D tasks, such as point-cloud comparison, point-cloud interpolation, image style transfer, and training deep point-cloud autoencoders, to demonstrate the favorable performance of the proposed QSW and RQSW variants.

Quasi-Monte Carlo for 3D Sliced Wasserstein

TL;DR

The paper tackles the computational bottleneck of the Sliced Wasserstein distance by introducing Quasi-Sliced Wasserstein (QSW) with low-discrepancy sphere point sets and Randomized Quasi-Sliced Wasserstein (RQSW) for unbiased optimization. It develops several sphere-sampling mappings (Gaussian-based, equal-area, generalized spiral, maximizing distance, minimizing Coulomb energy), proves asymptotic convergence of QSW and unbiasedness of RQSW, and demonstrates superior performance over standard Monte Carlo SW across 3D tasks such as point-cloud comparison/interpolation, image style transfer, and deep point-cloud autoencoding. The results show that QSW/RQSW yield better approximation accuracy and more stable gradients, with RCQSW and related randomized variants recommended for practical use. This work broadens the applicability of SW in 3D data contexts and lays groundwork for extending QSW/RQSW to higher dimensions and alternative SW forms.

Abstract

Monte Carlo (MC) integration has been employed as the standard approximation method for the Sliced Wasserstein (SW) distance, whose analytical expression involves an intractable expectation. However, MC integration is not optimal in terms of absolute approximation error. To provide a better class of empirical SW, we propose quasi-sliced Wasserstein (QSW) approximations that rely on Quasi-Monte Carlo (QMC) methods. For a comprehensive investigation of QMC for SW, we focus on the 3D setting, specifically computing the SW between probability measures in three dimensions. In greater detail, we empirically evaluate various methods to construct QMC point sets on the 3D unit-hypersphere, including the Gaussian-based and equal area mappings, generalized spiral points, and optimizing discrepancy energies. Furthermore, to obtain an unbiased estimator for stochastic optimization, we extend QSW to Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness in the discussed point sets. Theoretically, we prove the asymptotic convergence of QSW and the unbiasedness of RQSW. Finally, we conduct experiments on various 3D tasks, such as point-cloud comparison, point-cloud interpolation, image style transfer, and training deep point-cloud autoencoders, to demonstrate the favorable performance of the proposed QSW and RQSW variants.
Paper Structure (25 sections, 2 theorems, 20 equations, 13 figures, 6 tables, 4 algorithms)

This paper contains 25 sections, 2 theorems, 20 equations, 13 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Sliced Wasserstein distance and Monte Carlo estimation
  4. Quasi-Monte Carlo Methods
  5. Quasi-Monte Carlo for 3D Sliced Wasserstein
  6. Low-discrepancy sequences on the unit-hypersphere
  7. Quasi-Sliced Wasserstein
  8. Randomized Quasi-Sliced Wasserstein
  9. Experiments
  10. Approximation Error
  11. Point-cloud interpolation
  12. Image Style Transfer
  13. Deep Point-cloud Autoencoder
  14. conclusion
  15. Proofs
  16. ...and 10 more sections

Key Result

Proposition 1

With point sets constructed through the Gaussian-based mapping, the equal area mapping, the generalized spiral points, and minimizing Coulomb energy, we have $\widehat{\text{QSW}}^p_p(\mu,\nu;\theta_1,\ldots,\theta_L) \to \text{SW}_p^p(\mu,\nu)$ as $L \to \infty$.

Figures (13)

  • Figure 1: The error for approximation SW distances between empirical distributions over point-clouds.
  • Figure 2: Point-cloud interpolation from SW, CQSW, and RCQSW with $L=100$.
  • Figure 3: Style-transferred images from SW, CQSW, and RCQSW with $L=100$.
  • Figure 4: Reconstructed point-clouds from SW, CQSW, and RCQSW with $L=100$.
  • Figure 5: Optimization-based Orthogonal Monte Carlo point set and scaled mapping with Halton Sequence point set.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2