mlsanityData to 7 Oct 2026

Quasi-Monte Carlo for 3D Sliced Wasserstein

arXiv 2309.11713 ↗First posted 21 Sep 2023stat.ML cs.GR cs.LGcited by 22

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Abstract (arXiv)

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.

Key facts. Cited by 22 papers in the map (data through 7 Oct 2026). Builds on 25 of its 55 references that are in the map. First posted on arXiv 21 Sep 2023.

Builds on

25 of 55 references.

References · 25

  1. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,223
  2. 2015ShapeNet: An Information-Rich 3D Model Repositorycited by 2,880
  3. 2018Computational Optimal Transportcited by 765
  4. 2017Joint Distribution Optimal Transportation for Domain Adaptationcited by 280
  5. 2019Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptationcited by 235
  6. 2018Improving GANs Using Optimal Transportcited by 122
  7. 2020SoftFlow: Probabilistic Framework for Normalizing Flow on Manifoldscited by 97
  8. 2020Statistical and Topological Properties of Sliced Probability Divergencescited by 83
  9. 2020Distributional Sliced-Wasserstein and Applications to Generative Modelingcited by 69
  10. 2017Sliced Wasserstein Distance for Learning Gaussian Mixture Modelscited by 59
  11. 2017Multilevel Clustering via Wasserstein Meanscited by 48
  12. 2017Optimal Transport for Diffeomorphic Registrationcited by 39
13 more
  1. 2023Energy-Based Sliced Wasserstein Distancecited by 39
  2. 2021Accurate Point Cloud Registration with Robust Optimal Transportcited by 38
  3. 2022Meta Optimal Transportcited by 28
  4. 2021Efficient Gradient Flows in Sliced-Wasserstein Spacecited by 21
  5. 2022Sliced Wasserstein Variational Inferencecited by 17
  6. 2023Sliced Wasserstein Estimation with Control Variatescited by 16
  7. 2022Hilbert Curve Projection Distance for Distribution Comparisoncited by 9
  8. 2020Demystifying Orthogonal Monte Carlo and Beyondcited by 8
  9. 2023Lidar Upsampling with Sliced Wasserstein Distancecited by 8
  10. 2023Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstructioncited by 7
  11. 2023Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstructioncited by 7
  12. 2023Convergence of SGD for Training Neural Networks with Sliced Wasserstein Lossescited by 4
  13. 2020An enumerative formula for the spherical cap discrepancycited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 7; 2025: 13; 2026: 2.

22 papers.

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  1. 2025Multivariate Conformal Prediction using Optimal Transportcited by 21
  2. 2024Sliced Wasserstein with Random-Path Projecting Directionscited by 15
  3. 2025Lightspeed Geometric Dataset Distance via Sliced Optimal Transportcited by 10
  4. 2024Generalized Sobolev Transport for Probability Measures on a Graphcited by 9
  5. 2025A Novel Sliced Fused Gromov-Wasserstein Distancecited by 9
  6. 2024Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variatescited by 8
  7. 2024Tree-Sliced Wasserstein Distance: A Geometric Perspectivecited by 7
  8. 2024Fast Summation of Radial Kernels via QMC Slicingcited by 7
  9. 2025Slicing the Gaussian Mixture Wasserstein Distancecited by 7
  10. 2024Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributionscited by 6
  11. 2025Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distancescited by 5
  12. 2025Slicing Wasserstein Over Wasserstein Via Functional Optimal Transportcited by 5
10 more
  1. 2025Distance-Based Tree-Sliced Wasserstein Distancecited by 4
  2. 2025Tree-Sliced Wasserstein Distance with Nonlinear Projectioncited by 3
  3. 2024Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slicescited by 2
  4. 2026Amortized Optimal Transport from Sliced Potentialscited by 2
  5. 2025Unbiased Sliced Wasserstein Kernels for High-Quality Audio Captioningcited by 1
  6. 2025Repulsive Monte Carlo on the sphere for the sliced Wasserstein distancecited by 1
  7. 2025Streaming Sliced Optimal Transportcited by 0
  8. 2025Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimizationcited by 0
  9. 2025ReSWD: ReSTIR'd, not shaken. Combining Reservoir Sampling and Sliced Wasserstein Distance for Variance Reductioncited by 0
  10. 2026ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projectioncited by 0