# Quasi-Monte Carlo for 3D Sliced Wasserstein arXiv: https://arxiv.org/abs/2309.11713 ยท HTML: https://mlsanity.com/paper/2309.11713/ ## Key facts - Cited by 22 papers in the map (data through 14 Oct 2026). - Builds on 25 of its 55 references that are in the map. - First posted on arXiv 21 Sep 2023. ## Builds on (most cited first, 25 of 25) - [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016) - [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015) - [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018) - [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017) - [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019) - [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018) - [SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds](https://mlsanity.com/paper/2006.04604/) (2020) - [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020) - [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020) - [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017) - [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017) - [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017) - [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023) - [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021) - [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022) - [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021) - [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022) - [Sliced Wasserstein Estimation with Control Variates](https://mlsanity.com/paper/2305.00402/) (2023) - [Hilbert Curve Projection Distance for Distribution Comparison](https://mlsanity.com/paper/2205.15059/) (2022) - [Demystifying Orthogonal Monte Carlo and Beyond](https://mlsanity.com/paper/2005.13590/) (2020) - [Lidar Upsampling with Sliced Wasserstein Distance](https://mlsanity.com/paper/2301.13558/) (2023) - [Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction](https://mlsanity.com/paper/2301.04791/) (2023) - [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023) - [Convergence of SGD for Training Neural Networks with Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.11714/) (2023) - [An enumerative formula for the spherical cap discrepancy](https://mlsanity.com/paper/2012.10303/) (2020) ## Built on by (most cited first, 22 of 22) - [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025) - [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024) - [Lightspeed Geometric Dataset Distance via Sliced Optimal Transport](https://mlsanity.com/paper/2501.18901/) (2025) - [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024) - [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025) - [Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates](https://mlsanity.com/paper/2402.01493/) (2024) - [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024) - [Fast Summation of Radial Kernels via QMC Slicing](https://mlsanity.com/paper/2410.01316/) (2024) - [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025) - [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024) - [Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances](https://mlsanity.com/paper/2509.20508/) (2025) - [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025) - [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025) - [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025) - [Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices](https://mlsanity.com/paper/2411.10651/) (2024) - [Amortized Optimal Transport from Sliced Potentials](https://mlsanity.com/paper/2604.15114/) (2026) - [Unbiased Sliced Wasserstein Kernels for High-Quality Audio Captioning](https://mlsanity.com/paper/2502.05435/) (2025) - [Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance](https://mlsanity.com/paper/2509.10166/) (2025) - [Streaming Sliced Optimal Transport](https://mlsanity.com/paper/2505.06835/) (2025) - [Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimization](https://mlsanity.com/paper/2509.17405/) (2025) - [ReSWD: ReSTIR'd, not shaken. Combining Reservoir Sampling and Sliced Wasserstein Distance for Variance Reduction](https://mlsanity.com/paper/2510.01061/) (2025) - [ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projection](https://mlsanity.com/paper/2605.12879/) (2026) ## 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.