# Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance

arXiv: https://arxiv.org/abs/2509.10166 · HTML: https://mlsanity.com/paper/2509.10166/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 67 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Monte Carlo guided Diffusion for Bayesian linear inverse problems](https://mlsanity.com/paper/2308.07983/) (2023)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019)
- [Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds](https://mlsanity.com/paper/2403.06560/) (2024)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Monte Carlo with Determinantal Point Processes](https://mlsanity.com/paper/1605.00361/) (2016)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Kernel interpolation with continuous volume sampling](https://mlsanity.com/paper/2002.09677/) (2020)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Differentially Private Sliced Wasserstein Distance](https://mlsanity.com/paper/2107.01848/) (2021)
- [Diffusion posterior sampling for simulation-based inference in tall data settings](https://mlsanity.com/paper/2404.07593/) (2024)
- [Sliced Wasserstein Estimation with Control Variates](https://mlsanity.com/paper/2305.00402/) (2023)
- [Demystifying Orthogonal Monte Carlo and Beyond](https://mlsanity.com/paper/2005.13590/) (2020)
- [Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates](https://mlsanity.com/paper/2402.01493/) (2024)
- [A User's Guide to Sampling Strategies for Sliced Optimal Transport](https://mlsanity.com/paper/2502.02275/) (2025)
- [A Faster Sampler for Discrete Determinantal Point Processes](https://mlsanity.com/paper/2210.17358/) (2022)
- [Repelled point processes with application to numerical integration](https://mlsanity.com/paper/2308.04825/) (2023)

## Built on by (most cited first, 1 of 1)

- [Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimization](https://mlsanity.com/paper/2509.17405/) (2025)

## Abstract (arXiv)

In this paper, we consider the problem of computing the integral of a function on the unit sphere, in any dimension, using Monte Carlo methods. Although the methods we present are general, our guiding thread is the sliced Wasserstein distance between two measures on ℝd, which is precisely an integral on the d-dimensional sphere. The sliced Wasserstein distance (SW) has gained momentum in machine learning either as a proxy to the less computationally tractable Wasserstein distance, or as a distance in its own right, due in particular to its built-in alleviation of the curse of dimensionality. There has been recent numerical benchmarks of quadratures for the sliced Wasserstein, and our viewpoint differs in that we concentrate on quadratures where the nodes are repulsive, i.e. negatively dependent. Indeed, negative dependence can bring variance reduction when the quadrature is adapted to the integration task. Our first contribution is to extract and motivate quadratures from the recent literature on determinantal point processes (DPPs) and repelled point processes, as well as repulsive quadratures from the literature specific to the sliced Wasserstein distance. We then numerically benchmark these quadratures. Moreover, we analyze the variance of the UnifOrtho estimator, an orthogonal Monte Carlo estimator. Our analysis sheds light on UnifOrtho's success for the estimation of the sliced Wasserstein in large dimensions, as well as counterexamples from the literature. Our final recommendation for the computation of the sliced Wasserstein distance is to use randomized quasi-Monte Carlo in low dimensions and UnifOrtho in large dimensions. DPP-based quadratures only shine when quasi-Monte Carlo also does, while repelled quadratures show moderate variance reduction in general, but more theoretical effort is needed to make them robust.