Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance

2025-09-12

AI Summary

This work addresses efficient Monte Carlo estimation of integrals of functions over the high-dimensional unit sphere, with a focus on the sliced Wasserstein distance (SWD) as a canonical spherical integral problem. To reduce estimator variance, we propose a novel randomized quadrature method based on negatively correlated sampling: we construct spherical point sets exhibiting geometric repulsion using determinantal point processes (DPPs) and other repulsive point processes, and systematically analyze the asymptotic superiority of the UnifOrtho estimator in high dimensions. Experiments show that randomized quasi-Monte Carlo is optimal in low dimensions, whereas UnifOrtho significantly outperforms standard baselines in high dimensions; repulsive sampling consistently reduces variance, though its theoretical guarantees and robustness require further investigation. Our work establishes a new paradigm for high-dimensional spherical integration—grounded in rigorous analysis yet practically effective—offering both theoretical insight and empirical utility.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

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 $mathbb{R}^d$, 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 emph{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.

Problem

• Computing integrals on the unit sphere using Monte Carlo methods
• Evaluating sliced Wasserstein distance through repulsive quadrature techniques
• Analyzing variance reduction in orthogonal Monte Carlo estimators

Innovation

• Repulsive Monte Carlo for variance reduction
• Determinantal point processes for quadrature
• UnifOrtho estimator for large dimensions