Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance

TL;DR

This paper extracts and motivate quadratures from the recent literature on determinantal point processes and repelled point processes, as well as repulsive quadratures from the literature specific to the sliced Wasserstein distance, and analyzes the variance of the UnifOrtho estimator, an orthogonal Monte Carlo estimator.

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 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.

Figures (12)

• Figure 1: Realizations of three point processes
• Figure 2: Various point processes over the sphere
• Figure 3: Results for the Gaussian toy example, across $d=2, 10, 20$. The actual value of the $2$-sliced Wasserstein distance is estimated using Monte Carlo integration with $10^6$ projections.
• Figure 4: Two-dimensional projections of the various point clouds used in Section \ref{['s:point_clouds']}.
• Figure 5: Boxplots of the errors for three-dimensional point clouds. The boxplots are centered around a reference value of the sliced Wasserstein estimated using QMC with $10^5$ points.

Theorems & Definitions (5)

• Definition 1: Projection DPP
• Definition 2
• Definition 3: Spherical ensemble, Theorem 3 in krishnapur_random_2009
• Proposition 4
• Definition 5