Quasi-Monte Carlo for 3D Sliced Wasserstein

TL;DR

This paper introduces Quasi-Monte Carlo methods for approximating the 3D Sliced Wasserstein distance, improving accuracy over traditional Monte Carlo approaches and extending to unbiased stochastic optimization with theoretical guarantees.

Contribution

It proposes Quasi-Sliced Wasserstein (QSW) and Randomized QSW (RQSW) methods for 3D Wasserstein distance approximation, with convergence proofs and practical evaluations.

Findings

QSW outperforms Monte Carlo in approximation accuracy.

RQSW provides an unbiased estimator suitable for stochastic optimization.

Experimental results show improved performance in 3D point-cloud tasks.

Abstract

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.