Fast Approximation of the Generalized Sliced-Wasserstein Distance

TL;DR

This paper introduces fast, deterministic approximation methods for the generalized sliced Wasserstein distance, leveraging the concentration of random projections and Gaussian approximations to improve efficiency in high-dimensional settings.

Contribution

It proposes novel deterministic approximation techniques for the generalized sliced Wasserstein distance using polynomial, circular, and neural network functions.

Findings

Deterministic approximations are significantly faster than Monte Carlo methods.

Approximations work well in high-dimensional scenarios.

The approach leverages Gaussian behavior of projected high-dimensional vectors.

Abstract

Generalized sliced Wasserstein distance is a variant of sliced Wasserstein distance that exploits the power of non-linear projection through a given defining function to better capture the complex structures of the probability distributions. Similar to sliced Wasserstein distance, generalized sliced Wasserstein is defined as an expectation over random projections which can be approximated by the Monte Carlo method. However, the complexity of that approximation can be expensive in high-dimensional settings. To that end, we propose to form deterministic and fast approximations of the generalized sliced Wasserstein distance by using the concentration of random projections when the defining functions are polynomial function, circular function, and neural network type function. Our approximations hinge upon an important result that one-dimensional projections of a high-dimensional random vector are approximately Gaussian.