Sliced Gromov-Wasserstein

TL;DR

This paper introduces Sliced Gromov-Wasserstein, a new scalable divergence for comparing distributions in different metric spaces, which is faster to compute than traditional GW and effective in large-scale applications.

Contribution

It derives a closed-form solution for GW in 1D, proposes a slicing-based approximation called SGW, and demonstrates its efficiency and effectiveness in large-scale distribution comparison.

Findings

SGW is $O(n\log(n))$ to compute, significantly faster than GW.

SGW effectively compares distributions in various experiments.

SGW maintains similar performance to GW in practical tasks.

Abstract

Recently used in various machine learning contexts, the Gromov-Wasserstein distance (GW) allows for comparing distributions whose supports do not necessarily lie in the same metric space. However, this Optimal Transport (OT) distance requires solving a complex non convex quadratic program which is most of the time very costly both in time and memory. Contrary to GW, the Wasserstein distance (W) enjoys several properties (e.g. duality) that permit large scale optimization. Among those, the solution of W on the real line, that only requires sorting discrete samples in 1D, allows defining the Sliced Wasserstein (SW) distance. This paper proposes a new divergence based on GW akin to SW. We first derive a closed form for GW when dealing with 1D distributions, based on a new result for the related quadratic assignment problem. We then define a novel OT discrepancy that can deal with large scale distributions via a slicing approach and we show how it relates to the GW distance while being $O(n\log(n))$ to compute. We illustrate the behavior of this so called Sliced Gromov-Wasserstein (SGW) discrepancy in experiments where we demonstrate its ability to tackle similar problems as GW while being several order of magnitudes faster to compute.