Quantized Gromov-Wasserstein

TL;DR

This paper introduces Quantized Gromov-Wasserstein (qGW), a new metric and algorithmic approach that improves scalability and efficiency for comparing large-scale probability distributions across different metric spaces.

Contribution

The paper proposes qGW, a novel quantization-based framework that provides theoretical bounds and enables scalable GW matching on datasets with over 1 million points.

Findings

qGW offers significant speedups and memory reductions.

It enables GW matching on datasets an order of magnitude larger.

The method outperforms existing algorithms in large-scale scenarios.

Abstract

The Gromov-Wasserstein (GW) framework adapts ideas from optimal transport to allow for the comparison of probability distributions defined on different metric spaces. Scalable computation of GW distances and associated matchings on graphs and point clouds have recently been made possible by state-of-the-art algorithms such as S-GWL and MREC. Each of these algorithmic breakthroughs relies on decomposing the underlying spaces into parts and performing matchings on these parts, adding recursion as needed. While very successful in practice, theoretical guarantees on such methods are limited. Inspired by recent advances in the theory of quantization for metric measure spaces, we define Quantized Gromov Wasserstein (qGW): a metric that treats parts as fundamental objects and fits into a hierarchy of theoretical upper bounds for the GW problem. This formulation motivates a new algorithm for approximating optimal GW matchings which yields algorithmic speedups and reductions in memory complexity. Consequently, we are able to go beyond outperforming state-of-the-art and apply GW matching at scales that are an order of magnitude larger than in the existing literature, including datasets containing over 1M points.