Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces

TL;DR

This paper introduces a scalable framework for solving the Gromov-Wasserstein problem between point clouds in low-dimensional Euclidean spaces, enabling efficient global solutions for large-scale shape comparison tasks.

Contribution

It reformulates the Gromov-Wasserstein problem as a low-rank concave quadratic optimization, improving scalability and solution quality over existing methods.

Findings

The method scales well with thousands of points.

It outperforms state-of-the-art approaches on synthetic datasets.

Successfully applied to a computational biology problem.

Abstract

This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm. The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning. The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points. We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.