Gromov-Wasserstein at Scale, Beyond Squared Norms

2026-02-06

AI Summary

This work addresses the limitations of traditional optimal transport—its sensitivity to rotation—and the computational intractability and non-convexity of Gromov–Wasserstein (GW) methods. To overcome these challenges, the authors propose a novel framework that reformulates GW matching as an alignment task in an augmented feature space through a generalized distortion penalty. They introduce a differentiable iterative solver that achieves linear memory consumption and quadratic time complexity, enabling, for the first time, GW computations on point clouds with hundreds of thousands of points within minutes. Theoretical analysis further uncovers an intrinsic connection between local minima in the energy landscape and the symmetry of correspondences, substantially enhancing the scalability and practical utility of GW distances in geometric matching tasks.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point displacements under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing distortions via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.

Problem

• Gromov-Wasserstein
• optimal transport
• point set matching
• non-convex optimization
• rotation invariance

Innovation

• Gromov-Wasserstein
• scalable optimization
• lifted feature space
• quadratic time complexity
• differentiable solver