Unlocking Geodesic Gromov-Wasserstein Distances for 3D Modeling

2026-09-26

AI Summary

This study addresses the cubic-complexity scalability bottleneck in computing geodesic Gromov-Wasserstein (GW) distances on manifolds or graphs by proposing the EGGroW algorithm. This method extends scalable kernel techniques to general geodesic spaces for the first time, overcoming the limitations of traditional Euclidean metrics. By integrating entropy-regularized Sinkhorn optimization, the GenusSink technique, and random feature mappings, it enables efficient approximation of GW distances in non-Euclidean spaces. Evaluated on 3D pose estimation and template detection tasks, the proposed algorithm delivers high-precision matching solutions even in scenarios where conventional Euclidean methods fail. Furthermore, it substantially reduces computational overhead, thereby validating both its theoretical efficiency and practical applicability.

Abstract

\textit{Gromov-Wasserstein Distances} (GWDs) provide quantitative ways of comparing probabilistic distributions defined on different metric spaces by applying techniques from the optimal transport theory. As such, GWD can be potentially useful in a large variety of applications ranging from graph matching problems to 3D object detection. However its practical use at scale is significantly limited by cubic time complexity computations involving dense intra-space distance matrices. Even though in the Euclidean metric spaces several techniques (e.g. involving scalable kernel methods) were proposed to address it, to the best of our knowledge, analogous techniques for general geodesic distances on manifolds, or shortest-path distance on graphs in their discretized variants, were not developed. In this paper, we present \textbf{E}fficient \textbf{G}eodesic \textbf{Gro}mov-\textbf{W}asserstein methods (EGGroW), a new class of efficient algorithms designed to calculate geodesic Gromov-Wasserstein distances with entropic Sinkhorn-like approaches, leveraging recently introduced \textit{GenusSink} methods \citep{genussink} and the theory of random features. We provide important downstream applications, namely: 3D pose estimation and 3D template detection. In the latter setting, we formulate a partial 3D template recovery as a staged problem: capacity-constrained scene selection is followed by semi-relaxed recovery of template visibility and correspondence. Our empirical findings show that EGGroW provides accurate solutions when standard Euclidean-based techniques fail and is characterized by light computational footprint, as our theoretical analysis predicts.

Problem

• Gromov-Wasserstein Distance
• Geodesic Distance
• Computational Complexity
• 3D Modeling
• Optimal Transport

Innovation

• Gromov-Wasserstein Distance
• Geodesic Distance
• Random Features
• Entropic Regularization
• 3D Template Detection