# Unlocking Geodesic Gromov-Wasserstein Distances for 3D Modeling

arXiv: https://arxiv.org/abs/2609.32824 · HTML: https://mlsanity.com/paper/2609.32824/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [A Survey on Optimal Transport for Machine Learning: Theory and Applications](https://mlsanity.com/paper/2106.01963/) (2021)
- [Semi-relaxed Gromov-Wasserstein divergence with applications on graphs](https://mlsanity.com/paper/2110.02753/) (2021)
- [Quantized Gromov-Wasserstein](https://mlsanity.com/paper/2104.02013/) (2021)
- [Linear Time Sinkhorn Divergences using Positive Features](https://mlsanity.com/paper/2006.07057/) (2020)
- [Optimal Transport for Machine Learners](https://mlsanity.com/paper/2505.06589/) (2025)
- [Efficient Graph Field Integrators Meet Point Clouds](https://mlsanity.com/paper/2302.00942/) (2023)
- [A Survey on Algorithmic Developments in Optimal Transport Problem with Applications](https://mlsanity.com/paper/2501.06247/) (2025)
- [Near-Linear Time Generalized Sinkhorn Algorithms for Bounded Genus Graphs](https://mlsanity.com/paper/2605.09782/) (2026)
- [Sinkhorn Algorithm for Sequentially Composed Optimal Transports](https://mlsanity.com/paper/2412.03120/) (2024)
- [Manifold Random Features](https://mlsanity.com/paper/2602.03797/) (2026)

## Abstract (arXiv)

textitGromov-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 Efficient Geodesic Gromov-Wasserstein methods (EGGroW), a new class of efficient algorithms designed to calculate geodesic Gromov-Wasserstein distances with entropic Sinkhorn-like approaches, leveraging recently introduced textitGenusSink methods citepgenussink 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.