# A Novel Sliced Fused Gromov-Wasserstein Distance

arXiv: https://arxiv.org/abs/2508.02364 · HTML: https://mlsanity.com/paper/2508.02364/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 52 references that are in the map.
- First posted on arXiv 4 Aug 2025.

## Builds on (most cited first, 20 of 20)

- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [The Radon cumulative distribution transform and its application to image classification](https://mlsanity.com/paper/1511.03206/) (2015)
- [Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds](https://mlsanity.com/paper/2403.06560/) (2024)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces](https://mlsanity.com/paper/2002.01615/) (2020)
- [On a linear Gromov-Wasserstein distance](https://mlsanity.com/paper/2112.11964/) (2021)
- [Properties of Discrete Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.10352/) (2023)
- [Flowing Datasets with Wasserstein over Wasserstein Gradient Flows](https://mlsanity.com/paper/2506.07534/) (2025)
- [On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line](https://mlsanity.com/paper/2205.09006/) (2022)
- [Fast Summation of Radial Kernels via QMC Slicing](https://mlsanity.com/paper/2410.01316/) (2024)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Orthogonal Gromov-Wasserstein Discrepancy with Efficient Lower Bound](https://mlsanity.com/paper/2205.05838/) (2022)
- [Tangential Fixpoint Iterations for Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2403.08612/) (2024)

## Built on by (most cited first, 9 of 9)

- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [Binding Multiple Modalities via Multimodal Wasserstein Barycenter](https://mlsanity.com/paper/2609.33800/) (2026)
- [TRANSPORTER: Transferring Visual Semantics from VLM Manifolds](https://mlsanity.com/paper/2511.18359/) (2025)
- [Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition](https://mlsanity.com/paper/2512.08099/) (2025)
- [Convex Distance Operator Transport: A Convex and Geometry-Preserving Formulation](https://mlsanity.com/paper/2606.02047/) (2026)
- [k-Nearest Neighbors in Gromov--Wasserstein Space](https://mlsanity.com/paper/2606.10295/) (2026)
- [Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms](https://mlsanity.com/paper/2608.11016/) (2026)
- [Gromov-Monge Flow Matching for Equivariant Graph Generation](https://mlsanity.com/paper/2608.26961/) (2026)

## Abstract (arXiv)

The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.