# Entropic Partial Optimal Transport and Partial Gromov--Wasserstein Distance between Gaussian Mixtures

arXiv: https://arxiv.org/abs/2608.09265 · HTML: https://mlsanity.com/paper/2608.09265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 10 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Partial Gromov-Wasserstein Metric](https://mlsanity.com/paper/2402.03664/) (2024)
- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)
- [A Wasserstein-type metric for generic mixture models, including location-scatter and group invariant measures](https://mlsanity.com/paper/2301.07963/) (2023)
- [A Wasserstein-type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis](https://mlsanity.com/paper/2311.16988/) (2023)

## Abstract (arXiv)

Optimal transport and Gromov--Wasserstein distances are useful tools for comparing probability measures and metric measure spaces, but their balanced formulations force all mass to be matched. This constraint is often too strong for data with outliers, missing parts, or only partial overlap. In this paper, we develop entropic partial optimal transport for Gaussian mixture models and define a partial mixture Gromov--Wasserstein distance. For the finite entropic partial optimal transport problem, we prove the existence and uniqueness of the minimizer and establish quantitative large-penalty estimates. Moreover, the resulting entropic partial component couplings induce continuous partial transport plans through Gaussian optimal maps. We analyze their large-penalty and subsequent zero-entropy limits and construct the associated displacement interpolations and barycentric projection maps. In addition, by identifying each Gaussian mixture with a finite metric measure space of Gaussian components, we establish the metric property and large-penalty limit of the partial mixture Gromov--Wasserstein distance. Finally, numerical experiments on synthetic Gaussian mixtures and point clouds illustrate the effects of the penalty and entropic regularization and the robustness of partial matching to outliers.