# A Wasserstein-type distance in the space of Gaussian Mixture Models

arXiv: https://arxiv.org/abs/1907.05254 · HTML: https://mlsanity.com/paper/1907.05254/

## Key facts

- Cited by 75 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2019.

## Built on by (most cited first, 30 of 75)

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- [Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors](https://mlsanity.com/paper/2405.14250/) (2024)
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- [RegGS: Unposed Sparse Views Gaussian Splatting with 3DGS Registration](https://mlsanity.com/paper/2507.08136/) (2025)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Evaluating generative networks using Gaussian mixtures of image features](https://mlsanity.com/paper/2110.05240/) (2021)
- [A Bag-of-Prototypes Representation for Dataset-Level Applications](https://mlsanity.com/paper/2303.13251/) (2023)
- [Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians](https://mlsanity.com/paper/2406.04012/) (2024)
- [D2GS: Depth-and-Density Guided Gaussian Splatting for Stable and Accurate Sparse-View Reconstruction](https://mlsanity.com/paper/2510.08566/) (2025)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS](https://mlsanity.com/paper/2506.09534/) (2025)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies](https://mlsanity.com/paper/2305.10411/) (2023)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [Concept Heterogeneity-aware Representation Steering](https://mlsanity.com/paper/2603.02237/) (2026)
- [Lighter, Better, Faster Multi-Source Domain Adaptation with Gaussian Mixture Models and Optimal Transport](https://mlsanity.com/paper/2404.10261/) (2024)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)

## Abstract (arXiv)

In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We derive a very simple discrete formulation for this distance, which makes it suitable for high dimensional problems. We also study the corresponding multi-marginal and barycenter formulations. We show some properties of this Wasserstein-type distance, and we illustrate its practical use with some examples in image processing.