# Optimal Transport for Kernel Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2310.18586 · HTML: https://mlsanity.com/paper/2310.18586/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 28 Oct 2023.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey](https://mlsanity.com/paper/2106.08443/) (2021)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [A Geometric View of Optimal Transportation and Generative Model](https://mlsanity.com/paper/1710.05488/) (2017)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [A Two-round Variant of EM for Gaussian Mixtures](https://mlsanity.com/paper/1301.3850/) (2013)
- [Maximum Mean Discrepancy for Generalization in the Presence of Distribution and Missingness Shift](https://mlsanity.com/paper/2111.10344/) (2021)
- [Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes](https://mlsanity.com/paper/2011.07489/) (2020)
- [Gromov-Wasserstein Discrepancy with Local Differential Privacy for Distributed Structural Graphs](https://mlsanity.com/paper/2202.00808/) (2022)
- [Kernel Wasserstein Distance](https://mlsanity.com/paper/1905.09314/) (2019)
- [Gain with no Pain: Efficient Kernel-PCA by Nystr"om Sampling](https://mlsanity.com/paper/1907.05226/) (2019)
- [Orthogonal Gromov-Wasserstein Discrepancy with Efficient Lower Bound](https://mlsanity.com/paper/2205.05838/) (2022)

## Abstract (arXiv)

The Wasserstein distance from optimal mass transport (OMT) is a powerful mathematical tool with numerous applications that provides a natural measure of the distance between two probability distributions. Several methods to incorporate OMT into widely used probabilistic models, such as Gaussian or Gaussian mixture, have been developed to enhance the capability of modeling complex multimodal densities of real datasets. However, very few studies have explored the OMT problems in a reproducing kernel Hilbert space (RKHS), wherein the kernel trick is utilized to avoid the need to explicitly map input data into a high-dimensional feature space. In the current study, we propose a Wasserstein-type metric to compute the distance between two Gaussian mixtures in a RKHS via the kernel trick, i.e., kernel Gaussian mixture models.