# Scaling Limits of the Wasserstein information matrix on Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2309.12997 · HTML: https://mlsanity.com/paper/2309.12997/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 22 Sep 2023.

## Builds on (most cited first, 8 of 8)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019)
- [When Optimal Transport Meets Information Geometry](https://mlsanity.com/paper/2206.14791/) (2022)
- [Transport information Bregman divergences](https://mlsanity.com/paper/2101.01162/) (2021)
- [Scaling limits of discrete optimal transport](https://mlsanity.com/paper/1809.01092/) (2018)
- [On a prior based on the Wasserstein information matrix](https://mlsanity.com/paper/2202.03217/) (2022)
- [Master equations for finite state mean field games with nonlinear activations](https://mlsanity.com/paper/2212.05675/) (2022)

## Abstract (arXiv)

We consider the Wasserstein metric on the Gaussian mixture models (GMMs), which is defined as the pullback of the full Wasserstein metric on the space of smooth probability distributions with finite second moment. It derives a class of Wasserstein metrics on probability simplices over one-dimensional bounded homogeneous lattices via a scaling limit of the Wasserstein metric on GMMs. Specifically, for a sequence of GMMs whose variances tend to zero, we prove that the limit of the Wasserstein metric exists after certain renormalization. Generalizations of this metric in general GMMs are established, including inhomogeneous lattice models whose lattice gaps are not the same, extended GMMs whose mean parameters of Gaussian components can also change, and the second-order metric containing high-order information of the scaling limit. We further study the Wasserstein gradient flows on GMMs for three typical functionals: potential, internal, and interaction energies. Numerical examples demonstrate the effectiveness of the proposed GMM models for approximating Wasserstein gradient flows.