# A note on the relations between mixture models, maximum-likelihood and entropic optimal transport

arXiv: https://arxiv.org/abs/2501.12005 · HTML: https://mlsanity.com/paper/2501.12005/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 5 references that are in the map.
- First posted on arXiv 21 Jan 2025.

## Builds on (most cited first, 3 of 3)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal spectral transportation with application to music transcription](https://mlsanity.com/paper/1609.09799/) (2016)
- [A unified framework for hard and soft clustering with regularized optimal transport](https://mlsanity.com/paper/1711.04366/) (2017)

## Built on by (most cited first, 3 of 3)

- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)
- [Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models](https://mlsanity.com/paper/2605.28488/) (2026)
- [Guidance Without Labels: Flow Matching from Coupled Gaussian Mixtures](https://mlsanity.com/paper/2610.16654/) (2026)

## Abstract (arXiv)

This note aims to demonstrate that performing maximum-likelihood estimation for a mixture model is equivalent to minimizing over the parameters an optimal transport problem with entropic regularization. The objective is pedagogical: we seek to present this already known result in a concise and hopefully simple manner. We give an illustration with Gaussian mixture models by showing that the standard EM algorithm is a specific block-coordinate descent on an optimal transport loss.