# A Riemannian Newton Trust-Region Method for Fitting Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2104.14957 · HTML: https://mlsanity.com/paper/2104.14957/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 51 references that are in the map.
- First posted on arXiv 30 Apr 2021.

## Builds on (most cited first, 5 of 5)

- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization](https://mlsanity.com/paper/1706.03267/) (2017)
- [A Riemannian trust-region method for low-rank tensor completion](https://mlsanity.com/paper/1703.10019/) (2017)
- [MixEst: An Estimation Toolbox for Mixture Models](https://mlsanity.com/paper/1507.06065/) (2015)
- [Detecting anomalies in fibre systems using 3-dimensional image data](https://mlsanity.com/paper/1907.06988/) (2019)

## Built on by (most cited first, 2 of 2)

- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)
- [Riemannian Optimization for Variance Estimation in Linear Mixed Models](https://mlsanity.com/paper/2212.09081/) (2022)

## Abstract (arXiv)

Gaussian Mixture Models are a powerful tool in Data Science and Statistics that are mainly used for clustering and density approximation. The task of estimating the model parameters is in practice often solved by the Expectation Maximization (EM) algorithm which has its benefits in its simplicity and low per-iteration costs. However, the EM converges slowly if there is a large share of hidden information or overlapping clusters. Recent advances in manifold optimization for Gaussian Mixture Models have gained increasing interest. We introduce an explicit formula for the Riemannian Hessian for Gaussian Mixture Models. On top, we propose a new Riemannian Newton Trust-Region method which outperforms current approaches both in terms of runtime and number of iterations. We apply our method on clustering problems and density approximation tasks. Our method is very powerful for data with a large share of hidden information compared to existing methods.