  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.

Machine Learning;Machine Learning;Optimization and Control;Computation