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.

高斯混合模型是数据科学和统计学中的强大工具，主要用于聚类和密度近似。估计模型参数的任务在实践中通常由期望最大化 (EM) 算法解决，该算法的优点在于其简单性和低每次迭代成本。但是，如果存在大量隐藏信息或重叠集群，则 EM 收敛缓慢。高斯混合模型的流形优化的最新进展引起了越来越多的兴趣。我们为高斯混合模型引入了黎曼 Hessian 的显式公式。最重要的是，我们提出了一种新的 Riemannian Newton Trust-Region 方法，它在运行时间和迭代次数方面都优于当前的方法。我们将我们的方法应用于聚类问题和密度近似任务。