# Convergence of k-means or EM on Mixture of Gaussians

There are many algorithms for learning mixture of Gaussians but typically k-means/EM is used in practice. My question is related to the performance of k-means/EM for MoG.

Recently, I came across this paper "Statistical Guarantees for EM algorithm" (https://arxiv.org/abs/1408.2156) which gives some guarantees for learning mixture of Gaussians using EM. The main idea as I understood is if initialized close to the optimal solution, EM converges to the optimal solution.

My question is what other theoretical guarantees are known about the performance of k-means/EM for learning mixtures of Gaussians. Particularly, using k-means, assume it is initialized with one random center from each cluster. Is it guaranteed to converge to optimal solution?

Any other reference is highly appreciated.