The Fisher-EM algorithm has been recently proposed in (Bouveyron2011) for the simultaneous visualization and clustering of high-dimensional data. It is based on a latent mixture model which fits the data into a latent discriminative subspace with a low intrinsic dimension. Although the Fisher-EM algorithm is based on the EM algorithm, it does not respect at a first glance all conditions of the EM convergence theory. Its convergence toward a maximum of the likelihood is therefore questionable. The aim of this work is two folds. Firstly, the convergence of the Fisher-EM algorithm is studied from the theoretical point of view. It is in particular proved that the algorithm converges under weak conditions in the general case. Secondly, the convergence of the Fisher-EM algorithm is considered from the practical point of view. It is shown that the Fisher's criterion can be used as stopping criterion for the algorithm to improve the clustering accuracy. It is also shown that the Fisher-EM algorithm converges faster than both the EM and CEM algorithm.