A new family of 12 probabilistic models, introduced recently, aims to simultaneously cluster and visualize high-dimensional data. It is based on a mixture model which fits the data into a latent discriminative subspace with an intrinsic dimension bounded by the number of clusters. An estimation procedure, named the Fisher-EM algorithm has also been proposed and turns out to outperform other subspace clustering in most situations. Moreover the convergence properties of the Fisher-EM algorithm are discussed; in particular it is proved that the algorithm is a GEM algorithm and converges under weak conditions in the general case. Finally, a sparse extension of the Fisher-EM algorithm is proposed in order to perform a selection of the original variables which are discriminative.