Theoretical and practical considerations on the convergence properties of the Fisher-EM algorithm

Authors:
Charles Bouveyron;Camille Brunet
Affiliations:
Laboratoire SAMM, EA 4543, Université Paris 1 Panthéon-Sorbonne, 90 rue de Tolbiac, 75013 Paris, France;Equipe Modal'X, EA 3454, Université Paris Ouest Nanterre La Defense, 200 av. de la République, 92000 Nanterre, France
Venue:
Journal of Multivariate Analysis
Year:
2012

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Abstract

The Fisher-EM algorithm has been recently proposed in Bouveyron and Brunet (2012) [5] 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 twofold. First, the convergence of the Fisher-EM algorithm is studied from the theoretical point of view. In particular, it is proved that the algorithm converges under weak conditions in the general case. Second, the convergence of the Fisher-EM algorithm is considered from the practical point of view. It is shown that the Fisher criterion can be used as a 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.