Constrained EM trajectories for mixtures of normal distributions

Constrained formulations of the multivariate normal mixture model have been proposed in order to remove singularities and reduce the number of spurious maxima of the likelihood function. However such approaches require some a priori information about the eigenvalues of the covariance matrices that is not always available. In this paper we investigate the behaviour of the EM algorithm near a degenerated solution. The obtained theoretical results would suggest to estimate the unconstrained model by means of a constrained EM algorithm, where the dissimilarity between two consecutive updates of the eigenvalues of each covariance matrix is bounded above. The performances of such “dynamic” constraints are evaluated on the grounds of numerical experiments.