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

International audienceThe 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

Discriminative variable selection for clustering with the sparse Fisher-EM algorithm

The interest in variable selection for clustering has increased recently due to the growing need in clustering high-dimensional data. Variable selection allows in particular to ease both the clustering and the interpretation of the results. Existing approaches have demonstrated the efficiency of variable selection for clustering but turn out to be either very time consuming or not sparse enough in high-dimensional spaces. This work proposes to perform a selection of the discriminative variables by introducing sparsity in the loading matrix of the Fisher-EM algorithm. This clustering method has been recently proposed for the simultaneous visualization and clustering of high-dimensional data. It is based on a latent mixture model which fits the data into a low-dimensional discriminative subspace. Three different approaches are proposed in this work to introduce sparsity in the orientation matrix of the discriminative subspace through $\ell_{1}$-type penalizations. Experimental comparisons with existing approaches on simulated and real-world data sets demonstrate the interest of the proposed methodology. An application to the segmentation of hyperspectral images of the planet Mars is also presented

Discriminative variable selection for clustering with the sparse Fisher-EM algorithm

International audienceThe interest in variable selection for clustering has increased recently due to the growing need in clustering high-dimensional data. Variable selection allows in particular to ease both the clustering and the interpretation of the results. Existing approaches have demonstrated the efficiency of variable selection for clustering but turn out to be either very time consuming or not sparse enough in high-dimensional spaces. This work proposes to perform a selection of the discriminative variables by introducing sparsity in the loading matrix of the Fisher-EM algorithm. This clustering method has been recently proposed for the simultaneous visualization and clustering of high-dimensional data. It is based on a latent mixture model which fits the data into a low-dimensional discriminative subspace. Three different approaches are proposed in this work to introduce sparsity in the orientation matrix of the discriminative subspace through \ell_{1} -type penalizations. Experimental comparisons with existing approaches on simulated and real-world data sets demonstrate the interest of the proposed methodology. An application to the segmentation of hyperspectral images of the planet Mars is also presented