On the estimation of the latent discriminative subspace in the Fisher-EM algorithm

The Fisher-EM algorithm has been recently proposed in [2] for the simultaneous visualization and clustering of high-dimensional data. It is based on a discriminative latent mixture model which fits the data into a latent discriminative subspace with an intrinsic dimension lower than the dimension of the original space. The Fisher-EM algorithm includes an F-step which estimates the projection matrix whose columns span the discriminative latent space. This matrix is estimated via an optimization problem which is solved using a Gram-Schmidt procedure in the original algorithm. Unfortunately, this procedure suffers in some case from numerical instabilities which may result in a deterioration of the visualization quality or the clustering accuracy. Two alternatives for estimating the latent subspace are proposed to overcome this limitation. The optimization problem of the F-step is first recasted as a regression-type problem and then reformulated such that the solution can be approximated with a SVD. Experiments on simulated and real datasets show the improvement of the proposed alternatives for both the visualization and the clustering of data.