  In this paper, we formulate the problem of inferring a Finite Mixture Model
from discrete data as an optimal transport problem with entropic regularization
of parameter $\lambda\geq 0$. Our method unifies hard and soft clustering, the
Expectation-Maximization (EM) algorithm being exactly recovered for
$\lambda=1$. The family of clustering algorithm we propose rely on the
resolution of nonconvex problems using alternating minimization. We study the
convergence property of our generalized $\lambda-$EM algorithms and show that
each step in the minimization process has a closed form solution when inferring
finite mixture models of exponential families. Experiments highlight the
benefits of taking a parameter $\lambda>1$ to improve the inference performance
and $\lambda\to 0$ for classification.
