A Unified Framework for Hard and Soft Clustering with Regularized Optimal Transport

We formulate the inference of 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 relies on the resolution of nonconvex problems using alternating minimization. We study the convergence property of our generalized $\lambda-\text{EM}$ algorithms and show that each algorithmic step 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 $\lambda\rightarrow 0$ for classification.