Reinterpreting EMML as Mirror Descent for Constrained Maximum Likelihood Estimation

The Expectation--Maximization Maximum Likelihood (EMML) algorithm belongs to the Expectation--Maximization family and is widely used for image reconstruction problems under Poisson noise. In this paper, we reinterpret EMML as a mirror descent method applied to a reparametrized objective function. This perspective allows us to incorporate convex constraints into the algorithm through appropriately chosen Bregman projections, while preserving the multiplicative structure of the EMML updates to ensure computational efficiency. We then establish the convergence of the resulting algorithm toward a solution of the constrained maximum-likelihood problem. Numerical experiments on hyperspectral unmixing problems demonstrate that the constrained EMML converges in fewer iterations than the classical EMML.