Reinterpreting EMML as Mirror Descent for Constrained Maximum Likelihood Estimation

TL;DR

This paper reinterprets EMML as a mirror descent method, enabling the incorporation of convex constraints and improving convergence in image reconstruction tasks under Poisson noise.

Contribution

It introduces a new perspective on EMML as mirror descent, allowing for constrained optimization with preserved efficiency and proven convergence.

Findings

Constrained EMML converges faster than classical EMML in hyperspectral unmixing.

The new approach effectively incorporates convex constraints into EMML.

The method maintains computational efficiency through Bregman projections.

Abstract

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.