The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis

The expectation-maximization (EM) algorithm is a convenient tool for approximating maximum likelihood estimators in situations when available data are incomplete, as is the case for many inverse problems. Our focus here is on the continuous version of the EM algorithm for a Poisson model, which is known to perform unstably when applied to ill-posed integral equations. We interpret and analyse the EM algorithm as a regularization procedure. We show weak convergence of the iterates to a solution of the equation when exact data are considered. In the case of perturbed data, similar results are established by employing a stopping rule of discrepancy type under boundedness assumptions on the problem data.