Asymptotic Convergence Properties of EM Type Algorithms

We analyze the asymptotic convergence properties of a general class of EM type algorithms for es timating an unknown parameter via alternating estimation and maximization As examples this class includes ML EM penalized ML EM Green s OSL EM and many other approximate EM al gorithms A theorem is given which provides conditions for monotone convergence with respect to a given norm and speci es an asymptotic rate of convergence for an algorithm in this class By investigating di erent parameterizations the condition for monotone convergence can be used to establish norms under which the distance between successive iterates and the limit point of the EM type algorithm approaches zero monotonically We apply these results to a modi ed ML EM algorithm with stochastic complete incomplete data mapping and establish global monotone conver gence for a linear Gaussian observation model We then establish that in the nal iterations the unpenalized and quadratically penalized ML EM algorithms for PET image reconstruction converge monotonically relative to two di erent norms on the logarithm of the images