Statistica Sinica 5(1995), 41-54

CONVERGENCE IN NORM FOR ALTERNATING

EXPECTATION-MAXIMIZATION (EM) TYPE ALGORITHMS

Alfred O. Hero and Jeffrey A. Fessler

The University of Michigan

Abstract: We provide a sufficient condition for convergence of a general class of alternating estimation-maximization (EM) type continuous-parameter estimation algorithms with respect to a given norm. This class includes EM, penalized EM, Green's OSL-EM, and other approximate EM algorithms. The convergence analysis can be extended to include alternating coordinate-maximization EM algorithms such as Meng and Rubin's ECM and Fessler and Hero's SAGE. 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. For illustration, we apply our results to estimation of Poisson rate parameters in emission tomography and establish that in the final iterations the logarithm of the EM iterates converge monotonically in a weighted Euclidean norm.

Key words and phrases: Penalized and approximate EM, convergence rates, norm reducing property, applications to tomographic imaging.