Convergence analysis for a multiplicatively relaxed EM algorithm

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Abstract
The expectation maximization (EM) algorithm is an iterative procedure used to determine maximum likelihood estimators in situations of incomplete data. In the case of independent Poisson variables it converges to a solution of a problem of the form min ∑[〈ai,x〉 −bilog 〈ai,x〉] such thatx⩾0. Thus, it can be used to solve systems of the formAx=b,x⩾0 (withAstochastic andbpositive.) It converges to a feasible solution if it exists and to an approximate one otherwise (the one that minimizesd(b,Ax), wheredis the Kullback–Leibler information divergence). We study the convergence of the multiplicatively relaxed version proposed by Tanaka for use in positron emission tomography. We prove global convergence in the underrelaxed and unrelaxed cases. In the overrelaxed case we present a local convergence theorem together with two partial global results: the sequence generated by the algorithm is bounded and, if it converges, its limit point is a solution of the aforementioned problem.

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