On the stabilization inherent in the EMS algorithm

Abstract

In an attempt to overcome the EM (estimate, maximize) algorithm's lack of inherent stabilization for the solution of underdetermined linear equations, a multiplicative smoothing step was added as an essential component in the iteration, thus creating the EMS (estimate, maximize, smooth) algorithm. This paper gives an analysis of the stabilization produced through the introduction of this smoothing step. The nonlinear EMS equation are given a data smoothing only interpretation in the sense that their fixed points are solutions of a nonlinear system which takes the form of the original linear equations with only the right-hand side replaced by a nonlinear function of the fixed points. It is found that the stabilization inherent in this reformulation depends crucially on the structure of the smoothing introduced into EMS. Two types of nonuniqueness for EMS (namely the persistence of spurious solutions, associated with the underdeterminedness of the original linear system, and the artifactual presence of extraneous EM/EMS solutions, induced by the smoothing's structure) are shown to be eliminated by the use of irreducible smoothing, and thus the fate of the inherent stabilization lies with the structure of the smoothing. An exact nonnegative solution of the underlying linear system (in the case that one exists) can be obtained from the EMS algorithm if this solution is an eigenvector of the introduced smoothing matrix corresponding to the eigenvalue one. Hence, to choose the smoothing appropriately requires a priori knowledge of the desired solution.