Consistency issues in Gaussian Mixture Models reduction algorithms

In many contexts Gaussian Mixtures (GM) are used to approximate probability distributions, possibly time-varying. In some applications the number of GM components exponentially increases over time, and reduction procedures are required to keep them reasonably limited. The GM reduction (GMR) problem can be formulated by choosing different measures of the dissimilarity of GMs before and after reduction, like the Kullback-Leibler Divergence (KLD) and the Integral Squared Error (ISE). Since in no case the solution is obtained in closed form, many approximate GMR algorithms have been proposed in the past three decades, although none of them provides optimality guarantees. In this work we discuss the importance of the choice of the dissimilarity measure and the issue of consistency of all steps of a reduction algorithm with the chosen measure. Indeed, most of the existing GMR algorithms are composed by several steps which are not consistent with a unique measure, and for this reason may produce reduced GMs far from optimality. In particular, the use of the KLD, of the ISE and normalized ISE is discussed and compared in this perspective.