Kullback-Leibler Approach to Gaussian Mixture Reduction

A common problem in multi-target tracking is to approximate a Gaussian mixture by one containing fewer components; similar problems can arise in integrated navigation. A common approach is successively to merge pairs of components, replacing the pair with a single Gaussian component whose moments up to second order match those of the merged pair. Salmond [1] and Williams [2, 3] have each proposed algorithms along these lines, but using different criteria for selecting the pair to be merged at each stage. The paper shows how under certain circumstances each of these pair-selection criteria can give rise to anomalous behaviour, and proposes that a key consideration should the the Kullback-Leibler (KL) discrimination of the reduced mixture with respect to the original mixture. Although computing this directly would normally be impractical, the paper shows how an easily computed upper bound can be used as a pair-selection criterion which avoids the anomalies of the earlier approaches. The behaviour of the three algorithms is compared using a high-dimensional example drawn from terrain-referenced navigation.