Information distance estimation between mixtures of multivariate Gaussians

Abstract

There are e cient software programs for extracting from large data sets and imagesequences certain mixtures of probability distributions, such as multivariate Gaussians, to representthe important features and their mutual correlations needed for accurate document retrieval fromdatabases. This note describes a method to use information geometric methods for distance measuresbetween distributions in mixtures of arbitrary multivariate Gaussians. There is no general analyticsolution for the information geodesic distance between two k-variate Gaussians, but for many purposesthe absolute information distance may not be essential and comparative values su ce for proximitytesting and document retrieval. Also, for two mixtures of di erent multivariate Gaussians we mustresort to approximations to incorporate the weightings. In practice, the relation between a reasonableapproximation and a true geodesic distance is likely to be monotonic, which is adequate for manyapplications. Here we consider some choices for the incorporation of weightings in distance estimationand provide illustrative results from simulations of di erently weighted mixtures of multivariateGaussians.

Keywords

• information geometry| multivariate spatial covariance| Gaussian mixtures| geodesicdistance| approximations