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The Jeffreys divergence is a renown symmetrization of the orientedKullback-Leibler divergence broadly used in information sciences. Since theJeffreys divergence between Gaussian mixture models is not available inclosed-form, various techniques with pros and cons have been proposed in theliterature to either estimate, approximate, or lower and upper bound thisdivergence. In this paper, we propose a simple yet fast heuristic toapproximate the Jeffreys divergence between two univariate Gaussian mixtureswith arbitrary number of components. Our heuristic relies on converting themixtures into pairs of dually parameterized probability densities belonging toan exponential family. In particular, we consider the versatile polynomialexponential family densities, and design a divergence to measure in closed-formthe goodness of fit between a Gaussian mixture and its polynomial exponentialdensity approximation. This goodness-of-fit divergence is a generalization ofthe Hyv\"arinen divergence used to estimate models with computationallyintractable normalizers. It allows us to perform model selection by choosingthe orders of the polynomial exponential densities used to approximate themixtures. We demonstrate experimentally that our heuristic to approximate theJeffreys divergence improves by several orders of magnitude the computationaltime of stochastic Monte Carlo estimations while approximating reasonably wellthe Jeffreys divergence, specially when the mixtures have a very small numberof modes. Besides, our mixture-to-exponential family conversion techniques mayprove useful in other settings.

Fast approximations of the Jeffreys divergence between univariate Gaussian mixture models via exponential polynomial densities