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arXiv · 2508.05066

Two tales for a geometric Jensen--Shannon divergence

Frank Nielsen
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Abstract

The geometric Jensen--Shannon divergence (G-JSD) gained popularity in machine learning and information sciences thanks to its closed-form expression between Gaussian distributions. In this work, we introduce an alternative definition of the geometric Jensen--Shannon divergence tailored to positive densities which does not normalize geometric mixtures. This novel divergence is termed the extended G-JSD as it applies to the more general case of positive measures. We report explicitly the gap between the extended G-JSD and the G-JSD when considering probability densities, and show how to express the G-JSD and extended G-JSD using the Jeffreys divergence and the Bhattacharyya distance or Bhattacharyya coefficient. The extended G-JSD is proven to be a $f$-divergence which is a separable divergence satisfying information monotonicity and invariance in information geometry. We derive corresponding closed-form formula for the two types of G-JSDs when considering the case of multivariate Gaussian distributions often met in applications. We consider Monte Carlo stochastic estimations and approximations of the two types of G-JSD using the projective $γ$-divergences. Although the square root of the JSD yields a metric distance, we show that this is not anymore the case for the two types of G-JSD. Finally, we explain how these two types of geometric JSDs can be interpreted as regularizations of the ordinary JSD.

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Frank Nielsen. 2025-09-18. Two tales for a geometric Jensen--Shannon divergence. https://arxiv.org/abs/2508.05066

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