{"ID":2885029,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.05066","arxiv_id":"2508.05066","title":"Two tales for a geometric Jensen--Shannon divergence","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.","short_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...","url_abs":"https://arxiv.org/abs/2508.05066","url_pdf":"https://arxiv.org/pdf/2508.05066v3","authors":"[\"Frank Nielsen\"]","published":"2025-08-07T06:34:39Z","proceeding":"cs.IT","tasks":"[\"cs.IT\",\"cs.LG\"]","methods":"[]","has_code":false}