# Two tales for a geometric Jensen--Shannon divergence

arXiv: https://arxiv.org/abs/2508.05066 · HTML: https://mlsanity.com/paper/2508.05066/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 61 references that are in the map.
- First posted on arXiv 7 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means](https://mlsanity.com/paper/1904.04017/) (2019)
- [Multimodal Generative Learning Utilizing Jensen-Shannon-Divergence](https://mlsanity.com/paper/2006.08242/) (2020)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [Constraining Variational Inference with Geometric Jensen-Shannon Divergence](https://mlsanity.com/paper/2006.10599/) (2020)
- [q-Paths: Generalizing the Geometric Annealing Path using Power Means](https://mlsanity.com/paper/2107.00745/) (2021)
- [Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means](https://mlsanity.com/paper/1401.4788/) (2014)
- [Stability Based Generalization Bounds for Exponential Family Langevin Dynamics](https://mlsanity.com/paper/2201.03064/) (2022)
- [Jensen-Shannon Divergence Based Novel Loss Functions for Bayesian Neural Networks](https://mlsanity.com/paper/2209.11366/) (2022)
- [NP-SemiSeg: When Neural Processes meet Semi-Supervised Semantic Segmentation](https://mlsanity.com/paper/2308.02866/) (2023)
- [On the Computation of the Gaussian Rate-Distortion-Perception Function](https://mlsanity.com/paper/2311.09190/) (2023)
- [EMPERROR: A Flexible Generative Perception Error Model for Probing Self-Driving Planners](https://mlsanity.com/paper/2411.07719/) (2024)
- [Geometric Jensen-Shannon Divergence Between Gaussian Measures On Hilbert Space](https://mlsanity.com/paper/2506.10494/) (2025)

## Abstract (arXiv)

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.