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On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means

arXiv 1904.04017 ↗First posted 8 Apr 2019cs.IT cs.LG math.ITcited by 61

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Abstract (arXiv)

The Jensen-Shannon divergence is a renown bounded symmetrization of the unbounded Kullback-Leibler divergence which measures the total Kullback-Leibler divergence to the average mixture distribution. However the Jensen-Shannon divergence between Gaussian distributions is not available in closed-form. To bypass this problem, we present a generalization of the Jensen-Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using generalized statistical mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen-Shannon divergence between probability densities of the same exponential family, and (ii) the geometric JS-symmetrization of the reverse Kullback-Leibler divergence. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen-Shannon divergence between scale Cauchy distributions. We also define generalized Jensen-Shannon divergences between matrices (e.g., quantum Jensen-Shannon divergences) and consider clustering with respect to these novel Jensen-Shannon divergences.

Key facts. Cited by 61 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Apr 2019.

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Citing papers per year: 2021: 7; 2022: 5; 2023: 11; 2024: 13; 2025: 21; 2026: 4.

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