article published yes Arseniy Akopyan author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Ziga Virk author 2E36B656-F248-11E8-B48F-1D18A9856A87 Hubert Wagner author 379CA8B8-F248-11E8-B48F-1D18A9856A87 HeEd department Alpha Shape Theory Extended project Mathematics, Computer Science project Persistence and stability of geometric complexes project Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence. https://research-explorer.ista.ac.at/download/20293/20309/2025_Entropy_Akopyan.pdf application/pdfno MDPI2025 eng 1099-4300 40870326 00155747600000110.3390/e27080854 278 Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href="https://doi.org/10.3390/e27080854">10.3390/e27080854</a>. Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href="https://doi.org/10.3390/e27080854">10.3390/e27080854</a> A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” <i>Entropy</i>, vol. 27, no. 8. MDPI, 2025. Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>. MDPI, 2025. <a href="https://doi.org/10.3390/e27080854">https://doi.org/10.3390/e27080854</a>. Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href="https://doi.org/10.3390/e27080854">https://doi.org/10.3390/e27080854</a> Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854. A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025). 202932025-09-07T22:01:33Z2025-09-30T14:32:31Z