---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '20293'
abstract:
- lang: eng
  text: 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.
acknowledgement: "This research received partial funding from the European Research
  Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation
  programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF),
  grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and
  the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis
  of Neural Networks’. The APC was waived."
article_number: '854'
article_processing_charge: Yes
article_type: original
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Ziga
  full_name: Virk, Ziga
  id: 2E36B656-F248-11E8-B48F-1D18A9856A87
  last_name: Virk
- first_name: Hubert
  full_name: Wagner, Hubert
  id: 379CA8B8-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
citation:
  ama: 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>
  apa: 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>
  chicago: 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>.
  ieee: 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.
  ista: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the
    Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.
  mla: 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>.
  short: A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).
corr_author: '1'
date_created: 2025-09-07T22:01:33Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-09-30T14:32:31Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.3390/e27080854
ec_funded: 1
external_id:
  isi:
  - '001557476000001'
  pmid:
  - '40870326'
file:
- access_level: open_access
  checksum: 65c5399c4015d9c8abb8c7a96f3d7836
  content_type: application/pdf
  creator: dernst
  date_created: 2025-09-08T07:55:48Z
  date_updated: 2025-09-08T07:55:48Z
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file_date_updated: 2025-09-08T07:55:48Z
has_accepted_license: '1'
intvolume: '        27'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Entropy
publication_identifier:
  eissn:
  - 1099-4300
publication_status: published
publisher: MDPI
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds between the Jensen–Shannon divergence and the minmax divergence
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 27
year: '2025'
...