{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/tight-lower-bounds-for-a-divergences-under","title":"Tight Lower Bounds for $α$-Divergences Under Moment Constraints and Relations Between Different $α$","arxiv_id":"2105.12972","date":"2021-05-27","proceeding":null,"authors":["Tomohiro Nishiyama"],"abstract":"The $\\alpha$-divergences include the well-known Kullback-Leibler divergence, Hellinger distance and $\\chi^2$-divergence. In this paper, we derive differential and integral relations between the $\\alpha$-divergences that are generalizations of the relation between the Kullback-Leibler divergence and the $\\chi^2$-divergence. We also show tight lower bounds for the $\\alpha$-divergences under given means and variances. In particular, we show a necessary and sufficient condition such that the binary divergences, which are divergences between probability measures on the same $2$-point set, always attain lower bounds. Kullback-Leibler divergence, Hellinger distance, and $\\chi^2$-divergence satisfy this condition.","url_abs":"https://arxiv.org/abs/2105.12972v4","url_pdf":"https://arxiv.org/pdf/2105.12972v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"tight-lower-bounds-for-a-divergences-under","repo_url":"https://github.com/nissy220/TotalVariationDistance","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}