{"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/a-parametrized-family-of-tversky-metrics","title":"Interpolating between the Jaccard distance and an analogue of the normalized information distance","arxiv_id":"2111.02498","date":"2021-11-03","proceeding":null,"authors":["Bjørn Kjos-Hanssen"],"abstract":"Jim\\'enez, Becerra, and Gelbukh (2013) defined a family of \"symmetric Tversky ratio models\" $S_{\\alpha,\\beta}$, $0\\le\\alpha\\le 1$, $\\beta>0$. Each function $D_{\\alpha,\\beta}=1-S_{\\alpha,\\beta}$ is a semimetric on the powerset of a given finite set. We show that $D_{\\alpha,\\beta}$ is a metric if and only if $0\\le\\alpha \\le \\frac12$ and $\\beta\\ge 1/(1-\\alpha)$. This result is formally verified in the Lean proof assistant. The extreme points of this parametrized space of metrics are $\\mathcal V_1=D_{1/2,2}$, the Jaccard distance, and $\\mathcal V_{\\infty}=D_{0,1}$, an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma, and Vit\\'anyi (2004). As a second interpolation, in general we also show that $\\mathcal V_p$ is a metric, $1\\le p\\le\\infty$, where $$\\Delta_p(A,B)=(|B\\setminus A|^p+|A\\setminus B|^p)^{1/p},$$ $$\\mathcal V_p(A,B)=\\frac{\\Delta_p(A,B)}{|A\\cap B| + \\Delta_p(A,B)}.$$","url_abs":"https://arxiv.org/abs/2111.02498v2","url_pdf":"https://arxiv.org/pdf/2111.02498v2.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":"a-parametrized-family-of-tversky-metrics","repo_url":"https://github.com/bjoernkjoshanssen/jaccard","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2111.02498","atlas_url":"https://app.syntology.ai/?focus=2111.02498","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}