{"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/moment-based-uniform-deviation-bounds-for-k","title":"Moment-based Uniform Deviation Bounds for $k$-means and Friends","arxiv_id":"1311.1903","date":"2013-11-08","proceeding":null,"authors":["Matus Telgarsky","Sanjoy Dasgupta"],"abstract":"Suppose $k$ centers are fit to $m$ points by heuristically minimizing the\n$k$-means cost; what is the corresponding fit over the source distribution?\nThis question is resolved here for distributions with $p\\geq 4$ bounded\nmoments; in particular, the difference between the sample cost and distribution\ncost decays with $m$ and $p$ as $m^{\\min\\{-1/4, -1/2+2/p\\}}$. The essential\ntechnical contribution is a mechanism to uniformly control deviations in the\nface of unbounded parameter sets, cost functions, and source distributions. To\nfurther demonstrate this mechanism, a soft clustering variant of $k$-means cost\nis also considered, namely the log likelihood of a Gaussian mixture, subject to\nthe constraint that all covariance matrices have bounded spectrum. Lastly, a\nrate with refined constants is provided for $k$-means instances possessing some\ncluster structure.","url_abs":"http://arxiv.org/abs/1311.1903v1","url_pdf":"http://arxiv.org/pdf/1311.1903v1.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":"abstracts"},"code_links":[{"paper_slug":"moment-based-uniform-deviation-bounds-for-k","repo_url":"https://github.com/ffs97/k-means-info-theory-survey","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1311.1903","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}