[{"related_material":{"record":[{"id":"20563","status":"public","relation":"dissertation_contains"}]},"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Filippo","last_name":"Quattrocchi","full_name":"Quattrocchi, Filippo","orcid":"0009-0000-9773-1931","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308"}],"external_id":{"arxiv":["2408.12924"]},"arxiv":1,"OA_place":"repository","acknowledgement":"The author is thankful to Nicolas Clozeau, Lorenzo Dello Schiavo, Jan Maas, Dejan Slepčev,\r\nand Dario Trevisan for many fruitful discussions and comments. The author gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.","keyword":["optimal empirical quantization","vector quantization","Wasserstein distance","semidiscrete optimal transport","Zador’s Theorem","Pierce’s Lemma"],"status":"public","month":"08","citation":{"ama":"Quattrocchi F. Asymptotics for optimal empirical quantization of measures. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2408.12924\">10.48550/arXiv.2408.12924</a>","ieee":"F. Quattrocchi, “Asymptotics for optimal empirical quantization of measures,” <i>arXiv</i>. .","apa":"Quattrocchi, F. (n.d.). Asymptotics for optimal empirical quantization of measures. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2408.12924\">https://doi.org/10.48550/arXiv.2408.12924</a>","chicago":"Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of Measures.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2408.12924\">https://doi.org/10.48550/arXiv.2408.12924</a>.","mla":"Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of Measures.” <i>ArXiv</i>, 2408.12924, doi:<a href=\"https://doi.org/10.48550/arXiv.2408.12924\">10.48550/arXiv.2408.12924</a>.","short":"F. Quattrocchi, ArXiv (n.d.).","ista":"Quattrocchi F. Asymptotics for optimal empirical quantization of measures. arXiv, 2408.12924."},"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"date_created":"2025-10-28T13:12:22Z","_id":"20570","date_published":"2024-08-23T00:00:00Z","project":[{"name":"Taming Complexity in Partial Differential Systems","_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"F06504"}],"publication_status":"draft","doi":"10.48550/arXiv.2408.12924","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2408.12924"}],"abstract":[{"text":"We investigate the minimal error in approximating a general probability\r\nmeasure $\\mu$ on $\\mathbb{R}^d$ by the uniform measure on a finite set with\r\nprescribed cardinality $n$. The error is measured in the $p$-Wasserstein\r\ndistance. In particular, when $1\\le p<d$, we establish asymptotic upper and\r\nlower bounds as $n \\to \\infty$ on the rescaled minimal error that have the\r\nsame, explicit dependency on $\\mu$.\r\n  In some instances, we prove that the rescaled minimal error has a limit.\r\nThese include general measures in dimension $d = 2$ with $1 \\le p < 2$, and\r\nuniform measures in arbitrary dimension with $1 \\le p < d$. For some uniform\r\nmeasures, we prove the limit existence for $p \\ge d$ as well.\r\n  For a class of compactly supported measures with H\\\"older densities, we\r\ndetermine the convergence speed of the minimal error for every $p \\ge 1$.\r\n  Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper\r\nestimate of the minimal error when $1 \\le p < d$.\r\n  In the initial sections, we survey the state of the art and draw connections\r\nwith similar problems, such as classical and random quantization.","lang":"eng"}],"corr_author":"1","language":[{"iso":"eng"}],"article_number":"2408.12924","title":"Asymptotics for optimal empirical quantization of measures","year":"2024","OA_type":"green","day":"23","oa":1,"article_processing_charge":"No","date_updated":"2026-05-03T22:30:18Z","type":"preprint","publication":"arXiv"}]