---
OA_place: repository
OA_type: green
_id: '20570'
abstract:
- lang: eng
  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."
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."
article_number: '2408.12924'
article_processing_charge: No
arxiv: 1
author:
- first_name: Filippo
  full_name: Quattrocchi, Filippo
  id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
  last_name: Quattrocchi
  orcid: 0009-0000-9773-1931
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>
  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>.
  ieee: F. Quattrocchi, “Asymptotics for optimal empirical quantization of measures,”
    <i>arXiv</i>. .
  ista: Quattrocchi F. Asymptotics for optimal empirical quantization of measures.
    arXiv, 2408.12924.
  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.).
corr_author: '1'
date_created: 2025-10-28T13:12:22Z
date_published: 2024-08-23T00:00:00Z
date_updated: 2026-04-27T22:30:15Z
day: '23'
department:
- _id: GradSch
- _id: JaMa
doi: 10.48550/arXiv.2408.12924
external_id:
  arxiv:
  - '2408.12924'
keyword:
- optimal empirical quantization
- vector quantization
- Wasserstein distance
- semidiscrete optimal transport
- Zador’s Theorem
- Pierce’s Lemma
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2408.12924
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 260482E2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: F06504
  name: Taming Complexity in Partial Differential Systems
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20563'
    relation: dissertation_contains
    status: public
status: public
title: Asymptotics for optimal empirical quantization of measures
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...