Super-polynomial accuracy of multidimensional randomized nets using the median-of-means

We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is an...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	arXiv.org 2022-08
Hauptverfasser:	Pan, Zexin, Owen, Art B
Format:	Artikel
Sprache:	eng
Schlagworte:	Estimates Median (statistics) Polynomials Samples (statistical)
Online-Zugang:	Volltext
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!

container_end_page
container_issue
container_start_page
container_title	arXiv.org
container_volume
creator	Pan, Zexin Owen, Art B
description	We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any \(c
format	Article
fullrecord	<record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2700908197</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2700908197</sourcerecordid><originalsourceid>FETCH-proquest_journals_27009081973</originalsourceid><addsrcrecordid>eNqNjL0KwjAYAIMgWLTvEHAOpKm17SyKu4JjCc1XTclPzZcM9ent4AM43XDHrUgmyrJgzUGIDckRR865ONaiqsqMPG5pgsAmb2bnrZaGyr5PQfYz9QO1yUSttAWH2rtFBunUkn1AUQcRaULtnjS-gFpQWjrmB2ZBOtyR9SANQv7jluwv5_vpyqbg3wkwdqNPYVliJ2rOW94UbV3-V30B7iJCjw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2700908197</pqid></control><display><type>article</type><title>Super-polynomial accuracy of multidimensional randomized nets using the median-of-means</title><source>Free E- Journals</source><creator>Pan, Zexin ; Owen, Art B</creator><creatorcontrib>Pan, Zexin ; Owen, Art B</creatorcontrib><description>We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Estimates ; Median (statistics) ; Polynomials ; Samples (statistical)</subject><ispartof>arXiv.org, 2022-08</ispartof><rights>2022. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>780,784</link.rule.ids></links><search><creatorcontrib>Pan, Zexin</creatorcontrib><creatorcontrib>Owen, Art B</creatorcontrib><title>Super-polynomial accuracy of multidimensional randomized nets using the median-of-means</title><title>arXiv.org</title><description>We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.</description><subject>Estimates</subject><subject>Median (statistics)</subject><subject>Polynomials</subject><subject>Samples (statistical)</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><recordid>eNqNjL0KwjAYAIMgWLTvEHAOpKm17SyKu4JjCc1XTclPzZcM9ent4AM43XDHrUgmyrJgzUGIDckRR865ONaiqsqMPG5pgsAmb2bnrZaGyr5PQfYz9QO1yUSttAWH2rtFBunUkn1AUQcRaULtnjS-gFpQWjrmB2ZBOtyR9SANQv7jluwv5_vpyqbg3wkwdqNPYVliJ2rOW94UbV3-V30B7iJCjw</recordid><startdate>20220809</startdate><enddate>20220809</enddate><creator>Pan, Zexin</creator><creator>Owen, Art B</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20220809</creationdate><title>Super-polynomial accuracy of multidimensional randomized nets using the median-of-means</title><author>Pan, Zexin ; Owen, Art B</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_27009081973</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Estimates</topic><topic>Median (statistics)</topic><topic>Polynomials</topic><topic>Samples (statistical)</topic><toplevel>online_resources</toplevel><creatorcontrib>Pan, Zexin</creatorcontrib><creatorcontrib>Owen, Art B</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Pan, Zexin</au><au>Owen, Art B</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>Super-polynomial accuracy of multidimensional randomized nets using the median-of-means</atitle><jtitle>arXiv.org</jtitle><date>2022-08-09</date><risdate>2022</risdate><eissn>2331-8422</eissn><abstract>We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record>
fulltext	fulltext
identifier	EISSN: 2331-8422
ispartof	arXiv.org, 2022-08
issn	2331-8422
language	eng
recordid	cdi_proquest_journals_2700908197
source	Free E- Journals
subjects	Estimates Median (statistics) Polynomials Samples (statistical)
title	Super-polynomial accuracy of multidimensional randomized nets using the median-of-means
url	https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-06T16%3A06%3A13IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest&rft_val_fmt=info:ofi/fmt:kev:mtx:book&rft.genre=document&rft.atitle=Super-polynomial%20accuracy%20of%20multidimensional%20randomized%20nets%20using%20the%20median-of-means&rft.jtitle=arXiv.org&rft.au=Pan,%20Zexin&rft.date=2022-08-09&rft.eissn=2331-8422&rft_id=info:doi/&rft_dat=%3Cproquest%3E2700908197%3C/proquest%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2700908197&rft_id=info:pmid/&rfr_iscdi=true