Loading…

SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If X is the n × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{ma...

Full description

Saved in:
Bibliographic Details
Published in:The Annals of probability 2016-07, Vol.44 (4), p.2479-2506
Main Authors: Bandeira, Afonso S., van Handel, Ramon
Format: Article
Language:English
Subjects:
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3
cites cdi_FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3
container_end_page 2506
container_issue 4
container_start_page 2479
container_title The Annals of probability
container_volume 44
creator Bandeira, Afonso S.
van Handel, Ramon
description We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If X is the n × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{max}} \sqrt{\sum _{j} b^{2}_{ij}} + \underset{ij}{\text{max}}\left | b_{ij} \right | \sqrt{\log n}$. This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.
doi_str_mv 10.1214/15-AOP1025
format article
fullrecord <record><control><sourceid>jstor_proqu</sourceid><recordid>TN_cdi_proquest_journals_1814123989</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>24735373</jstor_id><sourcerecordid>24735373</sourcerecordid><originalsourceid>FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3</originalsourceid><addsrcrecordid>eNo9kMFLwzAUxoMoOKcX70LAm1DNS9ImOdaus4U1KW2GeippbcGhdrbbwf_eyoaH997h-_E-vg-hayD3QIE_gO-FJgdC_RM0oxBITyr-copmhCjwQCh5ji7GcUMICYTgM1SUSVjkWBsdlq9Zbo1NI_xo1npRYqOxTeJJKzJslrgI9cJkOAttkUZxiZ9Tm-BUL-I8npa2eJoijctLdNa5j7G9Ot45Wi9jGyXeyjylUbjyGubLnUedULyRb9L5gQiCtqNBTZxUHaudz5ySrgXZOcoIE4px2VHo5JSicaymDa3ZHN0e_m6H_nvfjrtq0--Hr8myAgkcKFNSTdTdgWqGfhyHtqu2w_unG34qINVfZxX41bGzCb45wJtx1w__JOWC-Uww9gvg2V-r</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1814123989</pqid></control><display><type>article</type><title>SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES</title><source>Access via JSTOR</source><creator>Bandeira, Afonso S. ; van Handel, Ramon</creator><creatorcontrib>Bandeira, Afonso S. ; van Handel, Ramon</creatorcontrib><description>We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If X is the n × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{max}} \sqrt{\sum _{j} b^{2}_{ij}} + \underset{ij}{\text{max}}\left | b_{ij} \right | \sqrt{\log n}$. This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.</description><identifier>ISSN: 0091-1798</identifier><identifier>EISSN: 2168-894X</identifier><identifier>DOI: 10.1214/15-AOP1025</identifier><language>eng</language><publisher>Hayward: Institute of Mathematical Statistics</publisher><subject>Asymptotic methods ; Eigenvalues ; Geometrical optics ; Mathematical constants ; Mathematical theorems ; Matrices ; Matrix ; Matrix theory ; Normal distribution ; Phase transitions ; Proof theory ; Random variables ; Scalars ; Semicircles ; Spectrum analysis ; Studies ; Symmetry ; Vertices</subject><ispartof>The Annals of probability, 2016-07, Vol.44 (4), p.2479-2506</ispartof><rights>Copyright © 2016 Institute of Mathematical Statistics</rights><rights>Copyright Institute of Mathematical Statistics Jul 2016</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3</citedby><cites>FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/24735373$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/24735373$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,58238,58471</link.rule.ids></links><search><creatorcontrib>Bandeira, Afonso S.</creatorcontrib><creatorcontrib>van Handel, Ramon</creatorcontrib><title>SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES</title><title>The Annals of probability</title><description>We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If X is the n × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{max}} \sqrt{\sum _{j} b^{2}_{ij}} + \underset{ij}{\text{max}}\left | b_{ij} \right | \sqrt{\log n}$. This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.</description><subject>Asymptotic methods</subject><subject>Eigenvalues</subject><subject>Geometrical optics</subject><subject>Mathematical constants</subject><subject>Mathematical theorems</subject><subject>Matrices</subject><subject>Matrix</subject><subject>Matrix theory</subject><subject>Normal distribution</subject><subject>Phase transitions</subject><subject>Proof theory</subject><subject>Random variables</subject><subject>Scalars</subject><subject>Semicircles</subject><subject>Spectrum analysis</subject><subject>Studies</subject><subject>Symmetry</subject><subject>Vertices</subject><issn>0091-1798</issn><issn>2168-894X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNo9kMFLwzAUxoMoOKcX70LAm1DNS9ImOdaus4U1KW2GeippbcGhdrbbwf_eyoaH997h-_E-vg-hayD3QIE_gO-FJgdC_RM0oxBITyr-copmhCjwQCh5ji7GcUMICYTgM1SUSVjkWBsdlq9Zbo1NI_xo1npRYqOxTeJJKzJslrgI9cJkOAttkUZxiZ9Tm-BUL-I8npa2eJoijctLdNa5j7G9Ot45Wi9jGyXeyjylUbjyGubLnUedULyRb9L5gQiCtqNBTZxUHaudz5ySrgXZOcoIE4px2VHo5JSicaymDa3ZHN0e_m6H_nvfjrtq0--Hr8myAgkcKFNSTdTdgWqGfhyHtqu2w_unG34qINVfZxX41bGzCb45wJtx1w__JOWC-Uww9gvg2V-r</recordid><startdate>20160701</startdate><enddate>20160701</enddate><creator>Bandeira, Afonso S.</creator><creator>van Handel, Ramon</creator><general>Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20160701</creationdate><title>SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES</title><author>Bandeira, Afonso S. ; van Handel, Ramon</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Asymptotic methods</topic><topic>Eigenvalues</topic><topic>Geometrical optics</topic><topic>Mathematical constants</topic><topic>Mathematical theorems</topic><topic>Matrices</topic><topic>Matrix</topic><topic>Matrix theory</topic><topic>Normal distribution</topic><topic>Phase transitions</topic><topic>Proof theory</topic><topic>Random variables</topic><topic>Scalars</topic><topic>Semicircles</topic><topic>Spectrum analysis</topic><topic>Studies</topic><topic>Symmetry</topic><topic>Vertices</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bandeira, Afonso S.</creatorcontrib><creatorcontrib>van Handel, Ramon</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>The Annals of probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bandeira, Afonso S.</au><au>van Handel, Ramon</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES</atitle><jtitle>The Annals of probability</jtitle><date>2016-07-01</date><risdate>2016</risdate><volume>44</volume><issue>4</issue><spage>2479</spage><epage>2506</epage><pages>2479-2506</pages><issn>0091-1798</issn><eissn>2168-894X</eissn><abstract>We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If X is the n × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{max}} \sqrt{\sum _{j} b^{2}_{ij}} + \underset{ij}{\text{max}}\left | b_{ij} \right | \sqrt{\log n}$. This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.</abstract><cop>Hayward</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/15-AOP1025</doi><tpages>28</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0091-1798
ispartof The Annals of probability, 2016-07, Vol.44 (4), p.2479-2506
issn 0091-1798
2168-894X
language eng
recordid cdi_proquest_journals_1814123989
source Access via JSTOR
subjects Asymptotic methods
Eigenvalues
Geometrical optics
Mathematical constants
Mathematical theorems
Matrices
Matrix
Matrix theory
Normal distribution
Phase transitions
Proof theory
Random variables
Scalars
Semicircles
Spectrum analysis
Studies
Symmetry
Vertices
title SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-24T21%3A34%3A36IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_proqu&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=SHARP%20NONASYMPTOTIC%20BOUNDS%20ON%20THE%20NORM%20OF%20RANDOM%20MATRICES%20WITH%20INDEPENDENT%20ENTRIES&rft.jtitle=The%20Annals%20of%20probability&rft.au=Bandeira,%20Afonso%20S.&rft.date=2016-07-01&rft.volume=44&rft.issue=4&rft.spage=2479&rft.epage=2506&rft.pages=2479-2506&rft.issn=0091-1798&rft.eissn=2168-894X&rft_id=info:doi/10.1214/15-AOP1025&rft_dat=%3Cjstor_proqu%3E24735373%3C/jstor_proqu%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c358t-2a794c8d8a56766ef26b0a89f3ba53a98ae18fa230379348f21f8091ca3b2c2b3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=1814123989&rft_id=info:pmid/&rft_jstor_id=24735373&rfr_iscdi=true