Document Zbl 1534.68203

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1534.68203 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1534.68203?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1534.68203" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/hertrich.johannes" title="Author Profile">Hertrich, Johannes</a>; <a href="/authors/neumayer.sebastian" title="Author Profile">Neumayer, Sebastian</a>; <a href="/authors/steidl.gabriele" title="Author Profile">Steidl, Gabriele</a></div> <h2 class="title"> <strong>Convolutional proximal neural networks and plug-and-play algorithms.</strong> <i>(English)</i> <a class="label nowrap" href="/1534.68203">Zbl 1534.68203</a> </h2> <div class="source"> <a href="/serials/1605" title="Journal Profile">Linear Algebra Appl.</a> <a href="/?q=in%3A471724" title="Articles in this Issue">631, 203-234 (2021)</a>. </div> <div class="abstract">Summary: In this paper, we introduce convolutional proximal neural networks (cPNNs), which are by construction averaged operators. For filters with full length, we propose a stochastic gradient descent algorithm on a submanifold of the Stiefel manifold to train cPNNs. In case of filters with limited length, we design algorithms for minimizing functionals that approximate the orthogonality constraints imposed on the operators by penalizing the least squares distance to the identity operator. Then, we investigate how scaled cPNNs with a prescribed Lipschitz constant can be used for denoising signals and images, where the achieved quality depends on the Lipschitz constant. Finally, we apply cPNN based denoisers within a Plug-and-Play framework and provide convergence results for the corresponding PnP forward-backward splitting algorithm based on an oracle construction.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7418878">Cited in <strong>18</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T05" title="MSC2020">68T05</a> </td> <td class="space"> Learning and adaptive systems in artificial intelligence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46N10" title="MSC2020">46N10</a> </td> <td class="space"> Applications of functional analysis in optimization, convex analysis, mathematical programming, economics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47H05" title="MSC2020">47H05</a> </td> <td class="space"> Monotone operators and generalizations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K10" title="MSC2020">65K10</a> </td> <td class="space"> Numerical optimization and variational techniques </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A08" title="MSC2020">94A08</a> </td> <td class="space"> Image processing (compression, reconstruction, etc.) in information and communication theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A12" title="MSC2020">94A12</a> </td> <td class="space"> Signal theory (characterization, reconstruction, filtering, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aaveraged+operators">averaged operators</a>; <a href="/?q=ut%3Adenoising">denoising</a>; <a href="/?q=ut%3ALipschitz+networks">Lipschitz networks</a>; <a href="/?q=ut%3Amatrix+manifolds">matrix manifolds</a>; <a href="/?q=ut%3Aneural+networks">neural networks</a>; <a href="/?q=ut%3Aplug-and-play+algorithms">plug-and-play algorithms</a></div> <div class="software"> <h3>Software:</h3><a href="/software/14427">BSDS</a>; <a href="/software/22205">Adam</a>; <a href="/software/7521">UNLocBoX</a>; <a href="/software/39678">DnCNN</a>; <a href="/software/28266">BM3D</a>; <a href="/software/23743">mftoolbox</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1534.68203" data-ciurl="/ci/07418878" data-biburl="/bibtex/07418878.bib" data-amsurl="/amsrefs/07418878.bib" data-xmlurl="/xml/07418878.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07418878.pdf" title="Zbl 1534.68203 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.laa.2021.09.004" aria-label="DOI for “Convolutional proximal neural networks and plug-and-play algorithms”" title="10.1016/j.laa.2021.09.004">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2011.02281"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization Algorithms on Matrix Manifolds (2008), Princeton University Press · <a href="/1147.65043" class="nowrap">Zbl 1147.65043</a></td> </tr><tr> <td>[2]</td> <td class="space">Beck, A., First-Order Methods in Optimization, MOS-SIAM Series on Optimization, vol. 25 (2017), SIAM: SIAM Philadelphia · <a href="/1384.65033" class="nowrap">Zbl 1384.65033</a></td> </tr><tr> <td>[3]</td> <td class="space">Behrmann, J.; Vicol, P.; Wang, K.-C.; Grosse, R.; Jacobsen, J.-H., Understanding and mitigating exploding inverses in invertible neural networks (2020), arXiv preprint</td> </tr><tr> <td>[4]</td> <td class="space">Bini, D.; Capovani, M., Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53, 99-126 (1983) · <a href="/0549.15005" class="nowrap">Zbl 0549.15005</a></td> </tr><tr> <td>[5]</td> <td class="space">Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J., Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3, 1, 101-122 (2011) · <a href="/1229.90122" class="nowrap">Zbl 1229.90122</a></td> </tr><tr> <td>[6]</td> <td class="space">Braides, A., Γ-Convergence for Beginners (2002), Oxford University Press: Oxford University Press Oxford · <a href="/1198.49001" class="nowrap">Zbl 1198.49001</a></td> </tr><tr> <td>[7]</td> <td class="space">Burger, M.; Sawatzky, A.; Steidl, G., First order algorithms in variational image processing, (Operator Splittings and Alternating Direction Methods (2017), Springer) · <a href="/1372.65053" class="nowrap">Zbl 1372.65053</a></td> </tr><tr> <td>[8]</td> <td class="space">Byers, R.; Xu, H., A new scaling for Newton’s iteration for the polar decomposition and its backward stability, SIAM J. Matrix Anal. Appl., 30, 2, 822-843 (2008) · <a href="/1170.65019" class="nowrap">Zbl 1170.65019</a></td> </tr><tr> <td>[9]</td> <td class="space">Chan, S. H.; Wang, X.; Elgendy, O. A., Plug-and-play ADMM for image restoration: fixed-point convergence and applications, IEEE Trans. Comput. Imaging, 3, 84-98 (2016)</td> </tr><tr> <td>[10]</td> <td class="space">Combettes, P. L., Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53, 5-6, 475-504 (2004) · <a href="/1153.47305" class="nowrap">Zbl 1153.47305</a></td> </tr><tr> <td>[11]</td> <td class="space">Combettes, P. L., Monotone operator theory in convex optimization, Math. Program., 170, 1, 177-206 (2018) · <a href="/1471.47033" class="nowrap">Zbl 1471.47033</a></td> </tr><tr> <td>[12]</td> <td class="space">Combettes, P. L.; Pesquet, J.-C., Proximal thresholding algorithm for minimization over orthonormal bases, SIAM J. Optim., 18, 1351-1376 (2021) · <a href="/1167.90011" class="nowrap">Zbl 1167.90011</a></td> </tr><tr> <td>[13]</td> <td class="space">Combettes, P. L.; Pesquet, J.-C., Proximal splitting methods in signal processing, (Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl., vol. 49 (2011), Springer: Springer New York), 185-212 · <a href="/1242.90160" class="nowrap">Zbl 1242.90160</a></td> </tr><tr> <td>[14]</td> <td class="space">Combettes, P. L.; Pesquet, J.-C., Deep neural network structures solving variational inequalities, Set-Valued Var. Anal., 28, 491-518 (2020) · <a href="/1448.49014" class="nowrap">Zbl 1448.49014</a></td> </tr><tr> <td>[15]</td> <td class="space">Combettes, P. L.; Pesquet, J.-C., Fixed point strategies in data science (2020), arXiv preprint</td> </tr><tr> <td>[16]</td> <td class="space">Combettes, P. L.; Pesquet, J.-C., Lipschitz certificates for layered network structures driven by averaged activation operators, SIAM J. Math. Data Sci., 2, 2, 529-557 (2020) · <a href="/07468921" class="nowrap">Zbl 07468921</a></td> </tr><tr> <td>[17]</td> <td class="space">Combettes, P. L.; Wajs, V. R., Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4, 4, 1168-1200 (2005) · <a href="/1179.94031" class="nowrap">Zbl 1179.94031</a></td> </tr><tr> <td>[18]</td> <td class="space">Combettes, P. L.; Yamada, I., Compositions and convex combinations of averaged nonexpansive operators, J. Math. Anal. Appl., 425, 1, 55-70 (2015) · <a href="/1322.47051" class="nowrap">Zbl 1322.47051</a></td> </tr><tr> <td>[19]</td> <td class="space">Condat, L.; Kitahara, D.; Contreras, A.; Hirabayashi, A., Proximal splitting algorithms: relax them all! (2019), arXiv preprint</td> </tr><tr> <td>[20]</td> <td class="space">Dabov, K.; Foi, A.; Katkovnik, V.; Egiazarian, K., Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image Process., 16, 8, 2080-2095 (2007)</td> </tr><tr> <td>[21]</td> <td class="space">Danielyan, A.; Katkovnik, V.; Egiazarian, K., BM3D frames and variational image deblurring, IEEE Trans. Image Process., 21, 4, 1715-1728 (2012) · <a href="/1373.94096" class="nowrap">Zbl 1373.94096</a></td> </tr><tr> <td>[22]</td> <td class="space">Daubechies, I.; Defrise, M.; De Mol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57, 11, 1413-1457 (2004) · <a href="/1077.65055" class="nowrap">Zbl 1077.65055</a></td> </tr><tr> <td>[23]</td> <td class="space">Donoho, D. L.; Maleki, A.; Montanari, A., Message-passing algorithms for compressed sensing, Proc. Natl. Acad. Sci., 106, 45, 18914-18919 (2009)</td> </tr><tr> <td>[24]</td> <td class="space">Eckstein, J.; Bertsekas, D. P., On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55, 3, Ser. A, 293-318 (1992) · <a href="/0765.90073" class="nowrap">Zbl 0765.90073</a></td> </tr><tr> <td>[25]</td> <td class="space">Effland, A.; Kobler, E.; Kunisch, K.; Pock, T., Variational networks: an optimal control approach to early stopping variational methods for image restoration, J. Math. Imaging Vis., 62, 3, 396-416 (2020) · <a href="/1434.68626" class="nowrap">Zbl 1434.68626</a></td> </tr><tr> <td>[26]</td> <td class="space">Glowinski, R.; Osher, S. J.; Yin, W., Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation (2016), Springer: Springer Cham · <a href="/1362.65002" class="nowrap">Zbl 1362.65002</a></td> </tr><tr> <td>[27]</td> <td class="space">Gouk, H.; Frank, E.; Pfahringer, B.; Cree, M., Regularisation of neural networks by enforcing Lipschitz continuity (2018), arXiv preprint</td> </tr><tr> <td>[28]</td> <td class="space">Gupta, H.; Jin, K. H.; Nguyen, H. Q.; McCann, M. T.; Unser, M., CNN-based projected gradient descent for consistent CT image reconstruction, IEEE Trans. Med. Imaging, 37, 6, 1440-1453 (2018)</td> </tr><tr> <td>[29]</td> <td class="space">Hagemann, P.; Neumayer, S., Stabilizing invertible neural networks using mixture models (2020), arXiv preprint · <a href="/07375643" class="nowrap">Zbl 07375643</a></td> </tr><tr> <td>[30]</td> <td class="space">Hasannasab, M.; Hertrich, J.; Neumayer, S.; Plonka, G.; Setzer, S.; Steidl, G., Parseval proximal neural networks, J. Fourier Anal. Appl., 26, 59 (2020) · <a href="/1489.68224" class="nowrap">Zbl 1489.68224</a></td> </tr><tr> <td>[31]</td> <td class="space">Hertrich, J.; Neumayer, S.; Steidl, G., Convolutional proximal neural networks and plug-and-play algorithms (2020), arXiv preprint</td> </tr><tr> <td>[32]</td> <td class="space">Hertrich, J.; Steidl, G., Inertial stochastic PALM and its application for learning Student-t mixture models (2020), arXiv preprint</td> </tr><tr> <td>[33]</td> <td class="space">Higham, N. J., Computing the polar decomposition-with applications, SIAM J. Sci. Stat. Comput., 7, 4, 1160-1174 (1986) · <a href="/0607.65014" class="nowrap">Zbl 0607.65014</a></td> </tr><tr> <td>[34]</td> <td class="space">Higham, N. J., Functions of Matrices: Theory and Computation (2008), SIAM: SIAM Philadelphia · <a href="/1167.15001" class="nowrap">Zbl 1167.15001</a></td> </tr><tr> <td>[35]</td> <td class="space">Horn, R. A.; Johnson, C. R., Matrix Analysis (2013), Oxford University Press · <a href="/1267.15001" class="nowrap">Zbl 1267.15001</a></td> </tr><tr> <td>[36]</td> <td class="space">Huang, L.; Liu, X.; Lang, B.; Yu, A. W.; Wang, Y.; Li, B., Orthogonal weight normalization: solution to optimization over multiple dependent Stiefel manifolds in deep neural networks, (32nd AAAI Conference on Artificial Intelligence (2018))</td> </tr><tr> <td>[37]</td> <td class="space">Kingma, D. P.; Adam, J. Ba., A method for stochastic optimization (2014), arXiv preprint</td> </tr><tr> <td>[38]</td> <td class="space">Krasnoselskii, M. A., Two observations about the method of successive approximations, Usp. Mat. Nauk, 10, 123-127 (1955), in Russian · <a href="/0064.12002" class="nowrap">Zbl 0064.12002</a></td> </tr><tr> <td>[39]</td> <td class="space">Li, J.; Li, F.; Todorovic, S., Efficient Riemannian optimization on the Stiefel manifold via the Cayley transform, (International Conference on Learning Representations (2020))</td> </tr><tr> <td>[40]</td> <td class="space">Mann, W. R., Mean value methods in iteration, Proc. Am. Math. Soc., 16, 4, 506-510 (1953) · <a href="/0050.11603" class="nowrap">Zbl 0050.11603</a></td> </tr><tr> <td>[41]</td> <td class="space">Martin, D.; Fowlkes, C.; Tal, D.; Malik, J., A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, (Proc. Eighth IEEE International Conference on Computer Vision. ICCV 2001, vol. 2 (2001), IEEE), 416-423</td> </tr><tr> <td>[42]</td> <td class="space">Meinhardt, T.; Moeller, M.; Hazirbas, C.; Cremers, D., Learning proximal operators: using denoising networks for regularizing inverse imaging problems, (Proc. IEEE International Conference on Computer Vision (2017)), 1799-1808</td> </tr><tr> <td>[43]</td> <td class="space">Miyato, T.; Kataoka, T.; Koyama, M.; Yoshida, Y., Spectral normalization for generative adversarial networks, (International Conference on Learning Representations (2018))</td> </tr><tr> <td>[44]</td> <td class="space">Monga, V.; Li, Y.; Eldar, Y., Algorithm unrolling: interpretable, efficient deep learning for signal and image processing (2019), arXiv preprint</td> </tr><tr> <td>[45]</td> <td class="space">Moreau, J.-J., Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. Fr., 93, 273-299 (1965) · <a href="/0136.12101" class="nowrap">Zbl 0136.12101</a></td> </tr><tr> <td>[46]</td> <td class="space">Nishimori, Y.; Akaho, S., Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67, 106-135 (2005)</td> </tr><tr> <td>[47]</td> <td class="space">Ono, S., Primal-dual plug-and-play image restoration, IEEE Signal Process. Lett., 24, 8, 1108-1112 (2017)</td> </tr><tr> <td>[48]</td> <td class="space">Potts, D.; Steidl, G., Preconditioners for ill-conditioned Toeplitz matrices, BIT Numer. Math., 39, 3, 513-533 (1999) · <a href="/0938.65067" class="nowrap">Zbl 0938.65067</a></td> </tr><tr> <td>[49]</td> <td class="space">Rick Chang, J.; Li, C.-L.; Poczos, B.; Vijaya Kumar, B.; Sankaranarayanan, A. C., One network to solve them all-solving linear inverse problems using deep projection models, (Proc. of the IEEE International Conference on Computer Vision (2017)), 5888-5897</td> </tr><tr> <td>[50]</td> <td class="space">Romano, Y.; Elad, M.; Milanfar, P., The little engine that could: regularization by denoising (RED), SIAM J. Imaging Sci., 10, 4, 1804-1844 (2017) · <a href="/1401.62101" class="nowrap">Zbl 1401.62101</a></td> </tr><tr> <td>[51]</td> <td class="space">Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268 (1992) · <a href="/0780.49028" class="nowrap">Zbl 0780.49028</a></td> </tr><tr> <td>[52]</td> <td class="space">Russell Luke, D.; Thao, N. H.; Tam, M. K., Quantitative convergence analysis of iterated expansive, set-valued mappings, Math. Oper. Res., 43, 4, 1143-1176 (2018) · <a href="/1434.49012" class="nowrap">Zbl 1434.49012</a></td> </tr><tr> <td>[53]</td> <td class="space">Sedghi, H.; Gupta, V.; Long, P. M., The singular values of convolutional layers, (International Conference on Learning Representations (2019))</td> </tr><tr> <td>[54]</td> <td class="space">Setzer, S., Operator splittings, Bregman methods and frame shrinkage in image processing, Int. J. Comput. Vis., 92, 3, 265-280 (2011) · <a href="/1235.68314" class="nowrap">Zbl 1235.68314</a></td> </tr><tr> <td>[55]</td> <td class="space">Sommerhoff, H.; Kolb, A.; Moeller, M., Energy dissipation with plug-and-play priors, (NeurIPS 2019 Workshop (2019))</td> </tr><tr> <td>[56]</td> <td class="space">Sreehariand, S.; Venkatakrishnan, S. V.; Wohlberg, B., Plug-and-play priors for bright field electron tomography and sparse interpolation, IEEE Trans. Comput. Imaging, 2, 408-423 (2016)</td> </tr><tr> <td>[57]</td> <td class="space">Strang, G.; MacNamara, S., Functions of difference matrices are Toeplitz plus Hankel, SIAM Rev., 56, 3, 525-546 (2014) · <a href="/1304.65195" class="nowrap">Zbl 1304.65195</a></td> </tr><tr> <td>[58]</td> <td class="space">Sun, Y.; Wohlberg, B.; Kamilov, U. S., An online plug-and-play algorithm for regularized image reconstruction, IEEE Trans. Comput. Imaging, 5, 3, 395-408 (2019)</td> </tr><tr> <td>[59]</td> <td class="space">Sutour, C.; Deledalle, C.-A.; Aujol, J.-F., Estimation of the noise level function based on a nonparametric detection of homogeneous image regions, SIAM J. Imaging Sci., 8, 2622-2661 (2015) · <a href="/1343.94015" class="nowrap">Zbl 1343.94015</a></td> </tr><tr> <td>[60]</td> <td class="space">Teodoro, A.; Bioucas-Dias, J. M.; Figueiredo, M., Scene-adapted plug-and-play algorithm with convergence guarantees, (Proc. IEEE Int. Workshop on Machine Learning for Signal Processing (2007))</td> </tr><tr> <td>[61]</td> <td class="space">Terris, M.; Repetti, A.; Pesquet, J.; Wiaux, Y., Building firmly nonexpansive convolutional neural networks, (ICASSP 2020 - IEEE International Conference on Acoustics, Speech and Signal Processing (2020)), 8658-8662</td> </tr><tr> <td>[62]</td> <td class="space">Tsuzuku, Y.; Sato, I.; Sugiyama, M., Lipschitz-margin training: scalable certification of perturbation invariance for deep neural networks, (Advances in Neural Information Processing Systems, vol. 31 (2018), Curran Associates, Inc.), 6541-6550</td> </tr><tr> <td>[63]</td> <td class="space">Venkatakrishnan, S. V.; Bouman, C. A.; Wohlberg, B., Plug-and-play priors for model based reconstruction, (2013 IEEE Global Conference on Signal and Information Processing (2013), IEEE), 945-948</td> </tr><tr> <td>[64]</td> <td class="space">Wen, Z.; Yin, W., A feasible method for optimization with orthogonality constraints, Math. Program., 142, 1-2, 397-434 (2013) · <a href="/1281.49030" class="nowrap">Zbl 1281.49030</a></td> </tr><tr> <td>[65]</td> <td class="space">Zhang, K.; Zuo, W.; Chen, Y.; Meng, D.; Zhang, L., Beyond a Gaussian denoiser: residual learning of deep CNN for image denoising, IEEE Trans. Image Process., 26, 7, 3142-3155 (2017) · <a href="/1409.94754" class="nowrap">Zbl 1409.94754</a></td> </tr><tr> <td>[66]</td> <td class="space">Zhang, K.; Zuo, W.; Gu, S.; Zhang, L., Learning deep CNN denoiser prior for image restoration, (Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (2017)), 3929-3938</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1534.68203 - zbMATH Open