Document Zbl 1189.62115

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Stat.</a> <a href="/?q=in%3A261945" title="Articles in this Issue">38, No. 3, 1287-1319 (2010)</a>. </div> <div class="abstract">Summary: We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on $\ell _{1}$-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $\ell_{1}$-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes $p$ and maximum neighborhood size $d$ are allowed to grow as a function of the number of observations $n$. Our main results provide sufficient conditions on the triple $(n, p, d)$ and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. <br class="zbmathjax-paragraph">With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes $n=\Omega (d^{3} \log p)$ with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of $n=\Omega (d^{2} \log p)$ suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5712422">Cited in <strong>175</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J12" title="MSC2020">62J12</a> </td> <td class="space"> Generalized linear models (logistic models) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62F12" title="MSC2020">62F12</a> </td> <td class="space"> Asymptotic properties of parametric estimators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C90" title="MSC2020">05C90</a> </td> <td class="space"> Applications of graph theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T99" title="MSC2020">68T99</a> </td> <td class="space"> Artificial intelligence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62M40" title="MSC2020">62M40</a> </td> <td class="space"> Random fields; image analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Agraphical+models">graphical models</a>; <a href="/?q=ut%3AMarkov+random+fields">Markov random fields</a>; <a href="/?q=ut%3Astructure+learning">structure learning</a>; <a href="/?q=ut%3A%5C%28%5Cell+_1%5C%29-regularization">$\ell _1$-regularization</a>; <a href="/?q=ut%3Amodel+selection">model selection</a>; <a href="/?q=ut%3Aconvex+risk+minimization">convex risk minimization</a>; <a href="/?q=ut%3Ahigh-dimensional+asymptotics">high-dimensional asymptotics</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4502">spatial</a>; <a href="/software/6072">pcalg</a>; <a href="/software/12177">TETRAD</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 1189.62115" data-ciurl="/ci/05712422" data-biburl="/bibtex/05712422.bib" data-amsurl="/amsrefs/05712422.bib" data-xmlurl="/xml/05712422.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05712422.pdf" title="Zbl 1189.62115 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/10.1214/09-AOS691" aria-label="DOI for “High-dimensional Ising model selection using $\ell _{1}$-regularized logistic regression”" title="10.1214/09-AOS691">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1010.0311"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">Abbeel, P., Koller, D. and Ng, A. 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