Document Zbl 1216.62090

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Stat.</a> <a href="/?q=in%3A281671" title="Articles in this Issue">39, No. 2, 1069-1097 (2011)</a>. </div> <div class="abstract">Summary: We study an instance of high-dimensional inference where the goal is to estimate a matrix $\Theta^*\in \mathbb R^{m_1\times m_2}$ on the basis of $N$ noisy observations. The unknown matrix $\Theta^*$ is assumed to be either exactly low rank, or “near” low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider a standard $M$-estimator based on regularization by the nuclear or trace norm over matrices, and analyze its performance under high-dimensional scaling. We define the notion of restricted strong convexity (RSC) for the loss function, and use it to derive non-asymptotic bounds on the Frobenius norm error that hold for a general class of noisy observation models, and apply to both exactly low-rank and approximately low rank matrices. We then illustrate consequences of this general theory for a number of specific matrix models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes and recovery of low-rank matrices from random projections. These results involve nonasymptotic random matrix theory to establish that the RSC condition holds, and to determine an appropriate choice of the regularization parameter. Simulation results show excellent agreement with the high-dimensional scaling of the error predicted by our theory.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5914741">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5914741">Cited in <strong>149</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H12" title="MSC2020">62H12</a> </td> <td class="space"> Estimation in multivariate analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62F30" title="MSC2020">62F30</a> </td> <td class="space"> Parametric inference under constraints </td> </tr><tr> <td class="mono"> 65C60 </td> <td class="space"> Computational problems in statistics (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahigh-dimensional+inference">high-dimensional inference</a>; <a href="/?q=ut%3Arank+constraints">rank constraints</a>; <a href="/?q=ut%3Anuclear+norm">nuclear norm</a>; <a href="/?q=ut%3Atrace+norm">trace norm</a>; <a href="/?q=ut%3A%5C%28M%5C%29-estimators">$M$-estimators</a>; <a href="/?q=ut%3Arandom+matrix+theory">random matrix theory</a></div> <div class="software"> <h3>Software:</h3><a href="/software/5148">PDCO</a>; <a href="/software/7432">glasso</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 1216.62090" data-ciurl="/ci/05914741" data-biburl="/bibtex/05914741.bib" data-amsurl="/amsrefs/05914741.bib" data-xmlurl="/xml/05914741.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05914741.pdf" title="Zbl 1216.62090 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/10.1214/10-AOS850" aria-label="DOI for “Estimation of (near) low-rank matrices with noise and high-dimensional scaling”" title="10.1214/10-AOS850">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/0912.5100"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">Abernethy, J., Bach, F., Evgeniou, T. and Stein, J. 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