Document Zbl 1476.62066

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Math. Anal. Appl.</a> <a href="/?q=in%3A471209" title="Articles in this Issue">505, No. 2, Article ID 125561, 32 p. (2022)</a>. </div> <div class="abstract">Summary: Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance in nonparametric estimation, whereas the results are suboptimal due to a redundant logarithmic sacrifice. In this paper, we show that such log-factors are not necessary. We derive upper bounds for the $L^2$ minimax risk in nonparametric estimation. Sufficient conditions on network architectures are provided such that the upper bounds become optimal (without log-sacrifice). Our proof relies on an explicitly constructed network estimator based on tensor product B-splines. We also derive asymptotic distributions for the constructed network and a relating hypothesis testing procedure. The testing procedure is further proved as minimax optimal under suitable network architectures.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7413024">Cited in <strong>7</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G05" title="MSC2020">62G05</a> </td> <td class="space"> Nonparametric estimation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G10" title="MSC2020">62G10</a> </td> <td class="space"> Nonparametric hypothesis testing </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62M45" title="MSC2020">62M45</a> </td> <td class="space"> Neural nets and related approaches to inference from stochastic processes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65D07" title="MSC2020">65D07</a> </td> <td class="space"> Numerical computation using splines </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T07" title="MSC2020">68T07</a> </td> <td class="space"> Artificial neural networks and deep learning </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adeep+neural+network">deep neural network</a>; <a href="/?q=ut%3Anonparametric+inference">nonparametric inference</a>; <a href="/?q=ut%3Atensor+product+B-splines">tensor product B-splines</a>; <a href="/?q=ut%3Aoptimal+minimax+risk+bound">optimal minimax risk bound</a>; <a href="/?q=ut%3Aasymptotic+distribution">asymptotic distribution</a>; <a href="/?q=ut%3Anonparametric+testing">nonparametric testing</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 1476.62066" data-ciurl="/ci/07413024" data-biburl="/bibtex/07413024.bib" data-amsurl="/amsrefs/07413024.bib" data-xmlurl="/xml/07413024.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07413024.pdf" title="Zbl 1476.62066 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.jmaa.2021.125561" aria-label="DOI for “Optimal nonparametric inference via deep neural network”" title="10.1016/j.jmaa.2021.125561">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1902.01687"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">Bianchini, M.; Scarselli, F., On the complexity of neural network classifiers: a comparison between shallow and deep architectures, IEEE Trans. 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