Document Zbl 1218.65037

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Math.</a> <a href="/?q=in%3A273308" title="Articles in this Issue">117, No. 2, 219-249 (2011)</a>. </div> <div class="abstract">It is well-known that an approximate solution $x_{opt}$ of an overdetermined system of linear equations<br class="zbmathjax-paragraph">\[ Ax = b, \quad A \in \mathbb{R}^{n \times d}, \quad b \in \mathbb{R}^n, \quad n \geq d, \tag{1} \]<br class="zbmathjax-paragraph">is commonly found by the method of least-squares solving the optimization problem<br class="zbmathjax-paragraph">\[ Z = \min_{x \in \mathbb{R}^d} {\| Ax-b \|}_2 \]<br class="zbmathjax-paragraph">by direct methods, such as QR decomposition, in $O(nd^2)$ time.<br class="zbmathjax-paragraph">The authors now present two randomized algorithms with which one can compute an accurate relative-error approximation ${\tilde x}_{\text{opt}}$ to the Euclidian norm solution vector $x_{\text{opt}}$ for a large class of overdetermined least-squares problems, i.e., for $n \gg d$, $n < e^d$, in $O(nd \ln d)$ time.<br class="zbmathjax-paragraph">In particular, the authors prove the following main theorem (Theorem 1 in the paper):<br class="zbmathjax-paragraph">Suppose (1), and let $\varepsilon \in (0,1)$. Then, there exists a randomized algorithm that returns a vector ${\tilde x}_{\text{opt}} \in \mathbb{R}^d$ with probability at least 0.8, such that the following two claims hold: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">(a)</dt><dd class="zbmathjax-item1">${\|A{{\tilde{x}}_{\text{opt}}} - b \|}_2 \leq (1 + \varepsilon) Z$, </dd><dt class="zbmathjax-item1">(b)</dt><dd class="zbmathjax-item1">$\| x_{\text{opt}} - {\tilde x}_{\text{opt}}\|_2 \leq \sqrt{(\varepsilon)} \kappa(A) \sqrt{{\gamma}^{-2}-1} \|x_{\text{opt}}\|_2$, </dd></dl><br class="zbmathjax-paragraph">where $\kappa(A)$ is condition number of $A$ and $\gamma \in [0,1]$ is the fraction of the norm of $b$ that lies in the column space of $A$.<br class="zbmathjax-paragraph">Then the authors formulate the two randomized algorithms and prove that the claims of the main theorem are fulfilled. Both algorithms preprocess the data with the randomized Hadamard transform, introduced by <span class="zbmathjax-textit">N. Ailon</span> and <span class="zbmathjax-textit">B. Chazelle</span> [“Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform,” Proceedings of the 38th Annual Symposium on Theory of Computing, 557–563 (2006)].<br class="zbmathjax-paragraph">Roughly speaking, after the multiplication of $A$ and $b$ by the matrix product of the randomized Hadamard transform the first algorithm samples uniformly $r$ constraints from the preprocessed problem and exactly solves the smaller $r \times d$ least-squares problem getting the approximate solution ${\tilde x}_{\text{opt}}$ that fulfills the claims of the main theorem. A formula for $r$ is provided.<br class="zbmathjax-paragraph">The second algorithm also constructs a smaller problem using a projection-based randomized algorithm with the help of a well-defined sparse matrix (sparse projection). <br class="zbmathjax-paragraph">It should be mentioned that both algorithms practically compute only those rows of the preprocessed matrix and only those elements of the right-hand sides that need to be accessed. <br class="zbmathjax-paragraph">For the empirical performance the authors refer to publications in which real world examples are provided using algorithms that are based on the ideas of this paper.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10878">Georg Hebermehl (Berlin)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5851401">Cited in <strong>115</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F20" title="MSC2020">65F20</a> </td> <td class="space"> Numerical solutions to overdetermined systems, pseudoinverses </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoverconstrained+least-squares+problem">overconstrained least-squares problem</a>; <a href="/?q=ut%3Amethod+of+least+squares">method of least squares</a>; <a href="/?q=ut%3Aoptimization">optimization</a>; <a href="/?q=ut%3Arandomized+Hadamard+transform">randomized Hadamard transform</a>; <a href="/?q=ut%3Asampling-based+randomized+algorithm">sampling-based randomized algorithm</a>; <a href="/?q=ut%3Aprojection-based+randomized+algorithm">projection-based randomized algorithm</a>; <a href="/?q=ut%3Aaccurate+relative-error+approximation">accurate relative-error approximation</a>; <a href="/?q=ut%3Aoverdetermined+system">overdetermined system</a>; <a href="/?q=ut%3Adirect+methods">direct methods</a>; <a href="/?q=ut%3AQR+decomposition">QR decomposition</a></div> <div class="software"> <h3>Software:</h3><a href="/software/9210">Blendenpik</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 1218.65037" data-ciurl="/ci/05851401" data-biburl="/bibtex/05851401.bib" data-amsurl="/amsrefs/05851401.bib" data-xmlurl="/xml/05851401.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05851401.pdf" title="Zbl 1218.65037 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/10.1007/s00211-010-0331-6" aria-label="DOI for “Faster least squares approximation”" title="10.1007/s00211-010-0331-6">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/0710.1435"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">Ailon, N., Chazelle, B.: Approximate nearest neighbors and the fast Johnson–Lindenstrauss transform. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 557–563 (2006) · <a href="/1301.68232" class="nowrap">Zbl 1301.68232</a></td> </tr><tr> <td>[2]</td> <td class="space">Ailon, N., Liberty, E.: Fast dimension reduction using Rademacher series on dual BCH codes. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1–9, (2008) · <a href="/1192.68334" class="nowrap">Zbl 1192.68334</a></td> </tr><tr> <td>[3]</td> <td class="space">Arora, S., Hazan, E., Kale, S.: A fast random sampling algorithm for sparsifying matrices. In: Proceedings of the 10th International Workshop on Randomization and Computation, pp. 272–279 (2006) · <a href="/1155.68567" class="nowrap">Zbl 1155.68567</a></td> </tr><tr> <td>[4]</td> <td class="space">Avron, H., Maymounkov, P., Toledo, S.: Blendenpik: Supercharging LAPACK’s least-squares solver. 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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>

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