Document Zbl 08010468

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Process. Lett.</a> <a href="/?q=in%3A525636" title="Articles in this Issue">189, Article ID 106542, 7 p. (2025)</a>. </div> <div class="abstract">Summary: Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work by the author in [ibid. 169, Article ID 106115, 7 p. (2021; <a href="/1516.68036">Zbl 1516.68036</a>)] explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P20" title="MSC2020">68P20</a> </td> <td class="space"> Information storage and retrieval of data </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P10" title="MSC2020">68P10</a> </td> <td class="space"> Searching and sorting </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T09" title="MSC2020">68T09</a> </td> <td class="space"> Computational aspects of data analysis and big data </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ainformation+retrieval">information retrieval</a>; <a href="/?q=ut%3Aconcentration+of+measure">concentration of measure</a>; <a href="/?q=ut%3Anearest+neighbor+search">nearest neighbor search</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1516.68036">Zbl 1516.68036</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 08010468" data-ciurl="/ci/08010468" data-biburl="/bibtex/08010468.bib" data-amsurl="/amsrefs/08010468.bib" data-xmlurl="/xml/08010468.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/08010468.pdf" title="Zbl 08010468 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.ipl.2024.106542" aria-label="DOI for “Instability results for cosine-dissimilarity-based nearest neighbor search on high dimensional Gaussian data”" title="10.1016/j.ipl.2024.106542">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Aryal, S.; Ting, K.; Haffari, G.; Washio, T., mp-Dissimilarity: a data dependent dissimilarity measure, (Proceedings of the IEEE Conference on Data Mining (ICDM), 2014), 707-712</td> </tr><tr> <td>[2]</td> <td class="space">Aryal, S.; Ting, K.; Washio, T.; Haffari, G., A comparative study of data-dependent approaches without learning in measuring similarities of data objects, Data Min. Knowl. Discov., 34, 124-162, 2020 · <a href="/1458.68198" class="nowrap">Zbl 1458.68198</a></td> </tr><tr> <td>[3]</td> <td class="space">Batir, N., Bounds for the gamma function, Result. Math., 72, 865-874, 2017 · <a href="/1371.33003" class="nowrap">Zbl 1371.33003</a></td> </tr><tr> <td>[4]</td> <td class="space">Beyer, K.; Goldstein, J.; Ramakrishnan, R.; Shaft, U., When is ‘nearest neighbor’ meaningful?, (Beeri, C.; Buneman, P., Database Theory — ICDT’99. Database Theory — ICDT’99, Lecture Notes in Computer Science, vol. 1540, 1999, Springer-Verlag), 217-235</td> </tr><tr> <td>[5]</td> <td class="space">Giannella, C., Instability results for Euclidean distance, nearest neighbor search on high dimensional Gaussian data, Inf. Process. Lett., 169, Article 106115 pp., 2021 · <a href="/1516.68036" class="nowrap">Zbl 1516.68036</a></td> </tr><tr> <td>[6]</td> <td class="space">Henzi, A.; Dümbgen, L., Some new inequalities for beta distributions, Stat. Probab. Lett., 195, Article 109783 pp., 2023 · <a href="/1512.62024" class="nowrap">Zbl 1512.62024</a></td> </tr><tr> <td>[7]</td> <td class="space">Kaban, A., Non-parametric detection of meaningless distances in high dimensional data, Stat. Comput., 22, 2, 375-385, 2012 · <a href="/1322.62145" class="nowrap">Zbl 1322.62145</a></td> </tr><tr> <td>[8]</td> <td class="space">Li, S., Concise formulas for the area and volume of a hyperspherical cap, Asian J. Math. Stat., 4, 1, 66-70, 2011</td> </tr><tr> <td>[9]</td> <td class="space">Mansouri, J.; Khademi, M., Multiplicative distance: a method to alleviate distance instability for high-dimensional data, Knowl. Inf. Syst., 45, 783-805, 2015</td> </tr><tr> <td>[10]</td> <td class="space">Manning, C.; Raghavan, P.; Schutze, H., An Introduction to Information Retrieval, 2008, Cambridge University Press · <a href="/1160.68008" class="nowrap">Zbl 1160.68008</a></td> </tr><tr> <td>[11]</td> <td class="space">Pestov, V., On the geometry of similarity search: dimensionality curse and concentration of measure, Inf. Process. Lett., 73, 1-2, 47-51, 2000 · <a href="/1339.68245" class="nowrap">Zbl 1339.68245</a></td> </tr><tr> <td>[12]</td> <td class="space">Petrov, V., On lower bounds for tail probabilities, J. Stat. Plan. Inference, 137, 2703-2705, 2007 · <a href="/1120.60016" class="nowrap">Zbl 1120.60016</a></td> </tr><tr> <td>[13]</td> <td class="space">Ting, K.; Zhu, Y.; Carman, M.; Zhu, Y.; Zhou, Z.-H., Overcoming key weaknesses of distance-based neighbourhood methods using a data dependent dissimilarity measure, (Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), 2016), 1205-1214</td> </tr><tr> <td>[14]</td> <td class="space">Vandaele, R.; Kang, B.; De Bie, T.; Saeys, Y., The curse revisited: when are distances informative for the ground truth in noisy high-dimensional data?, (Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS), 2022), 2158-2172</td> </tr><tr> <td>[15]</td> <td class="space">Vershynin, R., High-Dimensional Probability: An Introduction with Applications to Data Science, 2018, Cambridge University Press · <a href="/1430.60005" class="nowrap">Zbl 1430.60005</a></td> </tr><tr> <td>[16]</td> <td class="space">Zhu, X.; Su, S.; Fu, M.; Liu, J.; Zhu, L.; Yang, W.; Jing, G.; Guo, Y., A cosine similarity algorithm method for fast and accurate monitoring of dynamic droplet generation processes, Sci. Rep., 8, 9967, 2018</td> </tr><tr> <td>[17]</td> <td class="space">Zhu, S.; Wu, J.; Xiong, H.; Xia, G., Scaling up top-K cosine similarity search, Data Knowl. Eng., 70, 1, 60-83, 2011</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>

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