{"id":349001,"date":"2017-01-09T02:15:21","date_gmt":"2017-01-09T10:15:21","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=349001"},"modified":"2018-10-16T20:05:02","modified_gmt":"2018-10-17T03:05:02","slug":"embedding-approximately-low-dimensional-l_22-metrics-l_1-2","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/embedding-approximately-low-dimensional-l_22-metrics-l_1-2\/","title":{"rendered":"Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1"},"content":{"rendered":"<p>Goemans showed that any n points x_1,&#8230;, x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i<j}. Average distortion embedding suffices for applications such as the SPARSEST CUT problem. Our embedding gives an approximation algorithm for the SPARSEST CUT problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, in Proc. 17th APPROX, 2014]. Our ideas give a new perspective on l_{2}^{2} metric, an alternate proof of Goemans&#8217; theorem, and a simpler proof for average distortion sqrt{d}.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Goemans showed that any n points x_1,&#8230;, x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-349001","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik","msr_edition":"36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 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