{"id":186827,"date":"2011-09-29T00:00:00","date_gmt":"2011-10-01T09:04:19","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/sparsest-cut-discrete-differentiation-and-local-rigidity-of-sets-in-the-plane\/"},"modified":"2016-08-22T11:30:20","modified_gmt":"2016-08-22T18:30:20","slug":"sparsest-cut-discrete-differentiation-and-local-rigidity-of-sets-in-the-plane","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/sparsest-cut-discrete-differentiation-and-local-rigidity-of-sets-in-the-plane\/","title":{"rendered":"Sparsest Cut, Discrete Differentiation, and Local Rigidity of Sets in the Plane"},"content":{"rendered":"<div class=\"asset-content\">\n<p>I will briefly recall the connection between the Sparsest Cut problem in graphs and low-distortion embeddings of finite metric spaces into L<sub>1<\/sub>.  Then I will talk about a recent approach to lower bounds pioneered by Cheeger and Kleiner (2006), following a conjecture we made with Naor (2003).  The main idea is to develop a differentiation theory for L<sub>1<\/sub>-valued mappings.  I will discuss a discrete version of this theory (following Eskin-Fisher-Whyte and Lee-Raghavendra).<\/p>\n<p>I will then construct a doubling space whose n-point subsets rqeuire bi-lipschitz distortion ~ (log n)<sup>1\/2<\/sup> to embed into L<sub>1<\/sub>, matching the upper bound of Gupta-Krauthgamer-Lee (2003), and improving over the (log n)<sup>c<\/sup> bound of Cheeger, Kleiner, and Naor (2009).  This leads to nearly tight integrality gaps for some well studied semi-definite program relaxations.  Our lower bound space, developed jointly with Sidiropoulos, takes inspiration from both the 3-dimensional Heisenberg group and the diamond graphs.  The main technical difficulty involves approximately classifying certain weakly regular sets in the plane, a problem in &#8220;approximate&#8221; integral geometry that may be independently interesting.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I will briefly recall the connection between the Sparsest Cut problem in graphs and low-distortion embeddings of finite metric spaces into L1. Then I will talk about a recent approach to lower bounds pioneered by Cheeger and Kleiner (2006), following a conjecture we made with Naor (2003). The main idea is to develop a differentiation [&hellip;]<\/p>\n","protected":false},"featured_media":196399,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-186827","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/08HeyjtwJmg","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186827","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186827\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/196399"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=186827"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=186827"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=186827"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=186827"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=186827"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=186827"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=186827"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=186827"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=186827"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=186827"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}