{"id":184909,"date":"2010-05-28T00:00:00","date_gmt":"2010-06-03T19:31:56","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/kahn-kalai-linial-and-kruskal-katona\/"},"modified":"2016-08-22T11:28:02","modified_gmt":"2016-08-22T18:28:02","slug":"kahn-kalai-linial-and-kruskal-katona","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/kahn-kalai-linial-and-kruskal-katona\/","title":{"rendered":"Kahn-Kalai-Linial and Kruskal-Katona"},"content":{"rendered":"<div class=\"asset-content\">\n<p>The notable KKL Theorem of Kahn, Kalai, and Linial says that if f : 0,1<sup>n<\/sup> &#8211; 0,1 is a balanced boolean function, then there is at least one coordinate i with &#8220;influence&#8221; at least Omega(log n \/ n).<br \/>\nWe generalize this result to boolean functions on Cayley and Schreier graphs, whenever the generating set is a union of conjugacy classes and the associated Markov chain has a good log-Sobolev constant.  E.g., we show that if f is a balanced Boolean function on the set of n-bit strings with Hamming weight n\/2, then there exists some pair of indices i, j such that the influence on f of swapping  the ith and jth coordinates is at least Omega(log n \/ n).   As an application we prove a new &#8220;robustification&#8221; of the classical Kruskal-Katona Theorem from combinatorics.  We also discuss an application in computational learning theory. This talk is based on  two joint works with Karl Wimmer of Duquesne University.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The notable KKL Theorem of Kahn, Kalai, and Linial says that if f : 0,1n &#8211; 0,1 is a balanced boolean function, then there is at least one coordinate i with &#8220;influence&#8221; at least Omega(log n \/ n). We generalize this result to boolean functions on Cayley and Schreier graphs, whenever the generating set is [&hellip;]<\/p>\n","protected":false},"featured_media":195641,"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-184909","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/iFDleV9bzTQ","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184909","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\/184909\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195641"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184909"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184909"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184909"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184909"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184909"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184909"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184909"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184909"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184909"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184909"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}