{"id":184058,"date":"2005-07-13T00:00:00","date_gmt":"2009-10-31T13:20:15","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/singularity-of-random-bernoulli-matrices\/"},"modified":"2016-09-09T09:51:38","modified_gmt":"2016-09-09T16:51:38","slug":"singularity-of-random-bernoulli-matrices","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/singularity-of-random-bernoulli-matrices\/","title":{"rendered":"Singularity of random Bernoulli matrices"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let M<sub>n<\/sub> be an n by n random matrix, whose entries are i.i.d. Bernoulli random variables (taking  value 1 and -1 with probability half). Let p<sub>n<\/sub> be the probability that M<sub>n<\/sub> is singular. It has been conjectured for sometime that p<sub>n<\/sub>= (1\/2+o(1))<sup>n<\/sup> (basically the probability that there are two equal rows).<\/p>\n<p>Komlos showed, back in the 60s, that p<sub>n<\/sub>=o(1). Later he proved that p<sub>n<\/sub>=O(n<sup>-1\/2<\/sup>).<br \/>\nA breakthrough result of Kahn, Komlos and Szemeredi in early 90s gives p<sub>n<\/sub>= O(.999<sup>n<\/sup>). In this talk, we present a new result which improves the bound to (3\/4+o(1))<sup>n<\/sup>. The new main ingredient in this work is the so-called &#8220;inverse&#8221; technique from additive number theory.<\/p>\n<p>Joint work with T. Tao (UCLA)<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let Mn be an n by n random matrix, whose entries are i.i.d. Bernoulli random variables (taking value 1 and -1 with probability half). Let pn be the probability that Mn is singular. It has been conjectured for sometime that pn= (1\/2+o(1))n (basically the probability that there are two equal rows). Komlos showed, back in [&hellip;]<\/p>\n","protected":false},"featured_media":195301,"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-184058","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/l_zZW5HsgI4","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184058","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\/184058\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195301"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184058"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184058"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184058"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184058"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184058"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184058"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184058"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184058"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184058"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}