{"id":185738,"date":"2010-07-28T00:00:00","date_gmt":"2010-12-28T09:39:12","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/spectral-sparsification-of-graphs-and-approximations-of-matrices\/"},"modified":"2016-08-22T11:27:52","modified_gmt":"2016-08-22T18:27:52","slug":"spectral-sparsification-of-graphs-and-approximations-of-matrices","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/spectral-sparsification-of-graphs-and-approximations-of-matrices\/","title":{"rendered":"Spectral Sparsification of Graphs and Approximations of Matrices"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We introduce a notion of what it means for one graph to be a good spectral approximation of another.  This induces the problem of spectral sparsification: finding a sparse graph that is a good spectral approximation of a given graph.<\/p>\n<p>It turns out that Ramanujan expanders are the best sparse spectral approximations of complete graphs.  We show that every graph may be approximated almost as well by a sparse graph as the complete graphs are by Ramanujan graphs.  We also present an efficient randomized algorithm for construction sparse approximations which only uses a logarithmic factor more edges than optimal.<\/p>\n<p>Our algorithms follow from the solution of a problem in linear algebra.  Given a rank-n symmetric matrix A that is expressed as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.<\/p>\n<p>This is joint work with Joshua Batson, Nikhil Srivastava and Shang-Hua Teng.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We introduce a notion of what it means for one graph to be a good spectral approximation of another. This induces the problem of spectral sparsification: finding a sparse graph that is a good spectral approximation of a given graph. It turns out that Ramanujan expanders are the best sparse spectral approximations of complete graphs. [&hellip;]<\/p>\n","protected":false},"featured_media":280784,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[13561],"msr-video-type":[],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-185738","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/EFU4QtmLoDE","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185738"}],"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\/185738\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/280784"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185738"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185738"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185738"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185738"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185738"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}