{"version":"1.0","provider_name":"Microsoft Research","provider_url":"https:\/\/www.microsoft.com\/en-us\/research","author_name":"Jeff Running","author_url":"https:\/\/www.microsoft.com\/en-us\/research\/people\/jeffrunn\/","title":"Spectral Sparsification of Graphs and Approximations of Matrices - Microsoft Research","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"bSMJlx0frt\"><a href=\"https:\/\/www.microsoft.com\/en-us\/research\/video\/spectral-sparsification-of-graphs-and-approximations-of-matrices\/\">Spectral Sparsification of Graphs and Approximations of Matrices<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/video\/spectral-sparsification-of-graphs-and-approximations-of-matrices\/embed\/#?secret=bSMJlx0frt\" width=\"600\" height=\"338\" title=\"&#8220;Spectral Sparsification of Graphs and Approximations of Matrices&#8221; &#8212; Microsoft Research\" data-secret=\"bSMJlx0frt\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/08\/EFU4QtmLoDE.jpg","thumbnail_width":480,"thumbnail_height":360,"description":"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;]"}