{"id":186061,"date":"2011-03-23T00:00:00","date_gmt":"2011-03-24T17:51:02","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/higher-order-principal-components-complexity-and-applications\/"},"modified":"2016-08-22T11:32:55","modified_gmt":"2016-08-22T18:32:55","slug":"higher-order-principal-components-complexity-and-applications","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/higher-order-principal-components-complexity-and-applications\/","title":{"rendered":"Higher Order Principal Components: Complexity and Applications"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Standard Principal Components are directions that optimize second moments of a given set of points and have proven to be a powerful tool. Here we consider higher order principal components, i.e., directions that optimize higher moments of a data set (or the spectral norm of higher-dimensional arrays). They appear to be much less structured \u2014 there could be exponentially many, need not be pairwise orthogonal and it is NP-hard to find global maxima for arbitrary inputs.<br \/>\nWe discuss applications to combinatorial optimization and learning: (a) finding a planted clique in a random graph, where higher-order maxima even for semi-random inputs would be effective and (b) learning an unknown function of a low-dimensional subspace from labeled examples (a k-subspace junta, generalizing the well-known class of k-juntas), where *local* optima suffice and can be approximated efficiently for a wide class of input distributions.<br \/>\nMost of the talk is joint work with Ying Xiao.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Standard Principal Components are directions that optimize second moments of a given set of points and have proven to be a powerful tool. Here we consider higher order principal components, i.e., directions that optimize higher moments of a data set (or the spectral norm of higher-dimensional arrays). They appear to be much less structured \u2014 [&hellip;]<\/p>\n","protected":false},"featured_media":196051,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-186061","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/zVlIUBJ3S_s","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186061"}],"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\/186061\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/196051"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=186061"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=186061"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=186061"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=186061"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=186061"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=186061"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}