{"id":621057,"date":"2014-06-11T09:00:06","date_gmt":"2014-06-11T16:00:06","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=621057"},"modified":"2019-11-13T13:55:09","modified_gmt":"2019-11-13T21:55:09","slug":"provable-tensor-factorization-with-missing-data","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/provable-tensor-factorization-with-missing-data\/","title":{"rendered":"Provable Tensor Factorization with Missing Data"},"content":{"rendered":"

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode n<\/em>\u00d7n<\/em>\u00d7n<\/em> dimensional rank-r <\/em>tensor exactly from O<\/span><\/em>(<\/span>n<\/span><\/em>3<\/span>\/<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>r<\/span><\/em>5 <\/span><\/span>log<\/span>4 <\/span><\/span><\/span>n<\/span><\/em>)<\/span><\/span> randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in the process of analyzing the initialization step, we prove a generalization of a celebrated result by Szemer\u00e9die et al. on the spectrum of random graphs. Next, we prove global convergence of alternating minimization with a good initialization. Simulations suggest that the dependence of the sample size on dimensionality n<\/em> is indeed tight.<\/p>\n","protected":false},"excerpt":{"rendered":"

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13556],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-621057","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-artificial-intelligence","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2014-11-1","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/arxiv.org\/abs\/1406.2784","label_id":"243109","label":0}],"msr_related_uploader":"","msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"Prateek Jain","user_id":33278,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Prateek Jain"},{"type":"text","value":"S. Oh","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[171330],"publication":[],"video":[],"download":[],"msr_publication_type":"inproceedings","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/621057"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/621057\/revisions"}],"predecessor-version":[{"id":621060,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/621057\/revisions\/621060"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=621057"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=621057"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=621057"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=621057"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=621057"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=621057"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=621057"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=621057"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=621057"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=621057"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=621057"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=621057"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=621057"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=621057"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=621057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}