{"id":509798,"date":"2017-12-03T00:00:15","date_gmt":"2017-12-03T08:00:15","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=509798"},"modified":"2018-10-05T12:14:46","modified_gmt":"2018-10-05T19:14:46","slug":"iterative-collaborative-filtering-for-sparse-matrix-estimation","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/iterative-collaborative-filtering-for-sparse-matrix-estimation\/","title":{"rendered":"Iterative Collaborative Filtering for Sparse Matrix Estimation"},"content":{"rendered":"

The sparse matrix estimation problem consists of estimating the distribution of an\u00a0n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span>\u00a0matrix\u00a0Y<\/span><\/span><\/span><\/span>, from a sparsely observed single instance of this matrix where the entries of\u00a0Y<\/span><\/span><\/span><\/span>\u00a0are independent random variables. This captures a wide array of problems; special instances include matrix completion in the context of recommendation systems, graphon estimation, and community detection in (mixed membership) stochastic block models. Inspired by classical collaborative filtering for recommendation systems, we propose a novel iterative, collaborative filtering-style algorithm for matrix estimation in this generic setting. We show that the mean squared error (MSE) of our estimator converges to\u00a00<\/span><\/span><\/span><\/span>\u00a0at the rate of\u00a0O<\/span>(<\/span>d<\/span>2<\/span><\/span>(<\/span>p<\/span>n<\/span>)<\/span>\u2212<\/span>2<\/span>\/<\/span><\/span><\/span>5<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u00a0as long as\u00a0\u03c9<\/span>(<\/span>d<\/span>5<\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span>\u00a0random entries from a total of\u00a0n<\/span>2<\/span><\/span><\/span><\/span><\/span>\u00a0entries of\u00a0Y<\/span><\/span><\/span><\/span>\u00a0are observed (uniformly sampled),\u00a0E<\/span><\/span><\/span>[<\/span>Y<\/span>]<\/span><\/span><\/span><\/span>\u00a0has rank\u00a0d<\/span><\/span><\/span><\/span>, and the entries of\u00a0Y<\/span><\/span><\/span><\/span>\u00a0have bounded support. The maximum squared error across all entries converges to\u00a00<\/span><\/span><\/span><\/span>\u00a0with high probability as long as we observe a little more,\u00a0\u03a9<\/span>(<\/span>d<\/span>5<\/span><\/span>n<\/span>ln<\/span>2<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span>)<\/span><\/span><\/span><\/span>\u00a0entries. Our results are the best known sample complexity results in this generality.<\/p>\n","protected":false},"excerpt":{"rendered":"

The sparse matrix estimation problem consists of estimating the distribution of an\u00a0n\u00d7n\u00a0matrix\u00a0Y, from a sparsely observed single instance of this matrix where the entries of\u00a0Y\u00a0are independent random variables. This captures a wide array of problems; special instances include matrix completion in the context of recommendation systems, graphon estimation, and community detection in (mixed membership) stochastic […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2017-12-03","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1712.00710","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-509798","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2017-12-03","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":"https:\/\/arxiv.org\/abs\/1712.00710","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"https:\/\/arxiv.org\/abs\/1712.00710","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1712.00710"}],"msr-author-ordering":[{"type":"user_nicename","value":"Christian Borgs","user_id":31278,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Christian Borgs"},{"type":"user_nicename","value":"Jennifer Chayes","user_id":32200,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Jennifer Chayes"},{"type":"text","value":"Christina E. 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