{"id":152409,"date":"1999-11-01T00:00:00","date_gmt":"1999-11-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/estimating-the-support-of-a-high-dimensional-distribution\/"},"modified":"2018-10-16T19:57:23","modified_gmt":"2018-10-17T02:57:23","slug":"estimating-the-support-of-a-high-dimensional-distribution","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/estimating-the-support-of-a-high-dimensional-distribution\/","title":{"rendered":"Estimating the Support of a High-Dimensional Distribution"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a &#8220;simple&#8221; subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified v between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a preliminary theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a &#8220;simple&#8221; subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified v between 0 and 1. We propose a [&hellip;]<\/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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13563],"msr-publication-type":[193718],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-152409","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-data-platform-analytics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"1999-11-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"30","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"MSR-TR-99-87","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":"211113","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"tr-99-87.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/tr-99-87.pdf","id":211113,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":211113,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/tr-99-87.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"jplatt","user_id":32416,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=jplatt"},{"type":"text","value":"Bernhard Sch\u00f6lkopf","user_id":0,"rest_url":false},{"type":"text","value":"John Shawe-Taylor","user_id":0,"rest_url":false},{"type":"text","value":"Alex J. Smola","user_id":0,"rest_url":false},{"type":"text","value":"Robert C. Williamson","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[169805],"publication":[],"video":[],"download":[],"msr_publication_type":"techreport","related_content":{"projects":[{"ID":169805,"post_title":"Support Vector Machines","post_name":"support-vector-machines","post_type":"msr-project","post_date":"2001-11-05 12:17:42","post_modified":"2019-08-14 14:33:07","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/support-vector-machines\/","post_excerpt":"Support vector machines are a set of algorithms that learn from data by creating models that maximize their margin of error. Support vector machines\u00a0(SVMs) are a family of algorithms for\u00a0classification,\u00a0regression,\u00a0transduction, novelty detection, and\u00a0semi-supervised learning. They work by choosing a model that\u00a0maximizes the error margin of a training set. SVMs\u00a0were originally developed by\u00a0Vladimir Vapnik in 1963. Since the mid-90s, a energetic research community has grown around them. If you want to learn more about SVMs, you&hellip;","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-project\/169805"}]}}]},"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/152409","targetHints":{"allow":["GET"]}}],"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\/152409\/revisions"}],"predecessor-version":[{"id":514871,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/152409\/revisions\/514871"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=152409"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=152409"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=152409"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=152409"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=152409"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=152409"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=152409"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=152409"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=152409"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=152409"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=152409"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=152409"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=152409"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=152409"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=152409"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=152409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}