{"id":147878,"date":"2005-06-01T00:00:00","date_gmt":"2005-06-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/on-spectral-learning-of-mixtures-of-distributions\/"},"modified":"2018-10-16T20:23:08","modified_gmt":"2018-10-17T03:23:08","slug":"on-spectral-learning-of-mixtures-of-distributions","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/on-spectral-learning-of-mixtures-of-distributions\/","title":{"rendered":"On Spectral Learning of Mixtures of Distributions"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">We consider the problem of learning mixtures of distributions via spectral methods and derive a characterization of when such methods are useful. Specifically, given a mixture-sample, let \\({\\bar{\\mu }}_i,{\\bar{C}}_i,{\\bar{w}}_i\\) denote the empirical mean, covariance matrix, and mixing weight of the samples from the <em class=\"EmphasisTypeItalic \">i<\/em>-th component. We prove that a very simple algorithm, namely spectral projection followed by single-linkage clustering, properly classifies every point in the sample provided that each pair of means \\({\\bar{\\mu }}_i,{\\bar{\\mu }}_j\\) is well separated, in the sense that \\(\\| {\\bar{\\mu }}_i-{\\bar{\\mu }}_j{\\| }^2\\) is at least \\(\\| {\\bar{C}}_i{\\| }_2(1\/{\\bar{w}}_i+1\/{\\bar{w}}_j)\\) plus a term that depends on the concentration properties of the distributions in the mixture. This second term is very small for many distributions, including Gaussians, Log-concave, and many others. As a result, we get the best known bounds for learning mixtures of arbitrary Gaussians in terms of the required mean separation. At the same time, we prove that there are many Gaussian mixtures {(<em class=\"EmphasisTypeItalic \">\u03bc<\/em> <sub> <em class=\"EmphasisTypeItalic \">i<\/em> <\/sub> ,<em class=\"EmphasisTypeItalic \">C<\/em> <sub> <em class=\"EmphasisTypeItalic \">i<\/em> <\/sub> ,<em class=\"EmphasisTypeItalic \">w<\/em> <sub> <em class=\"EmphasisTypeItalic \">i<\/em> <\/sub>)} such that each pair of means is separated by ||<em class=\"EmphasisTypeItalic \">C<\/em> <sub> <em class=\"EmphasisTypeItalic \">i<\/em> <\/sub>||<sub>2<\/sub>(1\/<em class=\"EmphasisTypeItalic \">w<\/em> <sub> <em class=\"EmphasisTypeItalic \">i<\/em> <\/sub>\u2009+\u20091\/<em class=\"EmphasisTypeItalic \">w<\/em> <sub> <em class=\"EmphasisTypeItalic \">j<\/em> <\/sub>), yet upon spectral projection the mixture collapses completely, i.e., all means and covariance matrices in the projected mixture are identical.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider the problem of learning mixtures of distributions via spectral methods and derive a characterization of when such methods are useful. Specifically, given a mixture-sample, let denote the empirical mean, covariance matrix, and mixing weight of the samples from the i-th component. We prove that a very simple algorithm, namely spectral projection followed by [&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":"","msr-author-ordering":[{"type":"text","value":"Dimitris Achlioptas","user_id":0},{"type":"user_nicename","value":"mcsherry","user_id":"32863"}],"msr_publishername":"Springer","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":"458\u2013469","msr_page_range_start":"458","msr_page_range_end":"469","msr_series":"Lecture Notes in Computer Science","msr_volume":"3559","msr_copyright":"","msr_conference_name":"18th Annual Conference on Learning Theory (COLT 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