We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set \\(T\\) of \\(n\\) points in \\(R^d\\) and a parameter \\(0<\u03b1<{1 \\over 2}\\) such that an \\(\u03b1\\)-fraction of the points in \\(T\\) are i.i.d. samples from a well-behaved distribution \\(D\\) and the remaining \\((1\u2212\u03b1)\\)-fraction are arbitrary. The goal is to output a small list of vectors, at least one of which is close to the mean of \\(D\\). We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees, with running time \\(O(n^{1+\u03f5_0}d)\\), for any fixed \\(\u03f5_0>0\\). All prior algorithms for this problem had additional polynomial factors in \\(1 \\over \u03b1\\). We leverage this result, together with additional techniques, to obtain the first almost-linear time algorithms for clustering mixtures of \\(k\\) separated well-behaved distributions, nearly-matching the statistical guarantees of spectral methods. Prior clustering algorithms inherently relied on an application of \\(k\\)-PCA, thereby incurring runtimes of \\(\u03a9(ndk)\\). This marks the first runtime improvement for this basic statistical problem in nearly two decades.<\/p>\n
The starting point of our approach is a novel and simpler near-linear time robust mean estimation algorithm in the \\(\u03b1\u21921\\) regime, based on a one-shot matrix multiplicative weights-inspired potential decrease. We crucially leverage this new algorithmic framework in the context of the iterative multi-filtering technique of Diakonikolas et al. ’18, ’20, providing a method to simultaneously cluster and downsample points using one-dimensional projections — thus, bypassing the \\(k\\)-PCA subroutines required by prior algorithms.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set of points in and a parameter such that an -fraction of the points in are i.i.d. samples from a well-behaved distribution and the remaining -fraction are arbitrary. The goal is to […]<\/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":"STOC 2022","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":"2022-6-1","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"http:\/\/acm-stoc.org\/stoc2022\/","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":[13561,13556],"msr-publication-type":[193716],"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-850573","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":"2022-6-1","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\/2106.08537","label_id":"243109","label":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":[],"msr-author-ordering":[{"type":"text","value":"Ilias Diakonikolas","user_id":0,"rest_url":false},{"type":"text","value":"Daniel M. 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