{"id":324284,"date":"2016-11-18T15:02:45","date_gmt":"2016-11-18T23:02:45","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=324284"},"modified":"2018-10-16T20:38:31","modified_gmt":"2018-10-17T03:38:31","slug":"optimal-monotonicity-lipschitz-testers-hypercubes-hypergrids","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/optimal-monotonicity-lipschitz-testers-hypercubes-hypergrids\/","title":{"rendered":"Optimal Monotonicity and Lipschitz Testers over Hypercubes and Hypergrids"},"content":{"rendered":"<p>The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f:[k]<sup>n<\/sup> -> R (for some ordered range R). The hypergrid\/cube has a natural partial order given by coordinate-wise ordering, denoted by prec. A function is <i>monotone<\/i> if for all pairs x prec y, f(x) \u2264 f(y). The distance to monotonicity, \u03b5<sub>f<\/sub>, is the minimum fraction of values of f that need to be changed to make f monotone. For k=2 (the boolean hypercube), the usual tester is the <i>edge tester<\/i>, which checks monotonicity on adjacent pairs of domain points. It is known that the edge tester using O(\u03b5<sup>-1<\/sup>n log|R|) samples can distinguish a monotone function from one where \u03b5<sub>f<\/sub> > \u03b5. On the other hand, the best lower bound for monotonicity testing over general R is \u03a9(n). We resolve this long standing open problem and prove that O(n\/\u03b5) samples suffice for the edge tester. For hypergrids, known testers require O(\u03b5<sup>-1<\/sup>n log k log |R|) samples, while the best known (non-adaptive) lower bound is \u03a9(\u03b5<sup>-1<\/sup> n log k). We give a (non-adaptive) monotonicity tester for hypergrids running in O(\u03b5<sup>{-1} n log k)<\/sup> time.<\/p>\n<p>Our techniques lead to optimal property testers (with the same running time) for the natural <i>Lipschitz property<\/i> on hypercubes and hypergrids. (A <i>c<\/i>-Lipschitz function is one where |f(<i>x<\/i>) &#8211; f(<i>y<\/i>)| \u2264 <i>c<\/i>||<i>x-y<\/i>||<sub>1<\/sub>.) In fact, we give a general unified proof for <i>O<\/i>(\u03b5<sup>-1<\/sup><i>n<\/i>log <i>k<\/i>)-query testers for a class of &#8220;bounded-derivative&#8221; properties, a class containing both monotonicity and Lipschitz.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f:[k]n -> R (for some ordered range R). The hypergrid\/cube has a natural partial order given by coordinate-wise ordering, denoted by prec. A function is monotone if for [&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":null,"msr_publishername":"ACM New York, NY, USA","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_editors":"","msr_how_published":"","msr_isbn":"978-1-4503-2029-0","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"419-428","msr_page_range_start":"419","msr_page_range_end":"428","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_doi":"10.1145\/2488608.2488661","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":"2013-06-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488661","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":[13561,13546],"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-324284","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"ACM New York, NY, USA","msr_edition":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_affiliation":"","msr_published_date":"2013-06-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"419-428","msr_chapter":"","msr_isbn":"978-1-4503-2029-0","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":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488661","msr_doi":"10.1145\/2488608.2488661","msr_publication_uploader":[{"type":"url","title":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488661","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1145\/2488608.2488661","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":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488661"}],"msr-author-ordering":[{"type":"user_nicename","value":"dechakr","user_id":31593,"rest_url":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=dechakr"},{"type":"text","value":"C. 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