{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9383","name":"Testing k-Monotonicity","abstract":"A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions.\r\n\r\nMotivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. \r\n\r\nOur results include the following:\r\n\r\n1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}^d, for k >= 3;\r\n2. We demonstrate a separation between testing and learning on {0,1}^d, for k=\\omega(\\log d): testing k-monotonicity can be performed with 2^{O(\\sqrt d . \\log d . \\log{1\/\\eps})} queries, while learning k-monotone functions requires 2^{\\Omega(k . \\sqrt d .{1\/\\eps})} queries (Blais et al. (RANDOM 2015)).\r\n3. We present a tolerant test for functions f\\colon[n]^d\\to \\{0,1\\}$with complexity independent of n, which makes progress on a problem left open by Berman et al. (STOC 2014). \r\n\r\nOur techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.\r\n\r\n Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.","keywords":["Boolean Functions","Learning","Monotonicity","Property Testing"],"author":[{"@type":"Person","name":"Canonne, Cl\u00e9ment L.","givenName":"Cl\u00e9ment L.","familyName":"Canonne"},{"@type":"Person","name":"Grigorescu, Elena","givenName":"Elena","familyName":"Grigorescu"},{"@type":"Person","name":"Guo, Siyao","givenName":"Siyao","familyName":"Guo"},{"@type":"Person","name":"Kumar, Akash","givenName":"Akash","familyName":"Kumar"},{"@type":"Person","name":"Wimmer, Karl","givenName":"Karl","familyName":"Wimmer"}],"position":29,"pageStart":"29:1","pageEnd":"29:21","dateCreated":"2017-11-28","datePublished":"2017-11-28","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Canonne, Cl\u00e9ment L.","givenName":"Cl\u00e9ment L.","familyName":"Canonne"},{"@type":"Person","name":"Grigorescu, Elena","givenName":"Elena","familyName":"Grigorescu"},{"@type":"Person","name":"Guo, Siyao","givenName":"Siyao","familyName":"Guo"},{"@type":"Person","name":"Kumar, Akash","givenName":"Akash","familyName":"Kumar"},{"@type":"Person","name":"Wimmer, Karl","givenName":"Karl","familyName":"Wimmer"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2017.29","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/sublinear.info\/70","isPartOf":{"@type":"PublicationVolume","@id":"#volume6270","volumeNumber":67,"name":"8th Innovations in Theoretical Computer Science Conference (ITCS 2017)","dateCreated":"2017-11-28","datePublished":"2017-11-28","editor":{"@type":"Person","name":"Papadimitriou, Christos H.","givenName":"Christos H.","familyName":"Papadimitriou"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9383","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6270"}}}