{"id":185950,"date":"2011-02-23T00:00:00","date_gmt":"2011-02-25T15:45:06","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/on-the-fourier-spectrum-of-symmetric-boolean-functions\/"},"modified":"2016-08-22T11:31:13","modified_gmt":"2016-08-22T18:31:13","slug":"on-the-fourier-spectrum-of-symmetric-boolean-functions","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/on-the-fourier-spectrum-of-symmetric-boolean-functions\/","title":{"rendered":"On the Fourier Spectrum of Symmetric Boolean Functions"},"content":{"rendered":"<div class=\"asset-content\">\n<p>It is well-known that any Boolean function f:-1,+1<sup>n<\/sup> to  -1,+1 can be written uniquely as a polynomial f(x) = sum<sub>S subset [n]<\/sub> f<sub>s<\/sub> prod<sub>i in S<\/sub> x<sub>i<\/sub>. The collection of coefficients (f<sub>S<\/sub>&#8216;s) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in modern computer science by converting combinatorial and algorithmic questions about f into algebraic or analytic questions about the spectrum.<\/p>\n<p>In this talk I will focus on a basic feature of the Fourier spectrum, namely the minimal Fourier degree, or the size of the smallest non-empty set S such that f<sub>S<\/sub> is non-zero. For every symmetric function *except the parity function* we show that the minimal Fourier degree is at most O(Gamma(n)) where Gamma(m)  m<sup>0.525<\/sup> is the largest gap between consecutive prime numbers in 1,&#8230;,m.This improves the previous result of Kolountzakis et al. (Combinatorica &#8217;09) who showed that the minimal Fourier degree is at most k\/log k.<\/p>\n<p>As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. (STOC &#8217;03).<\/p>\n<p>This is a joint work with Avishay Tal.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>It is well-known that any Boolean function f:-1,+1n to -1,+1 can be written uniquely as a polynomial f(x) = sumS subset [n] fs prodi in S xi. The collection of coefficients (fS&#8216;s) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in [&hellip;]<\/p>\n","protected":false},"featured_media":195999,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[13561],"msr-video-type":[],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-185950","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/K_kHwshYs8A","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185950"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185950\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195999"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185950"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185950"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185950"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185950"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185950"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}