TY - JOUR

T1 - On the minimal fourier degree of symmetric Boolean functions

AU - Shpilka, Amir

AU - Tal, Avishay

N1 - Funding Information:
∗ A preliminary version appeared in CCC 2011 [13]. † This research was partially supported by the Israel Science Foundation (grant number 339/10). ‡ This research was partially supported by the Israel Science Foundation (grant number 339/10).

PY - 2014/6

Y1 - 2014/6

N2 - In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1}k → {0, 1} there exists a set (Formula presented) such that {pipe}S{pipe} = O(Γ(k)+√k, and f(S)̂ ≠ 0 where Γ(m)≤m 0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10]. Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that {pipe}S{pipe}=O(k=log k). We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.

AB - In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1}k → {0, 1} there exists a set (Formula presented) such that {pipe}S{pipe} = O(Γ(k)+√k, and f(S)̂ ≠ 0 where Γ(m)≤m 0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10]. Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that {pipe}S{pipe}=O(k=log k). We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.

UR - http://www.scopus.com/inward/record.url?scp=84903217065&partnerID=8YFLogxK

U2 - 10.1007/s00493-014-2875-z

DO - 10.1007/s00493-014-2875-z

M3 - מאמר

AN - SCOPUS:84903217065

VL - 34

SP - 359

EP - 377

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 3

ER -