
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
2011 26th Annual IEEE Conference on Computational Complexity
On the Minimal Fourier Degree of Symmetric Boolean Functions
San Jose, California USA
June 08June 11
ISBN: 9780769544113
ASCII Text  x  
Amir Shpilka, Avishay Tal, "On the Minimal Fourier Degree of Symmetric Boolean Functions," 2012 IEEE 27th Conference on Computational Complexity, pp. 200209, 2011 26th Annual IEEE Conference on Computational Complexity, 2011.  
BibTex  x  
@article{ 10.1109/CCC.2011.16, author = {Amir Shpilka and Avishay Tal}, title = {On the Minimal Fourier Degree of Symmetric Boolean Functions}, journal ={2012 IEEE 27th Conference on Computational Complexity}, volume = {0}, year = {2011}, issn = {10930159}, pages = {200209}, doi = {http://doi.ieeecomputersociety.org/10.1109/CCC.2011.16}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2012 IEEE 27th Conference on Computational Complexity TI  On the Minimal Fourier Degree of Symmetric Boolean Functions SN  10930159 SP200 EP209 A1  Amir Shpilka, A1  Avishay Tal, PY  2011 KW  Fourier spectrum KW  symmetric functions KW  learning juntas VL  0 JA  2012 IEEE 27th Conference on Computational Complexity ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.16
In this paper we give a new upper bound on the minimal degree of a nonzero Fouriercoefficient in any nonlinear symmetric Boolean function. Specifically, we prove that forevery nonlinear and symmetric $f:\B^{k}\to\B$ there exists a set $\emptyset\neqS\subset[k]$ such that $S=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S)\neq0$, where$\Gamma(m)\leq m^{0.525}$ is the largest gap between consecutive prime numbers in$\{1,\ldots,m\}$. As an application we obtain a new analysis of the PAC learningalgorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [JCSS,2004]. Namely, we show that the running time of their algorithm is at most$n^{O(k^{0.525})}\cdot\poly(n,2^{k},\log(1/\delta))$ where $n$ is the number ofvariables, $k$ is the size of the junta (i.e. number of relevant variables) and $\delta$is the error probability. In particular, for $k\geq\log(n)^{1/(10.525)}\approx\log(n)^{2.1}$ our analysis matches the lower bound $2^k$ (up to polynomial factors).\sloppy Our bound on the degree greatly improves the previous result of Kolountzakis etal. [Combinatorica, 2009] who proved that $S=O(k/\log k)$.
Index Terms:
Fourier spectrum, symmetric functions, learning juntas
Citation:
Amir Shpilka, Avishay Tal, "On the Minimal Fourier Degree of Symmetric Boolean Functions," ccc, pp.200209, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
Usage of this product signifies your acceptance of the Terms of Use.