NDLI: The Sign-Rank of AC$^0$

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sherstov, Alexander A. Razborov, Alexander A.
Copyright Year	2010
Abstract	The sign-rank of a matrix $A=[A_{ij}]$ with $\pm1$ entries is the least rank of a real matrix $B=[B_{ij}]$ with $A_{ij}B_{ij}>0$ for all $i,j$. We obtain the first exponential lower bound on the sign-rank of a function in $\mathsf{AC}^0$. Namely, let $f(x,y)=\bigwedge_{i=1,\dots,m}\bigvee_{j=1,\dots,m^2}(x_{ij}\wedge y_{ij})$. We show that the matrix $[f(x,y)]_{x,y}$ has sign-rank $\exp(\Omega(m))$. This in particular implies that $\Sigma_2^{cc}\not\subseteq\mathsf{UPP}^{cc}$, which solves a longstanding open problem in communication complexity posed by Babai, Frankl, and Simon [Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), 1986, pp. 337347]. Our result additionally implies a lower bound in learning theory. Specifically, let $\phi_1,\dots,\phi_r:\{0,1\}^n\to\mathbb{R}$ be functions such that every DNF formula $f:\{0,1\}^n\to\{-1,+1\}$ of polynomial size has the representation $f\equiv\mathrm{sgn}(a_1\phi_1+\dots+a_r\phi_r)$ for some reals $a_1,\dots,a_r$. We prove that then $r\geqslant\exp(\Omega(n^{1/3}))$, which essentially matches an upper bound of $\exp(\tilde{O}(n^{1/3}))$, due to Klivans and Servedio [J. Comput. System Sci., 68 (2004), pp. 303318]. Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in $\mathsf{AC}^0$. This substantially generalizes and strengthens the results of Krause and Pudlk [Theoret. Comput. Sci., 174 (1997), pp. 137156].
Starting Page	1833
Ending Page	1855
Page Count	23
File Format	PDF
ISSN	00975397
DOI	10.1137/080744037
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-01-22
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems constant-depth AND/OR/NOT circuits sign-rank and $\mathsf{UPP}^{cc}$ Complexity of computation $\Pi_2^{cc}$ complexity classes $\Sigma_2^{cc}$ communication complexity Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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