The Sign-Rank of AC$^0$

• Authors: Alexander A. Razborov;Alexander A. Sherstov
• Venue: SIAM Journal on Computing
• Year: 2010

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. 337-347]. 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. 303-318]. 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 Pudlák [Theoret. Comput. Sci., 174 (1997), pp. 137-156].