The Sign-Rank of AC^O

The sign-rank of a matrix A = [A
_ij
] with plusmn1 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 AC
⁰
. Namely, let f(x, y) = Lambda
_i=1

^m
Lambda
_j=1

<sup>m

2</sup>
(x
_ij
Lambda y
_ij
). We show that the matrix [f(x, y)]
<sub>x,

y</sub>
has sign-rank 2
^Omega(m)
. This in particular implies that Sigma
₂

^cc
nsubeUPP
^cc
, which solves a long-standing open problem posed by Babai, Frankl, and Simon (1986). Our result additionally implies a lower bound in learning theory. Specifically, let Phi
₁
,..., Phi
_r
: {0, 1}
ⁿ
rarrRopf be functions such that every DNF formula f : {0, 1}
ⁿ
rarr {-1, +1} of polynomial size has the representation f equiv sign(a
₁
Phi
₁
+ hellip + a
_r
Phi
_r
) for some reals a
₁
,..., a
_r
. We prove that then r ges 2
<sup>Omega(n

1/3

)</sup>
, which essentially matches an upper bound of 2
<sup>Otilde(n

1/3

)</sup>
due to Klivans and Servedio (2001). Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in AC
⁰
. This substantially generalizes and strengthens the results of Krause and Pudlak (1997).