Let
g:{−1,1}k→{−1,1} be any Boolean function and
q1,…,qk be any degree-2 polynomials over
{−1,1}n. We give a \emph{deterministic} algorithm which, given as input explicit descriptions of
g,q1,…,qk and an accuracy parameter $\eps>0$, approximates \[\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1]\] to within an additive $\pm \eps$. For any constant $\eps > 0$ and
k≥1 the running time of our algorithm is a fixed polynomial in
n. This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits.
Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function $\sign(q(x)),$ corresponding to the
k=1 case of our result.
Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the
k=1 case in \cite{DDS13:deg2count}. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for
k-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution. Finally, a third new ingredient is a for \emph{
k-tuples} of degree-
d polynomial threshold functions. This generalizes both the regularity lemmas of \cite{DSTW:10,HKM:09} and the regularity lemma of Gopalan et al \cite{GOWZ10}. Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.
Let
g:{−1,1}k→{−1,1} be any Boolean function and
q1,…,qk be any degree-2 polynomials over
{−1,1}n. We give a \emph{deterministic} algorithm which, given as input explicit descriptions of
g,q1,…,qk and an accuracy parameter $\eps>0$, approximates \[\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1]\] to within an additive $\pm \eps$. For any constant $\eps > 0$ and
k≥1 the running time of our algorithm is a fixed polynomial in
n. This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits. Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function $\sign(q(x)),$ corresponding to the
k=1 case of our result. Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the
k=1 case in \cite{DDS13:deg2count}. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for
k-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution. Finally, a third new ingredient is a "regularity lemma" for \emph{
k-tuples} of degree-
d polynomial threshold functions. This generalizes both the regularity lemmas of \cite{DSTW:10,HKM:09} and the regularity lemma of Gopalan et al \cite{GOWZ10}. Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.
Let g : {-1, 1}k→ {-1, 1} be any Boolean function and q1, . . . , qkbe any degree-2 polynomials over {-1, 1}n. We give a deterministic algorithm which, given as input explicit descriptions of g, q1, . . ., qkand an accuracy parameter ϵ > 0, approximates Prx~{-1, 1}n[g(sign(q1(x)), . . . , sign(qk(x))) = 1] to within an additive ±ϵ. For any constant ϵ > 0 and k ≥ 1 the running time of our algorithm is a fixed polynomial in n (in fact this is true even for some not-too-small ϵ = 0n(1) and not-too-large k = ωn(1)). This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits. Our algorithm extends a recent result [1] which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function sign(q(x)), corresponding to the k = 1 case of our result. Note that even in the k = 1 case it is NP-hard to determine whether Prx~{-1, 1}n[sign(q(x)) = 1] is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the k = 1 case in [1]. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for k-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution, i.e., an algorithm for estimating Prx~N(0, 1)n[g(sign(q1(x)), . . . , sign(qk(x))) = 1]. Finally, a third new ingredient is a “regularity lemma” for k-tuples of degree-d polynomial threshold functions. This generalizes both the regularity lemmas of [2], [3] (which apply to a single degree-d polynomial threshold function) and the regularity lemma of Gopalan et al [4] (which applies to a k-tuples of linear threshold functions, i.e., the case d = 1). Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.