Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deterministic Approximate Counting for Juntas of Degree-$2$ Polynomial
  Threshold Functions   Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$
be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic}
algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ 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 $k \geq 1$ 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.

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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: gr-qc