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• W2953085912 abstract "We give a deterministic algorithm for approximately counting satisfying assignments of a degree-$d$ polynomial threshold function (PTF). Given a degree-$d$ input polynomial $p(x_1,dots,x_n)$ over $R^n$ and a parameter $epsilon> 0$, our algorithm approximates $Pr_{x sim {-1,1}^n}[p(x) geq 0]$ to within an additive $pm epsilon$ in time $O_{d,epsilon}(1)cdot mathop{poly}(n^d)$. (Any sort of efficient multiplicative approximation is impossible even for randomized algorithms assuming $NPnot=RP$.) Note that the running time of our algorithm (as a function of $n^d$, the number of coefficients of a degree-$d$ PTF) is a emph{fixed} polynomial. The fastest previous algorithm for this problem (due to Kane), based on constructions of unconditional pseudorandom generators for degree-$d$ PTFs, runs in time $n^{O_{d,c}(1) cdot epsilon^{-c}}$ for all $c > 0$. The key novel contributions of this work are: A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. A new decomposition of low-degree multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such polynomial can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version." @default.
• W2953085912 created "2019-06-27" @default.
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• W2953085912 date "2013-11-27" @default.
• W2953085912 modified "2023-09-24" @default.
• W2953085912 title "Efficient deterministic approximate counting for low-degree polynomial threshold functions" @default.
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• W2953085912 doi "https://doi.org/10.48550/arxiv.1311.7178" @default.
• W2953085912 hasPublicationYear "2013" @default.
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