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May 2017 A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions
Daniel Kane
Ann. Probab. 45(3): 1612-1679 (May 2017). DOI: 10.1214/16-AOP1097

Abstract

We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_{0}$, which can be decomposed as some function of polynomials $q_{1},\ldots,q_{m}$ with $q_{i}$ normalized and $m=O_{d}(1)$, so that if $X$ is a Gaussian random variable, the probability distribution on $(q_{1}(X),\ldots,q_{m}(X))$ does not have too much mass in any small box.

Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.

Citation

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Daniel Kane. "A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions." Ann. Probab. 45 (3) 1612 - 1679, May 2017. https://doi.org/10.1214/16-AOP1097

Information

Received: 1 December 2013; Revised: 1 December 2014; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1377.60051
MathSciNet: MR3650411
Digital Object Identifier: 10.1214/16-AOP1097

Subjects:
Primary: 60G15
Secondary: 68R05

Keywords: anticoncentration , Gaussian chaos , invariance principle , Polynomial decompositions

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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