The Annals of Probability

A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions

Daniel Kane

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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.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1612-1679.

Dates
Received: December 2013
Revised: December 2014
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835227

Digital Object Identifier
doi:10.1214/16-AOP1097

Mathematical Reviews number (MathSciNet)
MR3650411

Zentralblatt MATH identifier
1377.60051

Subjects
Primary: 60G15: Gaussian processes
Secondary: 68R05: Combinatorics

Keywords
Polynomial decompositions Gaussian chaos anticoncentration invariance principle

Citation

Kane, Daniel. A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions. Ann. Probab. 45 (2017), no. 3, 1612--1679. doi:10.1214/16-AOP1097. https://projecteuclid.org/euclid.aop/1494835227


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