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

• Daniel Kane ^[1]
  1. [1] University of California
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 3, 2017, págs. 1612-1679
• Idioma: inglés
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• Resumen

  • We prove a structural result for degree-dd polynomials. In particular, we show that any degree-dd polynomial, pp can be approximated by another polynomial, p0p0, which can be decomposed as some function of polynomials q1,…,qmq1,…,qm with qiqi normalized and m=Od(1)m=Od(1), so that if XX is a Gaussian random variable, the probability distribution on (q1(X),…,qm(X))(q1(X),…,qm(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.