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A Structure Theorem for Poorly Anticoncentrated Gaussian Chaoses and Applications to the Study of Polynomial Threshold Functions

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1 Author(s)
Kane, D.M. ; Dept. of Math., Stanford Univ., Stanford, CA, USA

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, p0, which can be decomposed as some function of polynomials q1,· · ·, qm with qi normalized and m=Od(1), so that if X is a Gaussian random variable, the probability distribution on (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.

Published in:

Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on

Date of Conference:

20-23 Oct. 2012