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Volume 10 (2014) Article 2 pp. 27-53
A Regularity Lemma and Low-Weight Approximators for Low-Degree Polynomial Threshold Functions
Received: September 8, 2012
Revised: September 27, 2013
Published: March 25, 2014
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Keywords: complexity theory, Boolean functions, polynomials, threshold functions
ACM Classification: F.1.3, G.2.0
AMS Classification: 68Q15, 68R01

Abstract: [Plain Text Version]

$ \newcommand{\sign}{{\mathrm{sign}}} $

We give a “regularity lemma” for degree-$d$ polynomial threshold functions (PTFs) over the Boolean cube $\{-1,1\}^n$. Roughly speaking, this result shows that every degree-$d$ PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a “regular” PTF is a PTF $\sign(p(x))$ where the influence of each variable on the polynomial $p(x)$ is a small fraction of the total influence of $p$.

As an application of this regularity lemma, we prove that for any constants $d \geq 1, \epsilon > 0$, every degree-$d$ PTF over $n$ variables can be approximated to accuracy $\epsilon$ by a constant-degree PTF that has integer weights of total magnitude $O_{\epsilon,d}(n^d)$. This weight bound is shown to be optimal up to logarithmic factors.

A conference version of this paper appeared in the Proc. 25th Ann. IEEE Conf. on Computational Complexity, CCC 2010.