A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions

TL;DR

This paper introduces a regularity lemma for degree-d polynomial threshold functions (PTFs) over Boolean cubes, enabling decomposition into regular parts and approximation by low-weight PTFs with optimal bounds.

Contribution

It presents a novel regularity lemma for degree-d PTFs and demonstrates how to approximate any such PTF with a low-weight polynomial, improving understanding of their structure.

Findings

Decomposition of degree-d PTFs into regular subfunctions

Approximation of PTFs by low-weight polynomials with optimal bounds

Establishment of a regularity lemma for Boolean polynomial threshold functions

Abstract

We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. 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, \eps \geq 0, every degree-d PTF over n variables has can be approximated to accuracy eps by a constant-degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.