A regularity lemma, and low-weight approximators, for low-degree 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.
Cite
@article{arxiv.0909.4727,
title = {A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions},
author = {Ilias Diakonikolas and Rocco A. Servedio and Li-Yang Tan and Andrew Wan},
journal= {arXiv preprint arXiv:0909.4727},
year = {2015}
}
Comments
23 pages, 0 figures