Theory of Computing ------------------- Title : A Regularity Lemma and Low-Weight Approximators for Low-Degree Polynomial Threshold Functions Authors : Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, and Andrew Wan Volume : 10 Number : 2 Pages : 27-53 URL : https://theoryofcomputing.org/articles/v010a002 Abstract -------- 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 (http://dx.doi.org/10.1109/CCC.2010.28).