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

Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, Andrew Wan
2010 arXiv   pre-print
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
more » ... nstants d ≥ 1, ≥ 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.
arXiv:0909.4727v2 fatcat:xdezqawno5d4jmq24wuritcvsu