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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-dPTF can be decomposed into a constant number of subfunctions such that almostall of the subfunctions are close to being regular PTFs. Here a "regular PTF isa PTF sign(p(x)) where the influence of each variable on the polynomial p(x) isa 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 beapproximated to accuracy eps by a constant-degree PTF that has integer weightsof total magnitude O(n^d). This weight bound is shown to be optimal up toconstant factors.

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