Abstract
We give a 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 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 >= 1, eps > 0, every degree-d PTF over n variables 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 logarithmic factors.
Highlights
A polynomial threshold function ( polynomial threshold functions (PTFs)) is a Boolean function f : {−1, 1}n → {−1, 1}, f (x) = sign(p(x)), where p : {−1, 1}n → R is a polynomial with real coefficients
Low-degree PTFs are a natural generalization of linear threshold functions and are of significant interest in complexity theory, see e. g., [1, 4, 29, 28, 9, 13, 16, 25, 32, 34] and many other works
We showed that any constant-degree PTF is well-approximated by a constant-degree PTF with low integer weights
Summary
Regular polynomials and PTFs are useful because they inherit some nice properties of PTFs and polynomials over Gaussian (rather than Boolean) inputs; this intuition can be made precise using the “invariance principle” of Mossel et al [26]. This point of view has been useful in the d = 1 case for constructing pseudorandom generators [6], low-weight approximators [33, 10], and other results for LTFs [30, 24]
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