Bounding the Sensitivity of Polynomial Threshold Functions

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d we prove 1) The average sensitivity of f is at most O(n^{1-1/(4d+6)}) (we also give a combinatorial proof of the bound O(n^{1-1/2^d}). 2) The noise sensitivity of f with noise rate \delta is at most O(\delta^{1/(4d+6)}). Previously, only bounds for the linear case were known. Along the way we show new structural theorems about random restrictions of polynomial threshold functions obtained via hypercontractivity. These structural results may be of independent interest as they provide a generic template for transforming problems related to polynomial threshold functions defined on the Boolean hypercube to polynomial threshold functions defined in Gaussian space.

💡 Research Summary

The paper tackles a long‑standing open problem in the analysis of Boolean functions: obtaining non‑trivial upper bounds on the average sensitivity and the noise sensitivity of polynomial threshold functions (PTFs) of arbitrary degree d. A PTF is a Boolean function f : {−1,1}ⁿ → {−1,1} defined as the sign of a real multivariate polynomial p(x) of total degree d, i.e., f(x)=sign(p(x)). For linear threshold functions (d = 1) the classic results of Peres, Gotsman–Linial, and others give average sensitivity O(√n) and noise sensitivity O(√δ). Beyond the linear case, however, no sub‑linear bounds were known.

Main contributions

1. Average sensitivity bound. The authors prove that for any degree‑d PTF,
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