The Boolean surface area of polynomial threshold functions

TL;DR

This paper proves that polynomial threshold functions of degree d have polylogarithmic Boolean surface area, linking structural properties to complexity measures in Boolean functions.

Contribution

It establishes a new upper bound on the Boolean surface area of polynomial threshold functions, advancing understanding of their boundary complexity.

Findings

Degree-d PTFs have polylogarithmic Boolean surface area.

Provides a tail bound for pointwise sensitivity of PTFs.

Introduces a recursive argument for bounding surface area.

Abstract

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the weaker bound \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}). \]