Polynomial threshold functions, hyperplane arrangements, and random tensors

TL;DR

This paper determines the number of polynomial threshold functions of fixed degree, revealing their exponential growth rate in relation to the number of variables, and connects this problem to hyperplane arrangements and random tensors.

Contribution

It provides a complete asymptotic count for polynomial threshold functions of any fixed degree, extending previous results limited to degree 1, and introduces new connections to hyperplane arrangements and tensor theory.

Findings

Number of polynomial threshold functions grows as 2^{n * binom(n, ≤d)}

Established connections between threshold functions, hyperplane arrangements, and tensors

Utilized recent results on Reed-Muller codes to solve the counting problem

Abstract

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The partial case of this problem for degree $d=1$ was solved by Zuev in 1989, who showed that the number $T(n,1)$ of linear threshold functions satisfies $\log_2 T(n,1) \approx n^2$, up to smaller order terms. However the number of polynomial threshold functions for any higher degrees, including $d=2$, has remained open. We settle this problem for all fixed degrees $d \ge1$, showing that $ \log_2 T(n,d) \approx n \binom{n}{\le d}$. The solution relies on connections between the theory of Boolean threshold functions, hyperplane arrangements, and random tensors. Perhaps surprisingly, it uses also a recent result of E.Abbe, A.Shpilka, and A.Wigderson on Reed-Muller codes.