Polynomial Threshold Functions for Decision Lists

For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is an integer polynomial $p$ of degree $d$ and with sum of absolute coefficients $W$ such that $f(x) = \text{sign } p(x)$ for all $x \in S$. We study representation of decision lists as PTFs over Boolean cube $\{0,1\}^n$ and over Hamming ball $\{0,1\}^{n}_{\leq k}$. As our first result we show that for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ any decision list over $\{0,1\}^n$ can be represented by a PTF of degree $d$ and weight $2^{O(n/d^2)}$. This improves the result by Klivans and Servedio by a $\log^2 d$ factor in the exponent of the weight. Our bound is tight for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ due to the matching lower bound by Beigel. For decision lists over a Hamming ball $\{0,1\}^n_{\leq k}$ we show that the upper bound on the weight above can be drastically improved to $n^{O(\sqrt{k})}$ for $d = \Theta(\sqrt{k})$. We also show that similar improvement is not possible for smaller degree by proving the lower bound $W = 2^{\Omega(n/d^2)}$ for all $d = O(\sqrt{k})$.Comment: 14 page