§5.5: Highlight: Peres’s Theorem

Theorem 14 says that if $f$ is an unbiased linear threshold function $f(x) = \mathrm{sgn}(a_1 x_1 + \cdots + a_n x_n)$ in which all $a_i$’s are “small” then the noise stability $\mathbf{Stab}_\rho[f]$ is at least (roughly) $\frac{2}{\pi} \arcsin \rho$. Rephrasing in terms of noise sensitivity, this means $\mathbf{NS}_\delta[f]$ is at most (roughly) $\tfrac{2}{\pi} \sqrt{\delta} [...]

§5.1: Linear threshold functions and polynomial threshold functions

Recall from Chapter 2.1 that a linear threshold function (abbreviated LTF) is a boolean-valued function $f : \{-1,1\}^n \to \{-1,1\}$ that can be represented as \begin{equation} \label{eqn:generic-LTF} f(x) = \mathrm{sgn}(a_0 + a_1 x_1 + \cdots + a_n x_n) \end{equation} for some constants $a_0, a_1, \dots, a_n \in {\mathbb R}$.

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§2.1: Social choice functions

In this section we describe some rudiments of the mathematics of social choice, a topic studied by economists, political scientists, mathematicians, and computer scientists. The fundamental question in this area is how best to aggregate the opinions of many agents. Examples where this problem arises include citizens voting in an election, committees deciding on [...]