In this chapter we discuss various notions of pseudorandomness for boolean functions; by this we mean properties of a fixed boolean function which are in some way characteristic of randomly chosen functions. We will see some deterministic constructions of pseudorandom probability density functions with small support; these have algorithmic application in the field of derandomization. [...]
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Chow’s Theorem was proved by independently by Chow [Cho61] and by Tannenbaum [Tan61] in 1961; see also [Elg61]. [...] [...] 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} [...] In this section we prove two theorems about the degree-$1$ Fourier weight of boolean functions: \[ \mathbf{W}^{1}[f] = \sum_{i=1}^n \widehat{f}(i)^2. \] [...] In this section we will analyze the Fourier coefficients of $\mathrm{Maj}_n$. In fact, we give an explicit formula for them in Theorem 16 below. But most of the time this formula is not too useful; instead, it’s better to understand the Fourier coefficients of $\mathrm{Maj}_n$ asymptotically as $n \to \infty$. [...] Majority is one of the more important functions in boolean analysis and its study motivates the introduction of one of the more important tools: the Central Limit Theorem (CLT). [...] 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 $a_0, a_1, \dots, a_n \in {\mathbb R}$. [...] This chapter is devoted to linear threshold functions, their generalization to higher degrees, and their exemplar the majority function. The study of LTFs leads naturally to the introduction of the Central Limit Theorem and Gaussian random variables — important tools in analysis of boolean functions. We will first use these tools to analyze [...] |
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