The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]).
Continue reading Chapter 8 notes


The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]). A decision tree $T$ for $f : \{1,1\}^n \to \{1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they are equal, output their value; otherwise, query and output $x_3$.” For a worstcase input (one where $x_1 \neq x_2$) this algorithm has a cost of $3$, meaning it makes $3$ queries. The cost of the worstcase input is the depth of the decision tree. The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $\Omega = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $\Omega \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, \mathsf{Green}, \mathsf{Blue}\}^n \to {\mathbb R}$ then it’s best to just work abstractly with the orthogonal decomposition. However if there is a notion of, say, “addition” in $\Omega$ then there is a natural, canonical Fourier basis for $L^2(\Omega, \pi)$ when $\pi$ is the uniform distribution. Perhaps the most common generalized domain in analysis of boolean functions is the case of the hypercube with “biased” bits. In this section we describe a basisfree kind of “Fourier expansion” for functions on general product domains. We will refer to it as the orthogonal decomposition of $f \in L^2(\Omega^n, \pi^{\otimes n})$ though it goes by several other names in the literature: e.g., Hoeffding, Efron–Stein, or ANOVA decomposition. In this section we will revisit a number of combinatorial/probabilistic notions and show that for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, these notions have familiar Fourier formulas which don’t depend on the Fourier basis. We will now begin to discuss functions on (finite) product probability spaces. So far we have studied functions $f : \{0,1\}^n \to {\mathbb R}$. What about, say, $f : \{0,1,2\}^n \to {\mathbb R}$? In fact, very little of what we’ve done so far depends on the domain being $\{0,1\}^n$; what it has mostly depended on is our viewing the domain as a product probability distribution. Indeed, much of analysis of boolean functions carries over to the case of functions $f : \Omega_1 \times \cdots \times \Omega_n \to {\mathbb R}$ where the domain has a product probability distribution $\pi_1 \otimes \cdots \otimes \pi_n$. There are two main exceptions: the “derivative” operator $\mathrm{D}_i$ does not generalize to the case when $\Omega_i > 2$ (though the Laplacian operator $\mathrm{L}_i$ does); and, the important notion of hypercontractivity (introduced in the next chapter) depends strongly on the probability distributions $\pi_i$. In this chapter we focus on the case where all the $\Omega_i$’s are the same, as are the $\pi_i$’s. (This is just to save on notation; it will be clear that everything we do holds in the more general setting.) Important classic cases include functions on the $p$biased hypercube (Section 4) and functions on abelian groups (Section 5). For the issue of generalizing the range of functions — e.g., studying functions $f : \{0,1,2\}^n \to \{0,1,2\}$ — see the exercises. 

Copyright © 2014 Ryan O'Donnell  All Rights Reserved 
Recent comments