KKL, Kruskal--Katona, and Monotone Nets

We generalize the Kahn--Kalai--Linial (KKL) theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes. An example corollary is that for every $f : \binom{[n]}{k} \to \{0,1\}$ with ${\bf E}[f]$ and $k/n$ bounded away from $0$ and $1$, there is a pair $1 \leq i < j \leq n$ such that ${\cal I}_{ij}(f) \geq \Omega(\frac{\log n}{n})$. Here ${\cal I}_{ij}(f)$ denotes the "influence" on $f$ of swapping the $i$th and $j$th coordinates. Using this corollary we obtain a "robust" version of the Kruskal--Katona theorem: Given a constant-density subset $A$ of a middle slice of the Hamming $n$-cube, the density of $\partial A$ is greater by at least $\Omega(\frac{\log n}{n})$, unless $A$ is noticeably correlated with a single coordinate. As an application of these results, we show that the set of functions $\{0, 1, x_1, \dots, x_n, \mathrm{Maj}\}$ is a $(1/2 - \gamma)$-net for the set of all $n$-bit monotone Boolean functions, where $\gamma = \Omega(\frac{\log n}{\sqrt{n}})$. This distance is optimal for polynomial-size nets and gives an optimal weak-learning algorithm for monotone functions under the uniform distribution, solving a problem of Blum, Burch, and Langford. [PUBLICATION ABSTRACT]