Boolean functions on high-dimensional expanders

TL;DR

This paper develops a framework for analyzing Boolean functions on high-dimensional expanders, extending Fourier analysis and key theorems to this setting, with implications for derandomization and complexity theory.

Contribution

It introduces a random-walk based definition of high-dimensional expansion and a Fourier-like decomposition that extends to posets, enabling analysis of Boolean functions on these structures.

Findings

Extended Friedgut-Kalai-Naor theorem to high-dimensional expanders

Demonstrated high-dimensional expanders can model Boolean slices sparsely

Provided a new framework for Boolean function analysis on simplicial complexes

Abstract

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $\binom{n}{k}$ points in the $(k)$-slice (which consists of all $n$-bit strings with exactly $k$ ones).