Coset decision trees and the Fourier algebra

TL;DR

This paper establishes a bound on the complexity of computing 0-1 functions on finite groups using coset decision trees, linking the Fourier algebra norm to decision tree size.

Contribution

It introduces a bound on the size of coset decision trees needed to compute functions with bounded Fourier algebra norm on finite groups.

Findings

Functions with bounded Fourier algebra norm can be computed by extremely shallow coset decision trees.

The size of the decision tree grows double-exponentially with the Fourier algebra norm.

Functions computed by small coset decision trees have bounded Fourier algebra norm.

Abstract

We show that if G is a finite group and f is a {0,1}-valued function on G with Fourier algebra norm at most M then f may be computed by a coset decision tree (that is a decision tree in which at each vertex we query membership of a given coset) having at most \exp(\exp(\exp(O(M^2)))) leaves. A short calculation shows that any {0,1}-valued function which may be computed by a coset decision tree with m leaves has Fourier algebra norm at most \exp(O(m)).