Decision trees and influences of variables over product probability spaces

Abstract

A celebrated theorem of Friedgut says that every function $f:\{0,1\}^n \to \{0,1\}$ can be approximated by a function $g:\{0,1\}^n \to \{0,1\}$ with $\|f-g\|_2^2 \le ε$ which depends only on $e^{O(I_f/ε)}$ variables where $I_f$ is the sum of the influences of the variables of $f$. Dinur and Friedgut later showed that this statement also holds if we replace the discrete domain $\{0,1\}^n$ with the continuous domain $[0,1]^n$, under the extra assumption that $f$ is increasing. They conjectured that the condition of monotonicity is unnecessary and can be removed. We show that certain constant-depth decision trees provide counter-examples to Dinur-Friedgut conjecture. This suggests a reformulation of the conjecture in which the function $g:[0,1]^n \to \{0,1\}$ instead of depending on a small number of variables has a decision tree of small depth. In fact we prove this reformulation by showing that the depth of the decision tree of $g$ can be bounded by $e^{O(I_f/ε^2)}$. Furthermore we consider a second notion of the influence of a variable, and study the functions that have bounded total influence in this sense. We use a theorem of Bourgain to show that these functions have certain properties. We also study the relation between the two different notions of influence.