Decision Trees and Influences of Variables Over Product Probability Spaces

A celebrated theorem of Friedgut says that every function f : {0, 1}n → {0, 1} can be approximated by a function g : {0, 1}n → {0, 1} with ‖f − g‖2 ≤ ǫ which depends only on ef variables where If 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], 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 DinurFriedgut conjecture. This suggests a reformulation of the conjecture in which the function g : [0, 1] → {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 ef 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. AMS Subject Classification: 06E30 28A35