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Decision Trees and Influences of Variables Over Product Probability Spaces

Published online by Cambridge University Press:  01 May 2009

HAMED HATAMI
HAMED HATAMI*
Affiliation:
Department of Computer Science, University of Toronto, Ontario M5S 3G4, Canada (e-mail: hamed@cs.toronto.edu)
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Abstract

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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 $\|f-g\|_2^2 \leq \epsilon$, which depends only on eO(If / ε) 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]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 the Dinur–Friedgut conjecture. This suggests a reformulation of the conjecture in which the function g : [0, 1]n → {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 eO(If / ε2).

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.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 3 , May 2009, pp. 357 - 369
Copyright
Copyright © Cambridge University Press 2009

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