Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Boolean functions whose Fourier transform is concentrated on the first two levels
Advances in Applied Mathematics
Influences in Product Spaces: KKL and BKKKL Revisited
Combinatorics, Probability and Computing
Every decision tree has an in.uential variable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The Hardness of 3-Uniform Hypergraph Coloring
Combinatorica
Learning Monotone Decision Trees in Polynomial Time
SIAM Journal on Computing
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
On the influences of variables on boolean functions in product spaces
Combinatorics, Probability and Computing
A simple reduction from a biased measure on the discrete cube to the uniform measure
European Journal of Combinatorics
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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.