Homework 7
Due 2012/12/10
Fall 2012
Analytical Methods in CS
Oded Regev
Courant Institute
1. Stronger KKL theorem: Prove the following strengthening of the KKL theorem. There exists a c > 0
such that if f : cfw_0, 1n cfw_1, 1 is a balanced function with I
Fall 2012
Analytical Methods in CS
Homework 6
Due 2012/11/19
Oded Regev
Courant Institute
1. Total inuence of DNFs:
(a) Assume f can be expressed as a DNF of width w (i.e., each clause has at most
w literals). Show that I( f ) 2w. Hint: no need for Fourie
Fall 2012
Analytical Methods in CS
Homework 5
Due 2012/11/12
Oded Regev
Courant Institute
1. Learning juntas with queries: Show an algorithm for exactly learning k-juntas in time poly(n, 2k )
using query access, without using the Fourier transform.
2. Wea
Fall 2012
Analytical Methods in CS
Homework 4
Due 2012/10/29
Oded Regev
Courant Institute
Solve at least two out of the three questions (or all three for extra credit).
1. A hardness reduction that fails: Consider the following attempt to show that for an
Fall 2012
Analytical Methods in CS
Homework 3
Due 2012/10/22
Oded Regev
Courant Institute
1. Tribes function: For any k, l we dene the tribes function f : cfw_0, 1n cfw_1, 1 on
n = kl variables as
f ( x1 , . . . , xn ) = OR( AND ( x1 , . . . , xl ), AND (
Fall 2012
Analytical Methods in CS
Homework 2
Due 2012/10/8
Oded Regev
Courant Institute
1. Poincar inequality: Let f : cfw_0, 1n cfw_1, 1. Show that
e
Var( f ) = 4 Pr[ f ( x ) = 1] Pr[ f ( x ) = 1],
where Var( f ) is as dened in Homework 1. Dene the tota
Fall 2012
Analytical Methods in CS
Homework 1
Due 2012/10/1
Oded Regev
Courant Institute
Instructions
References: Try not to run to reference material to answer questions (this also includes the web!). Try to
think about the problem to see if you can solv
Analytical Methods
1
Friedgut and KKL
Lemma 1.1. For any f : cfw_0, 1n cfw_1, 1, 0 1,
Infi ( f ) =
|S| f(S)2 (Infi ( f )
2/(1+)
Si
Remarks:
Small inuence gets much smaller after noise
Not a Fourier statement
Means that small inuences cannot come from
H.110
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