Putting a Junta to the Test Joint work with Eldar Fischer & Guy Kindler

© S.Safra Boolean Functions and Juntas A boolean function Def: f is a j-Junta if there exists J  [n] where |J|≤ j, and s.t. for every x f(x) = f(x  J) f is ( , j)-Junta if  j-Junta f’ s.t. f is ( , j)-Junta if  j-Junta f’ s.t.

© S.Safra Junta Test Def: A Junta test is as follows: A distribution over l queries For each l-tuple, a local-test that either accepts or rejects:T[x 1, …, x l ]: {1, -1} l  {T,F} s.t. for a j-junta f whereas for any f which is not ( , j)-Junta

© S.Safra Functions as Inner-Product Space {-1,1} n f f n n 2n2n 2n2n f f n n 2n2n 2n2n

© S.Safra Fourier-Walsh Transform Consider all multiplicative functions, one for each character S  [n] Consider all multiplicative functions, one for each character S  [n] Given any function let the Fourier-Walsh coefficients of f be Given any function let the Fourier-Walsh coefficients of f be thus f can described as thus f can described as 2n2n 2n2n

© S.Safra Simple Observations Claim: Claim: Thm [Parseval]: Thm [Parseval]: Hence, for a boolean f Hence, for a boolean f 2n2n 2n2n

© S.Safra Variables` Influence The influence of an index i  [n] on a boolean function f:{1,-1} n  {1,-1} is The influence of an index i  [n] on a boolean function f:{1,-1} n  {1,-1} is Which can be expressed in terms of the Fourier coefficients of f Claim: Which can be expressed in terms of the Fourier coefficients of f Claim: 2n2n 2n2n

© S.Safra Fourier Representation of influence Proof: consider the I-average function which in Fourier representation is and

© S.Safra Fourier Representation of influence Proof: consider the influence function which in Fourier representation is and

© S.Safra High vs Low Frequencies Def: The section of a function f above k is and the low-frequency portion is

© S.Safra Subsets` Influence Def: The influence of a subset I  [n] on a boolean function f is and the low-frequency influence 2n2n 2n2n

© S.Safra Independence-Test The I-independence-test on a boolean function f is, for Lemma:

© S.Safra

Junta Test The junta-size test on a boolean function f is The junta-size test on a boolean function f is Randomly partition [n] to I 1,.., I r Randomly partition [n] to I 1,.., I r Run the independence-test t times on each I h Run the independence-test t times on each I h Accept if all but ≤j of the I h fail their independence-tests Accept if all but ≤j of the I h fail their independence-tests For r>>j 2 and t>>j 2 / 

© S.Safra Completeness Lemma: for a j-junta f Proof: only those sets which contain an index of the Junta would fail the independence-test

© S.Safra Soundness Lemma: Proof: Assume the premise. Fix  <<1/t and let

© S.Safra |J| ≤ j Prop: r >> j implies |J| ≤ j Proof: otherwise, J spreads among I h w.h.p. J spreads among I h w.h.p. and for any I h s.t. I h  J ≠  it must be that influence I (f) >  and for any I h s.t. I h  J ≠  it must be that influence I (f) > 

© S.Safra High Frequencies Contribute Little Prop: k >> r log r implies Proof: a character S of size larger than k spreads w.h.p. over all parts I h, hence contributes to the influence of all parts. If such characters were heavy (>  /4), then surely there would be more than j parts I h that fail the t independence-tests

© S.Safra Almost all Weight is on J Lemma: Proof: otherwise, since for a random partition w.h.p. (Chernoff bound) for every h however, since for any I the influence of every I h would be ≥  /100rk

© S.Safra Find the Close Junta Now, since consider the (non boolean) which, if rounded outside J is boolean and not more than  far from f

© S.Safra Open Problems Is there a characterization, via Fourier transform, of all efficiently testable properties? Is there a characterization, via Fourier transform, of all efficiently testable properties? What about tests that probe f only at two or three points? With applications to hardness of approximation. What about tests that probe f only at two or three points? With applications to hardness of approximation.