Analysis of Boolean Functions Fourier Analysis, Projections, Influence, Junta, Etc… And (some) applications Slides prepared with help of Ricky Rosen

Introduction Objectives: Objectives: To introduce Analysis of Boolean Functions and some of its applications. To introduce Analysis of Boolean Functions and some of its applications. Overview: Overview: Basic Definitions. Basic Definitions. Junta Test Junta Test Friedgut Theorem (F) Friedgut Theorem (F) Margulis-Russo Lemma (MR) Margulis-Russo Lemma (MR) Noise Sensitivity and juntas (KS) Noise Sensitivity and juntas (KS) … And more… … And more…

Boolean Functions Def: A Boolean function Def: A Boolean function Power set of [n] Choose the location of -1 Choose a sequence of -1 and 1

Functions as Vector-Spaces f f * * -1* 1* 11* 11-1* -1-1* -11* -11-1* -111* * -1-11* 111* 1-1* 1-1-1* 1-11* f f 2n2n 2n2n * * -1* 1* 11* 11-1* -1-1* -11* -11-1* -111* * -1-11* 111* 1-1* 1-1-1* 1-11* A function can be represented as a string of size (i.e.: it’s truth table) A function can be represented as a string of size 2 n (i.e.: it’s truth table)

Functions’ Vector-Space A functions f is a vector A functions f is a vector Addition: ‘f+g’(x) = f(x) + g(x) Addition: ‘f+g’(x) = f(x) + g(x) Multiplication by scalar Multiplication by scalar ‘c  f’(x) = c  f(x) ‘c  f’(x) = c  f(x) Inner product (normalized) Inner product (normalized)

Boolean function as voting system Consider n agents, each voting either “for” (T=-1) or “against” (F=1) Consider n agents, each voting either “for” (T=-1) or “against” (F=1) The system is not necessarily majority. The system is not necessarily majority. This is a boolean function over n variables. This is a boolean function over n variables

Def: the influence of i on f is the probability, over a random input x, that f changes its value when i is flipped Def: the influence of i on f is the probability, over a random input x, that f changes its value when i is flipped Voting and influence X represented as a set of variables

The influence of i on Majority is the probability, over a random input x, Majority changes with i The influence of i on Majority is the probability, over a random input x, Majority changes with i this happens when half of the n-1 coordinate (people) vote -1 and half vote 1. this happens when half of the n-1 coordinate (people) vote -1 and half vote 1. i.e. i.e. Majority :{1,-1} n  {1,-1} Majority :{1,-1} n  {1,-1} 1?

Parity : {1,-1} n  {1,-1} Parity : {1,-1} n  {1,-1} Always changes the value of parity

influence of i on Dictatorship i = 1. influence of i on Dictatorship i = 1. influence of j  i on Dictatorship i = 0. influence of j  i on Dictatorship i = 0. Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i (x)=x i Dictatorship i (x)=x i

Total Influence (Average Sensitivity) Def: the Average­ Sensitivity of f ( as ) is the sum of influences of all coordinates i  [n] : Def: the Average­ Sensitivity of f ( as ) is the sum of influences of all coordinates i  [n] : as (Majority) = O(n ½ ) as (Majority) = O(n ½ ) as (Parity) = n as (Parity) = n as (dictatorship) =1 as (dictatorship) =1

Example majority for majority for What is Average Sensitivity ? What is Average Sensitivity ? AS= ½+ ½+ ½= 1.5 AS= ½+ ½+ ½= Influence 2 3

When as (f)=1 Def: f is a balanced function if it equals -1 exactly half of the times: E x [f(x)]=0 Can a balanced f have as (f) < 1 ? What about as (f)=1 ? Beside dictatorships? Prop: f is balanced and as (f)=1  f is a dictatorship.

Representing f as a Polynomial What would be the monomials over x  P[n] ? What would be the monomials over x  P[n] ? All powers except 0 and 1 cancel out! All powers except 0 and 1 cancel out! Hence, one for each character S  [n] Hence, one for each character S  [n] These are all the multiplicative functions These are all the multiplicative functions

Fourier-Walsh Transform Consider all characters Consider all characters 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 be described as thus f can be described as

Norms Def: Expectation norm on the function Def: Summation norm on the transform Thm [Parseval]: Hence, for a Boolean f

We may think of the Transform as defining a distribution over the characters. We may think of the Transform as defining a distribution over the characters. Distribution over Characters

Characters and Multiplicative Claim: Characters are all the multiplicative functions Claim: Characters are all the multiplicative functions Proof: Proof: Let S={i | f({i})=-1 } we prove (f =  s ) Let S={i | f({i})=-1 } we prove (f =  s ) F is multiplicative function

Simple Observations Def: Def: For any function f whose range is {-1,0,1}: For any function f whose range is {-1,0,1}:

Variables` Influence Recall: influence of an index i  [n] on a Boolean function f:{1,-1} n  {1,-1} is Recall: 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: And the as: And the as:

Fourier Representation of influence Proof: consider the influence function which in Fourier representation is and

Restriction and Average Def: Let I  [n], x  P([n]\I), the restriction function is [n] I x y I I x  P[ [n]\I ]

Average function Def: the average function is Note: [n] I x y y y y y I I x  P[ [n]\I ]

In Fourier Expansion Prop: Prop: F I [x] is a functions only of the variables of I (since x  P[ [n]\I ] is fixed). F I [x] is a functions only of the variables of I (since x  P[ [n]\I ] is fixed). Representing it as a polynomial hence involves coefficient only to S  I,, each of which is the sum of all coefficient of characters whose intersection with I is S where the value is calculated according to the restriction x Representing it as a polynomial hence involves coefficient only to S  I,, each of which is the sum of all coefficient of characters whose intersection with I is S where the value is calculated according to the restriction x

In Fourier Expansion Recall: Recall: Since the expectation of a function is the coefficient of its empty character: Since the expectation of a function is the coefficient of its empty character: Cor 1: Cor 1: Cor 2: Cor 2: P[{i}] = {  ,{i} } A {i} [x]  {-1,0,1} P[{i}] = {  ,{i} } A {i} [x]  {-1,0,1} Parseval + corollary 1 + the sum of squares of the coefficients of a boolean function equals 1

Expectation and Variance Recall: Recall: Hence, for any f Hence, for any f

Balanced f s.t. as (f)=1 is Dict. Since f is balanced and So f is homogeneous & linear For any i s.t. If  s s.t |s|>1 and then as(f)>1 Only i has changed