On the Fourier Tails of On the Fourier Tails of

Bounded FunctionsBounded Functions

over the Discrete Cubeover the Discrete Cube

On the Fourier Tails of On the Fourier Tails of

Bounded FunctionsBounded Functions

over the Discrete Cubeover the Discrete Cube

Irit Dinur, Ehud Friedgut,Irit Dinur, Ehud Friedgut,

and Ryan O’Donnelland Ryan O’Donnell

Joint work with

Guy KindlerGuy KindlerMicrosoft ResearchMicrosoft Research

Fourier AnalysisFourier Analysis

Fourier AnalysisFourier Analysis

Fourier representation:Fourier representation:

can be written as a multilinear polynomialcan be written as a multilinear polynomial

is called the is called the SS Fourier coefficient of Fourier coefficient of ff..

Fourier AnalysisFourier Analysis

Fourier representation:Fourier representation:

can be written as a multilinear polynomialcan be written as a multilinear polynomial

is called the is called the SS Fourier coefficient of Fourier coefficient of ff..

Many structural properties of Many structural properties of ff can be inferred from its can be inferred from its

Fourier representation.Fourier representation.

Useful in:Useful in: hardness of approximation, circuit lower bounds, hardness of approximation, circuit lower bounds,

threshold phenomena, metric embeddings, algorithms, threshold phenomena, metric embeddings, algorithms,

learning, communication complexity, complexity,…learning, communication complexity, complexity,…

Boolean vs. Bounded functionsBoolean vs. Bounded functions

Often one needs to study Often one needs to study averagesaverages of Boolean functions. of Boolean functions.

Question:Question:

which properties persist for which properties persist for boundedbounded functions? functions?

Our initial motivation:Our initial motivation: coloring. coloring.

Ideas used in Ideas used in [KO 05][KO 05] and and [ABHKS 05][ABHKS 05]..

What next:What next:

Some technical backgroundSome technical background

Some symmetry breaking phenomena for Boolean Some symmetry breaking phenomena for Boolean

functionsfunctions

Main theorem: symmetry breaking for bounded Main theorem: symmetry breaking for bounded

functionsfunctions

Something about the proof.Something about the proof.

On weights and tailsOn weights and tails

k-tail of k-tail of ff::

Low-degree part of Low-degree part of ff::

Weight:Weight:

k-tail weight:k-tail weight:

Dinstance:Dinstance:

Parseval’s Parseval’s

identity. identity.

Parseval’s Parseval’s

identity. identity.

AA JJ-junta:-junta: a function a function ff that depends on at most that depends on at most JJ coordinates. coordinates.

Often:Often: having small having small kk-tail weight implies -tail weight implies ff is junta-ish. is junta-ish.

ff is an is an ((,J,J)-junta)-junta if if 99 a a JJ junta junta gg such that such that

[FKN 02][FKN 02] !! ff is an ( is an (O(O(),1),1)-junta.)-junta.

[B 02][B 02] !! ff is an ( is an (0.001,1000.001,100kk)-junta.)-junta.

For majority, the weight of the k-tail is . For majority, the weight of the k-tail is .

On Juntas and tailsOn Juntas and tails

Symmetry breaking. Symmetry breaking. Symmetry breaking. Symmetry breaking.

AA JJ-junta:-junta: a function a function ff that depends on at most that depends on at most JJ coordinates. coordinates.

Often:Often: having small having small kk-tail weight implies -tail weight implies ff is junta-ish. is junta-ish.

ff is an is an ((,J,J)-junta)-junta if if 99 a a JJ junta junta gg such that such that

[FKN 02][FKN 02] !! ff is an ( is an (O(O(),1),1)-junta.)-junta.

[B 02][B 02] !! ff is an ( is an (0.001,1000.001,100kk)-junta.)-junta.

For majority, the weight of the k-tail is . For majority, the weight of the k-tail is .

Tails of bounded functionsTails of bounded functions

Tails of bounded functionsTails of bounded functions

Is a threshold for Is a threshold for kk-tail bounded function?-tail bounded function?

No:No:

We have symmetric We have symmetric ff with with

Does there really exist a threshold ?? Does there really exist a threshold ??

Theorem:Theorem: If If then it is an then it is an

-junta. -junta.

what’s next:what’s next:

Some technical backgroundSome technical background

Some symmetry breaking phenomena for Boolean Some symmetry breaking phenomena for Boolean

functionsfunctions

Main theorem: symmetry breaking for bounded Main theorem: symmetry breaking for bounded

functionsfunctions

Something about the proof.Something about the proof.

Proof idea: use large deviationsProof idea: use large deviations

Theorem:Theorem: If If then it is an then it is an

-junta. -junta.

Idea:Idea:

If If ff<k<k is smeared over many coordinates then it must is smeared over many coordinates then it must

obtain large values. So obtain large values. So ff k k must also obtain large values, must also obtain large values,

and therefore have large weight.and therefore have large weight.

We need a lower-bound on large deviations for low-We need a lower-bound on large deviations for low-

degree functions.degree functions.

Large deviation lower boundsLarge deviation lower bounds

Linear case (folklore):Linear case (folklore): , ,

, and for all , and for all ii. Then . Then

Main lemma:Main lemma: , ,

, and for all , and for all ii. Then . Then

x N_0.1(x) N_0.2(x) N_0.3(x) N_0.4(x) N_0.5(x)

f^{=1}

Vague idea of the proofVague idea of the proof

x N_0.1(x) N_0.2(x) N_0.3(x) N_0.4(x) N_0.5(x)

f^{=1}

f^{=2}

f^{=3}

x N_0.1(x) N_0.2(x) N_0.3(x) N_0.4(x) N_0.5(x)

f^{=1}

f^{=2}

f^{=3}

f

Conclusions and questionsConclusions and questions

Bounded functions Bounded functions dodo show symmetry-breaking show symmetry-breaking

phenomena.phenomena.

This happens for different reasons and parameter-This happens for different reasons and parameter-

range than in the Boolean case.range than in the Boolean case.

Is there a generalization of Boolean functions where Is there a generalization of Boolean functions where

the same symmetry-breaking phenomena hold?the same symmetry-breaking phenomena hold?

Get other bounded-case analogues for Boolean Get other bounded-case analogues for Boolean

results. results.