On the Fourier Tails of Bounded Functions over

On the Fourier Tails of Bounded Functions over the Discrete Cube Guy Kindler Microsoft Research Joint work with Irit Dinur, Ehud Friedgut, and Ryan O’Donnell

Fourier Analysis

Fourier Analysis ª Fourier representation: can be written as a multilinear polynomial ª is called the S Fourier coefficient of f.

Fourier Analysis ª Fourier representation: can be written as a multilinear polynomial ª is called the S Fourier coefficient of f. ª Many structural properties of f can be inferred from its Fourier representation. ª Useful in: hardness of approximation, circuit lower bounds, threshold phenomena, metric embeddings, algorithms, learning, communication complexity, …

Boolean vs. Bounded functions ª Often one needs to study averages of Boolean functions. ª Question: which properties persist for bounded functions? ª Our initial motivation: coloring. ª Ideas used in [KO 05] and [ABHKS 05].

What next: Some technical background Some symmetry breaking phenomena for Boolean functions Main theorem: symmetry breaking for bounded functions Something about the proof.

On weights and tails ª k-tail of f: ª Low-degree part of f: ª Weight: ª k-tail weight: ª Dinstance: Parseval’s identity.

On Juntas and tails ª A J-junta: a function f that depends on at most J coordinates. ª Often: having small k-tail weight implies f is junta-ish. ª f is an ( , J)-junta if 9 a J junta g such that breaking. Symmetry ª [FKN 02] ª [B 02] ! f is an (O( ), 1)-junta. ! f is an (0. 001, 100 k)-junta. ª For majority, the weight of the k-tail is .

Tails of bounded functions ª A J-junta: a function f that depends on at most J coordinates. ª Often: having small k-tail weight implies f is junta-ish. ª f is an ( , J)-junta if 9 a J junta g such that ª [FKN 02] ª [B 02] ! f is an (O( ), 1)-junta. ! f is an (0. 001, 100 k)-junta. ª For majority, the weight of the k-tail is .

Tails of bounded functions ª Is a threshold for k-tail bounded function? ª No: ª We have symmetric f with ª Does there really exist a threshold ? ? ª Theorem: If then it is an -junta.

what’s next: Some technical background Some symmetry breaking phenomena for Boolean functions Main theorem: symmetry breaking for bounded functions Something about the proof.

Proof idea: use large deviations ª Theorem: If then it is an -junta. ª Idea: If f<k is smeared over many coordinates then it must obtain large values. So f k must also obtain large values, and therefore have large weight. ª We need a lower-bound on large deviations for low-degree functions.

Large deviation lower bounds , ª Linear case (folklore): and for all i. Then , ª Main lemma: and , for all i. Then ,

Vague idea of the proof

Conclusions and questions ª Bounded functions do show symmetry-breaking phenomena. ª This happens for different reasons and parameter-range than in the Boolean case. ª Is there a generalization of Boolean functions where the same symmetry-breaking phenomena hold? ª Get other bounded-case analogues for Boolean results.