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
 Fourier representation:
can be written as a multilinear polynomial

is called the S Fourier coefficient of f.
Fourier Analysis
 Fourier representation:
can be written as a multilinear polynomial

is called the S Fourier coefficient of f.
 Many structural properties of f can be inferred from its
Fourier representation.
 Useful in: hardness of approximation, circuit lower bounds,
threshold phenomena, metric embeddings, algorithms,
learning, communication complexity, complexity,…
Boolean vs. Bounded functions
 Often one needs to study averages of Boolean functions.
 Question:
which properties persist for bounded functions?
 Our initial motivation: coloring.
 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
 k-tail of f:
 Low-degree part of f:
 Weight:
 k-tail weight:
 Dinstance:
Parseval’s identity.
On Juntas and tails
 A J-junta: a function f that depends on at most J coordinates.
 Often: having small k-tail weight implies f is junta-ish.
 f is an (,J)-junta if 9 a J junta g such
that breaking.
Symmetry
 [FKN 02]
 [B 02]
! f is an (O(),1)-junta.
! f is an (0.001,100k)-junta.
 For majority, the weight of the k-tail is
.
Tails of bounded functions
 A J-junta: a function f that depends on at most J coordinates.
 Often: having small k-tail weight implies f is junta-ish.
 f is an (,J)-junta if 9 a J junta g such that
 [FKN 02]
 [B 02]
! f is an (O(),1)-junta.
! f is an (0.001,100k)-junta.
 For majority, the weight of the k-tail is
.
Tails of bounded functions
 Is
a threshold for k-tail bounded function?
 No:
 We have symmetric f with
 Does there really exist a threshold ??
 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
 Theorem: If
then it is an
-junta.
 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.
 We need a lower-bound on large deviations for low-degree
functions.
Large deviation lower bounds
,
 Linear case (folklore):
and
for all i. Then
,
 Main lemma:
and
,
for all i. Then
,
Vague idea of the proof
f^{=1}
f^{=1}
f^{=2}
f^{=1}
f^{=2}
f^{=3}
f^{=3}
f
x
N_0.1(x) N_0.2(x) N_0.3(x) N_0.4(x) N_0.5(x)
Conclusions and questions
 Bounded functions do show symmetry-breaking
phenomena.
 This happens for different reasons and parameter-range
than in the Boolean case.
 Is there a generalization of Boolean functions where the
same symmetry-breaking phenomena hold?
 Get other bounded-case analogues for Boolean results.