3.3 The Hypercontractive Inequality
3.3.1 Statement and Proof
Theorem 3.16
(Hypercontractive Inequality) LetP be a multilinear polynomial
over the real numbers of degreedinnvariables. Letx1, . . . , xn∼ {−1,1}
be uniformly distributed and independent, and consider the random variable F defined byP(x1, . . . , xn). Then for allq ≥p≥1 we have
||F||q≤
rq−1 p−1
d ||F||p.
Because of the comment preceding the statement, the quintessence of this inequality is that it gives a bound on||F||q from above, andin terms of ||F||p.
Thus, for a low-degree polynomial P,||F||q is not much bigger than ||F||p.
For our purposes the special case in which q= 4 and p= 2 suffices. This special case of the general theorem is also called the Bonami Lemma [97].
What is remarkable about this special case, is that it is by itself already strong enough to prove a lot of theorems from the theory of analysis of Boolean functions. For example, we will show the Kahn-Kalai-Linial (KKL) Theorem and Friedgut’s Theorem below. Both are corollaries of the Bonami Lemma. In the next chapter, about Arrow’s Theorem, the Bonami Lemma will be used to prove the Friedgut-Kalai-Naor (FKN) Theorem.
CHAPTER 3. INFLUENCE AND NOISE 40
Theorem 3.17
(Hypercontractive Inequality in case (q, p) = (4,2), Bonami’s Lemma) Let P be a multilinear polynomial over the real numbers of
degreedinnvariables. Letx1, . . . , xn∼ {−1,1}be uniformly distributed
bits. Consider the random variable F defined by P(x1, . . . , xn). Then
we have
E[F4]≤9dE[F2]2.
In other words, for anyf :{−1,1}n→R of degree at mostd, it holds
that
||f||4 ≤
√
3d||f||2.
The proof of this limited version is not particularly hard. The idea is to just proceed by induction onnand use some easy bounds, e.g., the Cauchy-Schwarz inequality. The details are as follows.
Proof of Theorem 3.19. The proof goes by induction on n.
The base casen= 0 is trivial, as then the random variableF is just a fixed constant.
Finally we show the induction step. Let n ≥ 1 and let P be the given polynomial of degreedin nvariablesx1, . . . , xn. SinceP is multilinear we can
write
P(x) =xnQ(x1, . . . , xn−1) +R(x1, . . . , xn−1)
whereQandR are real polynomials in onlyn−1 variablesx1, . . . , xn−1. Since
the degree of P is d, the degree of Qis must be at most d−1. The degree of R is at most d.
Let G be the random variable Q(x1, . . . , xn−1), andH beR(x1, . . . , xn−1),
where all xi are uniformly distributed and independent.
By simple algebra we calculate that E[F4] = E[(x
nG+H)4] = E[x4nG4+ 4x3nG3H+ 6x2nG2H2+ 4xnGH3+H4].
From linearity of expectation we obtain E[F4] = E[x4
nG4] + 4 E[x3nG3H] + 6 E[x2nG2H2] + 4 E[xnGH3] + E[H4].
Two terms immediately vanish: E[x3
nG3H] = 0 and E[xnGH3] = 0 because
xnis independent ofx1, . . . , xn−1(and thus also ofG3H andGH3) and E[xn] =
E[x3
n] = 0. Before we proceed, note that E[x2n] = E[x4n] = 1.We will use this
below.
We are left with three terms: • First, E[x4
nG4] = E[x4n] E[G4] = E[G4] again because of independence.
Noting that G is a random variable of polynomial form of degree at most d−1 and in only n−1 variables, the induction hypothesis implies E[x4
CHAPTER 3. INFLUENCE AND NOISE 41
• Second, E[x2
nG2H2] = E[x2n] E[G2H2] = E[G2H2] by independence. By
the Cauchy-Schwarz inequality we get E[x2
nG2H2] = E[G2H2]≤
p
E[G4]p E[H4].
Note that we can apply the induction hypothesis toG(which comes from a polynomial of degree of at mostd−1) as well asH (which comes from a polynomial of degree of at mostd), and we obtain that
E[x2
nG2H2]≤3d
−1E[G2]3dE[H2].
• Third, E[H4]≤9dE[H2]2 by the induction hypothesis.
Now all that remains is to put everything together. We have E[F4] = E[x4
nG4] + 4 E[x3nG3H] + 6 E[x2nG2H2] + 4 E[xnGH3] + E[H4]
≤ 9d−1(E[G2]2) + 6 3d−1E[G2]3dE[H2] + 9dE[H2]2
= 9d−1(E[G2]2) + 2 9dE[G2] E[H2] + 9dE[H2]2
≤ 9d(E[G2]2) + 2 E[G2] E[H2] + E[H2]2)
= 9d(E[G2] + E[H2])2 = 9d(E[G2] + 2 E[x nGH] E[x2nH2])2 = 9dE[(G+x nH)2]2 = 9dE[F2]2
Notice that the third last step is true because xn is independent of bothGH
(implying that E[xnGH] = E[xn] E[GH] = 0) as well as H2 (so E[x2nH2] =
E[x2
n] E[H2] = E[H2]).
For any f :{−1,1}n→
Rand natural number m, with f=m we mean the
Boolean function P
S:|S|=mfb(S)χS, i.e., f=m is obtained by taking only the degree-m terms of the Fourier expansion off. Since it is of degree preciselym, Bonami’s Lemma together with Theorem 3.15 implies
T√1/3(f =m) 4 = p 1/3mf=m 4= p 1/3m||f=m|| 4≤ ||f=m||2. (3.4) This expression hints at the following theorem, which involves the noise operator and can be shown to be equivalent with Theorem 3.16:
Theorem 3.18
(Hypercontractivity Theorem) Let p and q be reals satisfying 1≤p≤q, and let ρ be a real such that
ρ≤ r
p−1 q−1.
Then for all f :{−1,1}n→R we have
CHAPTER 3. INFLUENCE AND NOISE 42
One of the nice things about this formulation of hypercontractivity is that it involves the noise operator. Recall from the discussion in Subsection 3.2.3 that adding noise makes functions more flat. More precisely, Tρf will be a function
that looks somewhat like f but which is more “spread out”, i.e., its peaks are reduced.
More precisely the question is: how exactly does noise affect the p-norm
of a given Boolean function? The Hypercontractivity Theorem provides an answer (see [29] for a more in-depth discussion). Indeed, the interpretation goes as follows. First, it is not hard to see that ||Tρ(f)||p ≤ ||f||p for any p
andf. Second, even ifq is bigger thanp, provided a sufficient amount of noise is applied to the function (i.e., ρ≤p
(p−1)/(q−1)), then even in that case we still get ||Tρf||q ≤ ||f||p.