Hypercontractive inequalities via SOS,
and the Frankl-Rödl graph
Manuel Kauers (Johannes Kepler Universität)
Ryan O’Donnell (Carnegie Mellon University)
Li-Yang Tan
(Columbia University)
Yuan Zhou (Carnegie Mellon University)
The hypercontractive inequalities
• Real functions on Boolean cube: f : {-1,1}n ® R
• Noise operator: Tr f (x) = E [ f (y)]
y~ r x
y ~ r x means each yi is r -correlated with xi
Projection to deg-k
• The hypercontractive inequality:
Tr f q £ f p when 1 £ p £ q £ ¥, 0 £ r £
£k
p-1
q-1
P f £ (q -1) f 2 , "q ³ 2
Corollary.
q
£k
k/2
P f £3 f 2
24 hypercontractive ineq.
4
k/2
The hypercontractive inequalities
• The reverse hypercontractive inequaltiy:
Tr f q ³ f p when -¥ £ q £ p £1, 0 £ r £
Not norms
Corollary [MORSS12]. when
f, g : {-1,1} ® R , 0 £ q £1, 0 £ r £1- q
n
we have
³0
E [ f (x)g(y)] ³ E éë f (x)q ùû E éëg(y)q ùû
x~ r y
x
y
1/q
1/q
1-p
1-q
The hypercontractive inequalities
• The hypercontractive inequality:
p-1
Tr f q £ f p when 1 £ p £ q £ ¥, 0 £ r £ q-1
Applications: KKL theorem, Invariance Principle
• The reverse hypercontractive inequaltiy:
Tr f q ³ f p when -¥ £ q £ p £1, 0 £ r £
Applications: hardness of approximation,
quantitative social choice
1-p
1-q
The sum-of-squares (SOS) proof system
• Closely related to the Lasserre hierarchy
[BBHKSZ12, OZ13, DMN13]
– Used to prove that Lasserre succeeds on
specific instances
• In degree-d SOS proof system
To show a polynomial p(x1, x2,… , xn ) ³ 0,
2
write p(x1, x2 ,… , xn ) = åri (x1, x2,… , xn )
i
where each ri has degree
at most d
Previous works
• Deg-4 SOS proof of 24 hypercon. ineq. [BBHKSZ12]
– Level-2 Lasserre succeeds on known UniqueGames
instances
• Constant-deg SOS proof of KKL theorem [OZ13]
– Level-O(1) Lasserre succeeds on the
BalancedSeparator instances by [DKSV06]
• Constant-deg SOS proof of “2/π-theorem” [OZ13] and
Majority-Is-Stablest theorem [DMN13]
– Level-O(1) Lasserre succeeds on MaxCut instances
by [KV05, KS09, RS09]
Results in this work
• Deg-q SOS proof of the hypercon. ineq. when p=2, q=2s
n
f
,
f
,…
,
f
:{-1,1}
® R, r £ 1/ (q -1)
for
1 2
s
s
és
2ù
2ù
é
E
P
(T
f
(x))
£
P
E
f
(x)
we have
r i
i
ë
û
ú
xê
i=1
i=1
x
ë
û
• Deg-4k SOS proof of reverse hypercon. ineq. when q= 21k
for
f, g :{-1,1}n ® R³0 , 0 £ r £1- 2k1
2k
2k
1/(2k
)
1/(2k
)
ù E ég(y)
ù
we have E [ f (x)g(y)] ³ E éë f (x)
û ë
û
x~ r y
ff^(2k)
gg^(2k)
x
y
2k
2k
2k
2k ù
é
E ë f (x) g(y) û ³ E [ f (x)] E [ g(y)]
x~ r y
x
y
Application of the reverse hypercon.
ineq. to the Lasserre SDP
• Frankl-Rödl graphs
FRg (V, E) = FRg ({-1,1} ,{(x, y) : D(x, y) = (1- g )n})
n
n
n
• 3-Coloring
n
– c (FR1/4
[FR87,GMPT11]
) = w (1)
– SDPs by Kleinberg-Goemans fails to certify
n
c (FR1/4
)>3
[KMS98, KG98, Cha02]
– Arora and Ge [AG11]: level-poly(n) Lasserre SDP
n
certifies c (FR1/4
)³4
– Our SOS proof: level-2 Lasserre SDP certifies
n
c (FR1/4
) = w (1)
Application of the reverse hypercon.
ineq. to the Lasserre SDP
• Frankl-Rödl graphs
FRg (V, E) = FRg ({-1,1} ,{(x, y) : D(x, y) = (1- g )n})
n
n
n
• VertexCover
– Min-VC(FRgn ) >1- o(1) when g ³ .1
[FR87,GMPT11]
log n
n
n level-6
– Level-ω(1)
LS+SDP and
SA+SDP fails to certify
Min-VC(FR
) > 1 + W(1)
g
2
[BCGM11]
– The only known (2-o(1)) gap instances for SDP
relaxations
n
– Our Min-VC(FR
SOS proof: level-(1/γ)
1
g >>certifies
) >1- o(1)Lasserre SDP
g
logn
Proof sketches
• SOS proof of the normal hypercon. ineq.
– Induction
• SOS proof of the reverse hypercon. ineq.
– Induction + computer-algebra-assisted induction
• From reverse hypercon. ineq. to Frankl-Rödl graphs
– SOS-ize the “density” variation of the Frankl-Rödl
theorem due to Benabbas-Hatami-Magen [BHM12]
SOS proof (sketch) of the reverse
hypercon. ineq.
• Statement: for f, g :{-1,1} ® R , 0 £ r £1- 2k1
exists SOS proof of
n
³0
2k
2k
2k
2k ù
é
E ë f (x) g(y) û ³ E [ f (x)] E [ g(y)]
x~ r y
x
y
• Proof. Induction on n.
– Base case (n = 1). For F0 , F1,G0 ,G1 ³ 0
– Key challenge, will prove later.
• Statement: for f, g :{-1,1} ® R , 0 £ r £1- 2k1
exists SOS proof of
n
³0
2k
2k
2k
2k ù
é
E ë f (x) g(y) û ³ E [ f (x)] E [ g(y)]
x~ r y
x
y
• Proof. Induction on n.
– Base case (n = 1). For F0 , F1,G0 ,G1 ³ 0
– Induction step (n > 1). f0 (x1,… , xn-1) = f (x1,… , xn-1,-1)
f1 (x1,… , xn-1) = f (x1,… , xn-1,1)
Induction on (n-1) variables
• Statement: for f, g :{-1,1} ® R , 0 £ r £1- 2k1
exists SOS proof of
n
³0
2k
2k
2k
2k ù
é
E ë f (x) g(y) û ³ E [ f (x)] E [ g(y)]
x~ r y
x
y
• Proof. Induction on n.
– Base case (n = 1). For F0 , F1,G0 ,G1 ³ 0
– Induction step (n > 1). f0 (x1,… , xn-1) = f (x1,… , xn-1, 0)
f1 (x1,… , xn-1) = f (x1,… , xn-1,1)
Induction
by base
case
Proof of the base case
• Assume w.l.o.g. F0 + F1 = G0 +G1 =1
• “Two-point ineq.” (where r* =1- 2k1 ):
• Use the following substitutions: s = a + b, t = ab.
Pk (a, b)
where
To show SOS proof of
Pk (a, b)
where
³0
By the Fundamental Theorem of Algebra, a univariate
polynomial is SOS if it is nonnegative.
Only need to prove Qk,0 ,Q k,i are nonnegative.
To show SOS proof of
Pk (a, b)
where
³0
Proof of Qk,0 (t) ³ 0 :
Case 1. (t < 0) Straightforward.
Case 2. (t ³ 0) With the assistance of Zeilberger's alg.
[Zei90, PWZ97],
then its relatively easy to check Qk,0 (t) ³ 0
To show SOS proof of
Pk (a, b)
where
³0
Proof of Qk,i (t) ³ 0: By convexity, enough to prove
³0
By guessing the form of a polynomial recurrence and solving
via computer,
Finally prove the nonnegativity by induction.
Summary of the proof
• Induction on the number of variables
• Base case: “two-point” inequality
– Variable substitution
– Prove the nonnegativity of a class of univariate
polynomials
• Via the assistance of computer-algebra
algorithms, find out proper closed form or
recursions
• Prove directly or by induction
Open questions
• Is there constant-degree SOS proof for
Min-VC(FRgn ) >1- o(1) when g ³ .1 logn n ?
Recall that we showed a (1/γ)-degree proof
1
when g >> logn
.
Thanks!