Hypercontractive inequalities via SOS, and the Frankl-Rödl graph
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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)
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The hypercontractive inequalities
• Real functions on Boolean cube:• Noise operator: means each is -correlated with
• The hypercontractive inequality: when
Corollary.24 hypercontractive ineq.
Projection to deg-k
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The hypercontractive inequalities
• The reverse hypercontractive inequaltiy: when
Corollary [MORSS12]. when
we have
Not norms
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The hypercontractive inequalities
• The hypercontractive inequality: when Applications: KKL theorem, Invariance Principle
• The reverse hypercontractive inequaltiy: whenApplications: hardness of approximation, quantitative social choice
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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 , write where each has degree at most d
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Previous works
• Deg-4 SOS proof of 24 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]
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Results in this work
• Deg-q SOS proof of the hypercon. ineq. when p=2, q=2s for
we have
• Deg-4k SOS proof of reverse hypercon. ineq. when q= for we have
ff^(2k)gg^(2k)
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Application of the reverse hypercon. ineq. to the Lasserre SDP
• Frankl-Rödl graphs
• 3-Coloring – [FR87,GMPT11]– SDPs by Kleinberg-Goemans fails to certify [KMS98, KG98, Cha02]– Arora and Ge [AG11]: level-poly(n) Lasserre SDP
certifies– Our SOS proof: level-2 Lasserre SDP certifies
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Application of the reverse hypercon. ineq. to the Lasserre SDP
• Frankl-Rödl graphs
• VertexCover – when [FR87,GMPT11]– Level-ω(1) LS+SDP and level-6 SA+SDP fails to certify [BCGM11]– The only known (2-o(1)) gap instances for SDP
relaxations– Our SOS proof: level-(1/γ) Lasserre SDP certifies when
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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]
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SOS proof (sketch) of the reverse hypercon. ineq.
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• Statement: forexists SOS proof of
• Proof. Induction on n.– Base case (n = 1). For
– Key challenge, will prove later.
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• Statement: forexists SOS proof of
• Proof. Induction on n.– Base case (n = 1). For
– Induction step (n > 1).
Induction on (n-1) variables
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• Statement: forexists SOS proof of
• Proof. Induction on n.– Base case (n = 1). For
– Induction step (n > 1).
Induction by base
case
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Proof of the base case
• Assume w.l.o.g. • “Two-point ineq.” (where ):
• Use the following substitutions:
where
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To show SOS proof of
where
By the Fundamental Theorem of Algebra, a univariate polynomial is SOS if it is nonnegative.
Only need to prove are nonnegative.
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To show SOS proof of
where
Proof of : Case 1. ( ) Straightforward. Case 2. ( ) With the assistance of Zeilberger's alg. [Zei90, PWZ97],
then its relatively easy to check
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To show SOS proof of
where
Proof of : By convexity, enough to prove
By guessing the form of a polynomial recurrence and solving via computer,
Finally prove the nonnegativity by induction.
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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
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Open questions
• Is there constant-degree SOS proof for when ?
Recall that we showed a (1/γ)-degree proof when .
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Thanks!