Non Linear Invariance Principles

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Non Linear Invariance Principles with
Applications
Elchanan Mossel U.C. Berkeley http//stat.berkel
ey.edu/mossel
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Lecture Plan

• Background Noise Stability in Gaussian Spaces
• Noise Ornstein-Uhlenbeck process.
• Bubbles and half-spaces.
• Double Bubbles and the Peace Sign Conjecture.
• An invariance principle
• Half-Spaces Majorities are stablest
• Peace-signs Pluralities are stablest?
• Voting schemes.
• Computational hardness of graph coloring.

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Gaussian Noise

• Let 0 ? ? ? 1 and f, g Rn ? Rm.
• Define ltf, ggt? Eltf(N) , g(M) gt, where
  N,M Normal(0,I) with ENi Mj ? ?(i,j).
• For sets A,B let ltA,Bgt? lt1A,1Bgt?
• Let ?n standard Gaussian volume
• Let ?n Lebsauge measure.
• Let ?n-1, ?n-1 corresponding (n-1)-dims areas.

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Some isoperimetric results

• I. Ancient Among all sets with ?n(A) 1 the
  minimizer of ?n-1(? A) is A Ball.
• II. Recent (Borell, Sudakov-Tsierlson 70s) Among
  all sets with ?n(A) a the minimizer of ?n-1(?
  A) is A Half-Space.
• III. More recent (Borell 85) For all ?, among
  all sets with ?(A) a the maximizer of ltA,Agt? is
  given by A Half-Space.

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Double bubbles

• Thm1 (Double-Bubble)
• Among all pairs of disjoint sets A,B with ?n(A)
  ?n(B) a, the minimizer of ?n-1(? A ? ? B) is a
  Double Bubble
• Thm2 (Peace Sign)
• Among all partitions A,B,C of Rn with ?(A) ?(B)
  ?(C) 1/3 , the minimum of ?(? A ? ? B ? ? C)
  is obtained for the Peace Sign
• 1. Hutchings, Morgan, Ritore, Ros. Reichardt,
  Heilmann, Lai, Spielman 2. Corneli, Corwin,
  Hurder, Sesum, Xu, Adams, Dvais, Lee, Vissochi

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The Peace-Sign Conjecture

• Conj
• For all 0 ? ? ? 1,
• all n ? 2
• The maximum of
• ltA, Agt? ltB, Bgt? ltC, Cgt?
• among all partitions (A,B,C) of Rn with ?n(A)
  ?n(B) ?n(C) 1/3 is obtained for
• (A,B,C) Peace Sign

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Lecture Plan

• Background Noise Stability in Gaussian Spaces
• Noise Ornstein-Uhlenbeck operator.
• Bubbles and half-spaces.
• Double Bubbles and the Peace Sign Conjecture.
• An invariance principle
• Half-Spaces Majorities are stablest
• Peace-signs Pluralities are stablest?
• Voting schemes.
• Computational hardness of graph coloring.

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Influences and Noise in product Spaces

• Let X be a probability space.
• Let f ? L2(Xn,R). The ith influence of f is
  given by
• Ii(f) E Varf x1,,xi-1,xi1,,xn
• (Ben-Or,Kalai,Linial, Efron-Stein 80s)
• Given a reversible Markov operator T on X and
• f, g Xn ? R define the T - noise form by
• ltf, ggtT Ef T? n g
• The 2nd eigen-value ?(T) of T is defined by
• ?(T) max ? ? ? spec(T), ? lt 1

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Influences and Noise in product Spaces Example
1

• Let X -1, 1 with the uniform measure.
• For the dictator function xj Ii(xj) ?(i,j).
• For the majority m(x) sgn(?1 ? i ? n xi)
  function Ii(m) ? (2 ? n)-1/2.
• Let T be the Beckner Operator on X
• Ti,j ? ?(i,j) (1-?)/2.
• T xi ? xi and ltxi, xigtT ?.
• ltm, mgtT 2 arcsin(?) / ?
• ?(T) ?.

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Definition of Voting Schemes

• A population of size n is to choose between two
  options / candidates.
• A voting scheme is a function that associates to
  each configuration of votes an outcome.
• Formally, a voting scheme is a function f
  -1,1n ! -1,1.
• Assume below that
• f(-x1,,-xn) -f(x1,,xn)
• Two prime examples
• Majority vote,
• Electoral college.

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A voting model

• At the morning of the vote
• Each voter tosses a coin.
• The voters vote according
• to the outcome of the coin.

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A model of voting machines

• Which voting schemes are more robust against
  noise?
• Simplest model of noise The voting machine flips
  each vote independently with probability ?.
• ltf, fgt1-2 ? correlation of intended vote with
  actual outcome.

Registered vote
Intended vote
1
prob
e
-1
-1
prob
1
-
e
-1
prob
e
1
1
prob
1
-
e
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Majority and Electoral College

• ltm, mgt? ? 2 arcsin ? / ? n ? ? ? 1
  c(1-?)1/2 ? ? 1
• for m(x) majority(x) sgn(?i1n xi)
• Result is essentially due to Sheppard (1899) On
  the application of the theory of error to cases
  of normal distribution and normal correlation.
• For n1/2 n1/2 electoral college f
• ltf,fgt? ? 1- c (1-?)1/4 n ? ?, ? ? 1
• ltf,fgt-1/2 determined prob.
• of Condorcet Paradox (Kalai)

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An easy answer and a hard question

• Noise Theorem (folklore) Dictatorship, f(x) xi
  is the most stable balanced voting scheme.
• In other words, for all schemes, for all f
  -1,1n ? -1,1 with Ef 0 it holds that ltf,
  fgt? ? ? ltx1, x1gt?
• Harder question What is the stablest voting
  scheme not allowing dictatorships or Juntas?
• For example, consider only symmetric monotone f.
• More generally What is the stablest voting
  scheme f satisfying for all voters i Ii(f)
  Pf(x1,,xi,,xn) ? f(x1,,-xi,,xn) lt ? where n
  ? ? and ? ? 0.

X
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Influences and Noise in product Spaces Example
2

• Let X 0,1,2 with the uniform measure.
• Let Ti,j ½ ?(i ? j)
• Then ?(T) ½ and
• Claim (Colouring Graph) Consider Xn as a graph
  where (x,y) ? Edges(Xn) iff xi ? yi for all i.
• Let A,B ? Xn. Then ltA, BgtT 0 iff there are no
  edges between A and B. In particular, A is an
  independent set iff ltA, AgtT 0.
• Q How do large independent sets look like?

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Graph Colouring An Algorithmic Problem

• Let ?(G) min of colours needed
• to colour the vertices of a graph G
  so that no edge is
  monochramatic.
• ApxCol(q,Q)
• Given a graph G, is ?(G) ? q or ?(G) ? Q ?
• This is an algorithmic problem. How hard is it?
• For q2 easy simply check bipartiteness
• For q3, no efficient algorithms are known unless
  Q gtG0.1
• Efficient Running time that is polynomial in
  G.
• Also known that ?(3,4) and ?(3,5) are NP-hard.
• NP-hard Nobody believes polynomial time
  algorithms exist.
• What about ?(3,6) ?????

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Graph Colouring An Algorithmic Problem

• In 2002, Khot introduced a family of algorithmic
  problems called games. He speculated that these
  problems are NP-hard.
• These problems resisted multiple algorithmic
  attacks.
• Subhash games conjecture ?
• Claim Consider 0,1,2n as a graph G where
  (x,y) ? Edges(G) iff xi ? yi for
  all i.
• Let Q gt 3. Suppose that ? ? such that for all n
  if there are no edges between A and B ? 0,1,2n
  (ltA,BgtT 0) and A,B gt 3n/Q then there
  exists an i such that Ii(A) gt ? and Ii(B) gt ?.
• Then ApxColor(3,Q) is NP hard.

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Graph Colouring An Algorithmic Problem
u
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Graph Colouring An Algorithmic Problem
u
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Influences and Noise in product Spaces Example
3

• Let X 0,1,2 with the uniform measure.
• Let ?0, ?1, ?2 (1,0,0), (0,1,0),(0,0,1) ? R3.
• Let d Xn ? R3 defined by d(x) ?x(1)
• Let p Xn ? R defined by p(x) ?y
• where y is the most frequent value among the xi.
• Ii(d) 2/3 ?(i,1) Ii(p) ? c n-1/2.
• For 0 ? ? ? 1, let T be the Markov operator on X
  defined by Ti,j ? ?(i,j) (1-?)/3.
• ltd, dgtT ? Var(d).

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Gaussian Noise Bounds

• Def For a, b, ? ? 0,1 , let
• ?(a, b, r) sup lt F,G gt? F,G ? R, ?F
  a, ?G b
• ?(a, b, r) inf lt F,G gt? F,G ? R, ?F
  a, ?G b
• Thm Let X be a finite space. Let T be a
  reversible Markov operator on X with ? ?(T) lt
  1.
• Then ? ? gt 0 ? ? gt 0 such that for all n and all
  f,g Xn ? 0,1 satisfying maxi
  min(Ii(f), Ii(g)) lt ?
• It holds that ltf, ggtT ? ?(Ef, Eg, ?) ?
  and
• ltf, ggtT ? ?(Ef, Eg, ?)
  - ?
• M-ODonnell-Oleskiewicz-05 Dinur-M-Regev-06

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Example 1

• Taking T on -1,1 defined by Ti,j ? ?(i,j)
  (1-?)/2
• Thm ? Claim ? f -1,1n ? -1,1 with Ii(f) lt
  ? for all i and Ef 0 it holds that
• ltf, fgtT ? ltF, Fgt? ? where F(x) sgn(x)
• ltF, Fgt? 2 arcsin(?)/ ? (F is known by
  Borell-85)
• So Majority is Stablest Most Stable Voting
  Scheme among low influence ones.
• Weaker results obtained by Bourgain 2001.
• ? tight in-approximation result for MAX-CUT.
• Khot-Kindler-M-ODonnell-05

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Example 2

• Taking T on 0,1,2 defined by Ti,j ½ ?(i ? j)
• Thm ? Claim ? ? gt 0 ? ? gt 0 s.t. if A,B ?
  0,1,2n have no edges between them and PA,
  PB ? ? then
• There exists an i s.t. Ii(A), Ii(B) ? ?.
• Proof follows from Borell-85 showing ?(?,?,1/2) gt
  0.
• Claim ? Hardness of approximation
  result for graph-colouring
• For any constant K, it is NP hard to
• colour 3-colorable graphs using K colours.
• Dinur-M-Regev-06

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Example 3

• Taking T on 0,1,2 defined by Ti,j ? ?(i,j)
  (1-?)/3
• Recall ?0,?1,?2 (1,0,0),(0,1,0),(0,0,1)
• Thm Peace Sign Conjecture?
• Claim (Plurality is Stablest)
• ? f 0,1,2n ? ?0,?1,?2 with Ef
  (1/3,1/3,1/3) and Ii(f) lt ? for all i, it holds
  that
• ltf, fgtT ? limn ? ? ltp , pgtT ?, where
• p is the plurality function on n inputs
  (Plurality is Stablest)
• Claim ? Optimal Hardness of approximation
  result for MAX-3-CUT.

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More results

• More applied results use Noise-Stability bounds
• Social choice Kalai (Paradoxes).
• Hardness of approximation
• Dinur-Safra, Khot, Khot-Regev, Khot-Vishnoy etc.

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Gaussian Noise Bounds

• Def For a, b, ? ? 0,1 , let
• ?(a, b, r) sup lt F,G gt? F,G ? R, ?F
  a, ?G b
• ?(a, b, r) inf lt F,G gt? F,G ? R, ?F
  a, ?G b
• Thm Let X be a finite space. Let T be a
  reversible Markov operator on X with ? ?(T) lt
  1.
• Then ? ? gt 0 ? ? gt 0 such that for all n and all
  f,g Xn ? 0,1 satisfying maxi
  min(Ii(f), Ii(g)) lt ?
• It holds that ltf, ggtT ? ?(Ef, Eg, ?) ?
  and
• ltf, ggtT ? ?(Ef, Eg, ?)
  - ?
• M-ODonnell-Oleskiewicz-05 Dinur-M-Regev-06

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Gaussian Noise Bounds

• Proof Idea
• Low influence functions are close to functions in
  L2(?) L2(N1,N2,).
• Let H?a,b be
• ?n f Xn ? a, b ? i Ii(f) lt ?, Ef
  0, Ef2 1
• Then H? ? f ? L2(?) Ef 0, Ef2 1, a ?
  f ? b
• ? noise forms in H? a,b noise forms of a,
  b bounded functions in L2(?)

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An Invariance Principle

• For example, we prove
• Invariance Principle MODonnellOleszkiewicz(05)
  
• Let p(x) ?0 lt S k aS ?i 2 S xi be a degree
  k multi-linear polynomial with p2 1 and Ii(p)
  ? ? for all i.
• Let X (X1,,Xn) be i.i.d. PXi ? 1 1/2 .
• N (N1,,Nn) be i.i.d. Normal(0,1).
  
• Then for all t
• Pp(X) t - Pp(N) t O(k ?1/(4k))
• Note Noise form kills high order monomials.
• Proof works for any hyper-contractive random vars.

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Invariance Principle Proof Sketch

• Suffices to show that 8 smooth F (sup F(4) C
  ), EF(p(X1,,Xn) is close to EF(p(N1,,Nn)).

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Invariance Principle Proof Sketch

• Write p(X1,,Xi-1, Ni, Ni1,,Nn) R Ni S
• p(X1,,Xi-1, Xi, Ni1,,Nn) R Xi
  S
• F(RNi S) F(R) F(R) Ni S F(R) Ni2 S2/2
  F(3)(R) Ni3 S3/6 F(4)() Ni4
  S4/24
• EF(R Ni S) EF(R) EF(R) ENi2 /2
  EF(4)()Ni4S4/24
• EF(R Xi S) EF(R) EF(R) EXi2 /2
  EF(4)()Xi4 S4/24
• EF(R Ni S) EF(R Xi S) ? C ES4
• But, ES2 Ii(p).
• And by Hyper-Contractivity, ES4 ? 9k-1 ES2
• So EF(R Ni S) EF(R Xi S) ? C 9k Ii2

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Summary

• Prove the Peace Sign Conjecture (Isoperimetry)
• ? Plurality is Stablest (Low Inf Bounds)
• ? MAX-3-CUT hardness (CS) and voting.
• Other possible application of invariance
  principle
• To Convex Geometry?
• To Additive Number Theory?

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