Lecture 10: Hypercontractivity

Transcription

1 CS 880: Advanced Comlexity Theory /15/008 Lecture 10: Hyercontractivity Instructor: Dieter van Melkebeek Scribe: Baris Aydinlioglu This is a technical lecture throughout which we rove the hyercontractivity of the noise oerator, a result that will be used in later lectures. The reader may wish to review the notes of lecture 6 for a discussion of the noise oerator T α, the -norm of a function from the Boolean cube to the reals, and the notion of hyercontractivity. 1 Hyercontractivity Theorem Recall the definition of the noise oerator on functions f : { 1,1} n R: [ ] T α fx E fy, with ǫ 1 α y ǫx, where y ǫ x refers to the string y obtained by fliing each bit of x indeendently with robability ǫ. Also recall that T α fx S [n] α S ˆfSχS x. Intuitively, the noise oerator has a smoothening effect on a function, in the sense that the resulting function is a weighted average of the original function around some neighborhood of its argument. In the Fourier sectrum, the effect is that the higher freuencies get damened out. One conseuence of this is hyercontractivity. Theorem 1. For all,, α such that 1 and α 1 1, and for all functions f : { 1,1} n R, T α f f. 1 Remark: If α 0 then 1 always holds, for then Tα f E[f] E[ f ] 1 f, where the ineuality follows from Hölder s ineuality: E [ fg] fg, if with, 1. If α ±1 then 1 fails unless or f is constant in absolute value. This follows because T ±1 fx f±x, where x denotes x with all its bits flied, and because the only functions f for which f f for are those that are constant in absolute value. In roving the theorem we will arrive at the condition in the statement of the theorem on α as the weakest one that guarantees 1 to hold. Proof of the Hyercontractivity Theorem The roof is by induction on n. In the base case we develo the condition on α and in the inductive ste we maintain it. 1

2 .1 Base Case The base case is for n 1. Note that in this case we can reresent any function f : { 1,1} n R with the oint f 1,f1 in -sace. In what follows we say f to mean either the function or the oint that it corresonds to in -sace, which is clear in context. We want to show 1 1 min max {β : α β Tα f f f }. Let f 1 a and f1 b. Then T α f 1 1+α a + 1 α b, and T α f 1+α b + 1 α a. WLOG suose a,b > 0. Notice that as a oint in -sace T α f is a convex combination of the oints a,b and b,a, hence resides somewhere on the line segment joining a,b and b,a. As α gets closer to 1, T α f gets closer to a,b, which is the oint for f. As α gets closer to 1, T α f gets closer to b,a, which is the oint for T 1 f. For α 0, T α f is the midoint of the line segment [a,b,b,a]. Recall from Lecture 6 that the -norm of a vector corresonds to the amount that the unit -circle should be scaled so that the vector is on the scaled circle. From this it follows that given f, asking for the largest β for which ineuality 1 holds for all α β amounts to asking for the largest α such that, if we scale the unit -circle so that f is on it, and if we scale the unit -circle by the same amount, T α f remains within the scaled -circle. Note that for fixed a,b, T α f is linear in α, and therefore the last uestion is euivalent to asking for the fraction of the line segment [a,b,b,a] that falls within the -circle. b,a Tαf > f Tαf < f T αf f α 0 α 1 α 1 a,b Figure 1: T α resides on the line segment [a,b,b,a], where a,b f 1,f1. The outer circle reresents the unit -circle scaled by f, and the inner circle reresents the unit -circle scaled by the same amount. Note that the -circle and -circle meet on the diagonal. The figure is for and. We state without roof that this ratio decreases as we make a and b get closer while keeing f fixed. We can see that this is lausible if we take and ; see figure.1. Letting δ a b a+b, from the foregoing it follows that we want to find, as δ aroaches zero, the largest α such that T α f f. We do this in the rest of this section. First we write Tα f and f in terms of δ: T α f Tα f 1 + T α f1

3 0 b 1, a 1 b, a a, b a 1, b 1 Figure : An illustration of the fact, in the case for and, that as a and b get closer, max {β : α β Tα f f } decreases. As we move the line segment from [a 1,b 1,b 1,a 1 ] to [a,b,b,a ], the ortion of the line segment outside the -circle decreases at a slower rate than the ortion inside the -circle, due to convexity of the -circle. [ 1 a + b [ 1 a + b a + b and by setting α 1 and relacing by, Therefore T α f f iff + a b α + 1 a + b a b 1 a b a + b α 1 + a b a + b α δ α δ α, ]1 α ]1 f T1 f a + b δ δ δ α δ α 1 + δ δ. 3 Now, using Taylor s exansion 1 ± δα 1 ± δα+ 1 δα ±, we can rewrite the LHS of 3 as LHS δ 0 δ δα δα, 3

4 where the second line follows again from Taylor s exansion. Similarly, Putting together, RHS δ. δ 0 1 LHS < RHS α < 1 δ 0 1 α < 1, and by continuity LHS RHS δ 0 α 1 1, roving the base case.. Inductive Ste Consider x { 1,1} n and a nontrivial artition x x 1 x with x 1 k, 0 < k < n. Our first ste is to obtain an exression for T α fx in terms of T α alied to functions on fewer variables, so we can aly our induction hyothesis. T α fx 1 x α S ˆfSχS x 1 x S [n] artitioning S consistent with the way x is artitioned, S 1 [k] S {k+1,...,n} α S 1 α S ˆfS1 S χ S1 x 1 χ S x α S 1 α S ˆfS1 S χ S x χ S1 x 1 S 1 S } {{} + defining a function g x : { 1,1} k R such that ĝ x S 1 is +, S 1 α S 1 ĝ x S 1 χ S1 x 1 T α g x x1, 4 where T α in the last line is an oerator for functions on the Boolean k-cube. For future reference we oint out that for any fixed x 1, g x x 1 as a function of x can be exressed as the result of alying the noise oerator T α on the restriction of f to the Boolean n k-cube defined by x 1. We formalize this observation in the following lemma. 4

5 Lemma 1. Let g x : { 1,1} k R be the function such that its Fourier coefficient corresonding to S 1 [k] is given by ĝ x S 1 S α S ˆfS1 S χ S x. Then g x x 1 T α f R x, where R [k],x 1 restricts f to give f R : { 1,1} n k R with f R x fx 1 x. Proof. g x x 1 S 1 [k] S 1 ĝ x S 1 χ S1 x 1 S {k+1,...,n} α S ˆfS1 S χ S x χ S1 x 1 α S ˆfS1 S χ S1 x 1 χ S x. S S } 1 {{} # Recall that as art of the discussion in Lecture 9 on random restrictions, we obtained an exression for f R S, where R I,v and S [n]\i, as f R S S 1 I ˆfS 1 S χ S1 v. Setting I [k] and v x 1 in this exression gives recisely #, and so as claimed. g x x 1 S α S f R S χ S x T α f R x, Now we write Tα f in terms of Tα g x so that we can aly the induction hyothesis to the latter: Tα f [ [ E x Ex1 Tα fx 1 x ] ] by the definition of -norm [ [ E x Ex1 Tα g x x1 ] ] by 4 E x [ Tα g x ] E x [ gx ] the inner exectation is just the -th ower of a -norm. by the induction hyothesis Finally, we massage this last exression such that we get a -norm instead of a -norm so that by Lemma 1 we can aly the induction hyothesis again: T α f [ [ E x Ex1 g x x 1 ] ] 1 by the definition of -norm E x [ Ex1 [ g x x 1 ] } {{ } ] 5

6 viewing as a function of x, this entire exression is the -th root of the -norm of : {}}{ E x1 [ gx x 1 ] can be viewed as a convex combination of k functions of x. By linearity and the triangle ineuality, the -norm which is a valid norm, since 1 of the convex combination is at most the convex combination of the -norms of each term, thus yielding E x1 [ g x x 1 ] 1 E x1 [ E x [ g x x 1 ] [ gx E x1 x 1 ] 1 [ Tα E x1 f R E x1 [ f R ] ] 1 [ [ E x1 E f R x x ] ] ] 1 by the definition of -norm by Lemma 1 by the induction hyothesis by the definition of -norm [ E x1 E x [ fx1 x ] ] 1 f. 3 Next Lecture We will use the instantiation of this hyercontractivity result with to rove two theorems stated in lecture 1, namely that for every balanced function there is a variable with influence Ω, and that for every ǫ any function f is ǫ-close to a OIf/ǫ -junta. log n n 6