Free Talagrand Inequality, a Simple Proof. Ionel Popescu. Northwestern University & IMAR

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1 Free Talagrand Inequality, a Simple Proof Ionel Popescu Northwestern University & IMAR

2 A Joke If F : [0, 1] Ris a smooth convex function such that F(0)=F (0)=0, then F(t) 0 for any t [0, 1]. Proof. F is convex= F is nondecreasing

3 A Joke If F : [0, 1] Ris a smooth convex function such that F(0)=F (0)=0, then F(t) 0 for any t [0, 1]. Proof. F is convex= F is nondecreasing F (0)=0 ==== F is nondecreasing

4 A Joke If F : [0, 1] Ris a smooth convex function such that F(0)=F (0)=0, then F(t) 0 for any t [0, 1]. Proof. F is convex= F is nondecreasing F (0)=0 ==== F is nondecreasing==== F(0)=0 F 0.

5 The Wasserstein Distance onr n W(µ,ν) := inf π Π(µ,ν) x y 2 π(dx, dy), Π(µ,ν) : probability measures onr n R n with marginalsµ andν. This is a distance for the topology of weak convergence on the probability measures with second moment.

6 The Wasserstein Distance onr n W(µ,ν) := inf π Π(µ,ν) x y 2 π(dx, dy), Π(µ,ν) : probability measures onr n R n with marginalsµ andν. For givenµandν, there is a unique transport map T such thatπ=(t, Id) ν is the minimizer for W(µ,ν) and W(µ,ν) 2 = x T(x) 2 ν(dx).

7 The Wasserstein Distance onr n W(µ,ν) := inf π Π(µ,ν) x y 2 π(dx, dy), Π(µ,ν) : probability measures onr n R n with marginalsµ andν. For givenµandν, there is a unique transport map T such thatπ=(t, Id) ν is the minimizer for W(µ,ν) and W(µ,ν) 2 = x T(x) 2 ν(dx). µ t = ((1 t)x+tt(x)) ν is the geodesic path for W with W(µ t,ν)=tw(µ,ν).

8 Classical Talagrand Inequality Relative entropy: H(µ ν) = dµ f log( f )dν µ ν, f= µ ν. dν Theorem (Talagrand). Assumeν(dx)=e ξ(x) dx is a probability measure onr n such thatξ(x) ρ x 2 is a convex function. Then ρw(µ,ν) 2 H(µ ν). (T(ρ))

9 Classical Talagrand Inequality Relative entropy: H(µ ν) = dµ f log( f )dν µ ν, f= µ ν. dν Theorem (Talagrand). Assumeν(dx)=e ξ(x) dx is a probability measure onr n such thatξ(x) ρ x 2 is a convex function. Then ρw(µ,ν) 2 H(µ ν). (T(ρ)) The particular case of the Gaussian:ξ(x)= x 2 /2+C.

10 Classical Talagrand Inequality Relative entropy: H(µ ν) = dµ f log( f )dν µ ν, f= µ ν. dν Theorem (Talagrand). Assumeν(dx)=e ξ(x) dx is a probability measure onr n such thatξ(x) ρ x 2 is a convex function. Then ρw(µ,ν) 2 H(µ ν). (T(ρ)) The particular case of the Gaussian:ξ(x)= x 2 /2+C. Notice here the fact that n does not appear in T(ρ).

11 Classical Talagrand Inequality Relative entropy: H(µ ν) = dµ f log( f )dν µ ν, f= µ ν. Theorem (Talagrand). Assumeν(dx)=e ξ(x) dx is a probability measure onr n such thatξ(x) ρ x 2 is a convex function. Then dν ρw(µ,ν) 2 H(µ ν). (T(ρ)) The particular case of the Gaussian:ξ(x)= x 2 /2+C. Notice here the fact that n does not appear in T(ρ). T(ρ) extends to infinite dimensional case with the Wiener measure in place ofν.

12 Proofs were given by Talagrand, Otto and Villani, D. Cordero-Erausquin.

13 Proofs were given by Talagrand, Otto and Villani, D. Cordero-Erausquin. Proof. Take n=1 and the (smooth) map T such that µ=t ν= fν. Then T (x) f (T(x))e ξ(t(x)) = e ξ(x) E(µ ν) ρw(µ,ν) 2 = (ξ(t(x)) ξ(x) log(t (x)) ρ(x T(x)) 2 )ν(dx).

14 Proofs were given by Talagrand, Otto and Villani, D. Cordero-Erausquin. Proof. Take n=1 and the (smooth) map T such that µ=t ν= fν. Then T (x) f (T(x))e ξ(t(x)) = e ξ(x) E(µ ν) ρw(µ,ν) 2 = (ξ(t(x)) ξ(x) log(t (x)) ρ(x T(x)) 2 )ν(dx). Take T t (x)=(1 t)x+tt(x),µ t = (T t ) ν and F(t)=H[µ t ν] ρw(µ t,ν) = (ξ(t t ) ξ(x) log(t t (x)) ρ(x T t(x)) 2 )ν(dx).

15 F(t)=H[µ t ν] ρw(µ t,ν) 2 = (ξ(t t ) ξ(x) log(t t (x)) ρ(x T t(x)) 2 )ν(dx). Sinceξ(x) ρx 2 is convex, it follows that F(t) is convex.

16 F(t)=H[µ t ν] ρw(µ t,ν) 2 = (ξ(t t ) ξ(x) log(t t (x)) ρ(x T t(x)) 2 )ν(dx). Sinceξ(x) ρx 2 is convex, it follows that F(t) is convex. F(0)=0 and (ξ F (0)= (x)e ξ(x) (T(x) x) (T (x) 1)e ξ(x)) dx = ((x T(x))e ξ(x) ) dx=0.

17 F(t)=H[µ t ν] ρw(µ t,ν) 2 = (ξ(t t ) ξ(x) log(t t (x)) ρ(x T t(x)) 2 )ν(dx). Sinceξ(x) ρx 2 is convex, it follows that F(t) is convex. F(0)=0 and (ξ F (0)= (x)e ξ(x) (T(x) x) (T (x) 1)e ξ(x)) dx = ((x T(x))e ξ(x) ) dx=0. Now, joke to get that F(t) 0. In particular H(µ ν) ρw(µ,ν) 2 = F(1) 0.

18 Talagrand, Log Sobolev and HWI Inequalities A measureνis said to satisfy T(ρ) if for any probabilityµ ρw(µ,ν) 2 H(µ ν).

19 Talagrand, Log Sobolev and HWI Inequalities A measureνis said to satisfy T(ρ) if for any probabilityµ ρw(µ,ν) 2 H(µ ν). ν satisfies Log Sobolev with constantρ(lsi(ρ)) if for anyµ, H(µ ν) 1 4ρ I(µ ν) where I(µ ν)= log ( dµ dν) 2 dµ=4 dµ dν 2 dν.

20 Talagrand, Log Sobolev and HWI Inequalities A measureνis said to satisfy T(ρ) if for any probabilityµ ρw(µ,ν) 2 H(µ ν). ν satisfies Log Sobolev with constantρ(lsi(ρ)) if for anyµ, H(µ ν) 1 4ρ I(µ ν) where I(µ ν)= log ( dµ dν) 2 dµ=4 ν satisfies HWI(ρ) inequality if for anyµ: dµ dν 2 dν. H(µ ν) W(µ ν) I(µ ν) ρw(µ ν) 2.

21 Theorem (Otto and Villani). Assumeν=e ξ(x) dx is a probability measure onr n. 1. Ifξ(x) ρ x 2 is convex for someρ R, then HWI(ρ) holds true.

22 Theorem (Otto and Villani). Assumeν=e ξ(x) dx is a probability measure onr n. 1. Ifξ(x) ρ x 2 is convex for someρ R, then HWI(ρ) holds true. 2. Ifξ(x) ρ x 2 is convex for aρ>0, then T(ρ) implies LSI(ρ).

23 Theorem (Otto and Villani). Assumeν=e ξ(x) dx is a probability measure onr n. 1. Ifξ(x) ρ x 2 is convex for someρ R, then HWI(ρ) holds true. 2. Ifξ(x) ρ x 2 is convex for aρ>0, then T(ρ) implies LSI(ρ). 3. Ifξ(x) C x 2 is convex for a certain C R, then LSI(ρ) implies T(ρ).

24 Theorem (Otto and Villani). Assumeν=e ξ(x) dx is a probability measure onr n. 1. Ifξ(x) ρ x 2 is convex for someρ R, then HWI(ρ) holds true. 2. Ifξ(x) ρ x 2 is convex for aρ>0, then T(ρ) implies LSI(ρ). 3. Ifξ(x) C x 2 is convex for a certain C R, then LSI(ρ) implies T(ρ). 4. In particular forξ(x) ρ x 2 is convex forρ>0, then T(ρ) is equivalent with LSI(ρ).

25 The main idea The proof is based on the geometric/pde interpretation of the gradient of the entropy functional H(µ ν). Ifνis a fixed measure andµis another measure, then I(µ ν)= grad µ H( ν) 2. LSI(ρ) translates in this context H(µ ν) 1 4ρ grad µ H( ν) 2.

26 The gradient flow µ t : µ t = grad µt H( ν) with µ 0 =µ. This combined with LSI(ρ) implies that d dt H(µ t ν)= grad µt H( ν), µ t = grad µt H( ν) 2 4ρH(µ t ν) which yields that H(µ t ν) converges exponentially fast to 0, lim t µ t =ν and g(t)=w(µ,µ t )+ H(µ t ν)/ρ is nonincreasing in t. Therefore, 0= g( ) g(0) results with T(ρ): W(µ,ν) H(µ t ν)/ρ.

27 The Free Counterpart Free entropy with potential Q for probability measures onr: E(µ)= Q(x)µ(x) log x y µ(dx)µ(dy). There is a unique probability measureµ Q such that E(µ Q )=inf µ E(µ).

28 The Free Counterpart Free entropy with potential Q for probability measures onr: E(µ)= Q(x)µ(x) log x y µ(dx)µ(dy). There is a unique probability measureµ Q such that E(µ Q )=inf µ E(µ). The variational characterization ofµ Q : for a.e. x suppµ Q(x)=2 log x y µ Q (dx)

29 The Free Counterpart Free entropy with potential Q for probability measures onr: E(µ)= Q(x)µ(x) log x y µ(dx)µ(dy). There is a unique probability measureµ Q such that E(µ Q )=inf µ E(µ). The variational characterization ofµ Q : for a.e. x suppµ Q(x)=2 log x y µ Q (dx) Q 2 (x)= x y µ Q(dx).

30 If Q(x)=x 2 /2 the minimizer for the free entropy is µ Q (dx)=s(dx)= 1 2π 4 x 2 dx. The analog of the relative entropy is played by The relative free information: I(µ µ Q )= where Hµ(x)= 2 x y µ(dx). E(µ µ Q )=E(µ) E(µ Q ). (Hµ(x) Q (x)) 2 µ(dx)

31 Free Talagrand: Theorem (Biane and Voiculescu). For Q(x)=x 2 /2, 1 2 W(µ, s)2 E(µ s). The technique of the proof is similar to the one of Otto and Villani for the classical case. The Idea: If S is a semicircular free with X, then X(t)=e t/2 X+(1 e t ) 1/2 S, d dt H(X(t) S)=I(X(t) S) t W(X(t), S) 2H(X(t) S) is nondecreasing lim (W(X(t), S) 2H(X(t) S))=0. t

32 Random Matrices and Free Entropy OnM sa n (C) consider the probability measure P n Q (da)= 1 Z n (Q) e ntr nq(a) da. The distribution of the eigenvalues is given by Λ n (dx)= 1 n Z n (Q) e n i=1 Q(x i) = 1 Z n exp n2 1 n 1 i< j n n i=1 n (x i x j ) 2 dx i Q(x i ) 1 n 2 i=1 1 i j n log x i x j n dx i. i=1

33 B. Arous and A. Guionnet: Takeλ n to be the distribution on ofη n = 1 n n i=1 δ x i underλ n. Thenλ n satisfies a Large Deviation Principle with rate function given by H(µ µ Q ) i.e. for any measurable set A of probability measures onr, inf E(µ µ Q ) lim inf µ Å n In particularη n µ Q. lim sup n 1 n 2 log(λ n(a)) 1 n 2 log(λ n(a)) inf µ A E(µ µ Q )

34 Hiai, Petz and Ueda: P n Q (da)= 1 Z n (Q) e ntr nq(a) da. For a givenµ, take Q µ (x)= log x y µ(dy). If Q(x) ρx 2 is a convex function, then A ntr n Q(A) nρ A 2 is convex, and from classical Talagrand: nρw(p n Q µ,p n Q )2 H(P n Q µ P n Q ) W(µ,µ Q ) 2 lim inf n W(Pn Q µ,p n Q )2 /n lim n H(Pn Q µ P n Q )/n2 = E(µ µ Q ) ρw(µ,µ Q ) 2 E(µ µ Q ).

35 Free Talagrand, Log Sobolev and HWI Inequalities For a given potential Q, the free Talagrand T(ρ): ρw(µ,µ Q ) 2 E(µ µ Q ), for anyµ.

36 Free Talagrand, Log Sobolev and HWI Inequalities For a given potential Q, the free Talagrand T(ρ): ρw(µ,µ Q ) 2 E(µ µ Q ), for anyµ. Log Sobolev LSI(ρ) if for anyµ, E(µ µ Q ) 1 4ρ I(µ µ Q) where I(µ µ Q )= (Hµ(x) Q (x)) 2 µ(dx), Hµ(x)= 2 x y µ(dx).

37 Free Talagrand, Log Sobolev and HWI Inequalities For a given potential Q, the free Talagrand T(ρ): ρw(µ,µ Q ) 2 E(µ µ Q ), for anyµ. Log Sobolev LSI(ρ) if for anyµ, E(µ µ Q ) 1 4ρ I(µ µ Q) where I(µ µ Q )= (Hµ(x) Q (x)) 2 µ(dx), Hµ(x)= 2 x y µ(dx). HWI(ρ) inequality if for anyµ: E(µ µ Q ) W(µ,µ Q ) I(µ µ Q ) ρw(µ,µ Q ) 2.

38 The Simpler Proof of T(ρ) If Q(x) ρx 2 is convex for a certainρ>0, then T(ρ) holds true. Let T be the transport map ofµ Q intoµ, i.e.µ=t µ Q. T is nondecreasing. Set T t (x)=(1 t)x+tt(x),µ t = (T t ) µ Q and F(t)=E(µ t µ Q ) ρw(µ t,µ Q ) 2

39 The Simpler Proof of T(ρ) If Q(x) ρx 2 is convex for a certainρ>0, then T(ρ) holds true. Let T be the transport map ofµ Q intoµ, i.e.µ=t µ Q. T is nondecreasing. Set T t (x)=(1 t)x+tt(x),µ t = (T t ) µ Q and F(t)=E(µ t µ Q ) ρw(µ t,µ Q ) 2 (Q(Tt = E(µ Q )+ (x)) ρ(x T t (x)) 2) µ Q (dx) 2 x>y log(t t (x) T t (y))µ Q (dx)µ Q (dy)

40 F is convex; F(0)=0; From Q (x)=2 1 x y µ Q(dx), it follows that T(x) x F (0)= (T(x) x)q (x)µ Q (dx) 2 x y µ Q(dx)µ Q (dy) = 0.

41 F is convex; F(0)=0; From Q (x)=2 1 x y µ Q(dx), it follows that T(x) x F (0)= (T(x) x)q (x)µ Q (dx) 2 x y µ Q(dx)µ Q (dy) = 0. Therefore, F(t) 0 and then E(µ µ Q ) ρw(µ,µ Q ) 2 = F(1) 0.

42 F is convex; F(0)=0; From Q (x)=2 1 x y µ Q(dx), it follows that T(x) x F (0)= (T(x) x)q (x)µ Q (dx) 2 x y µ Q(dx)µ Q (dy) = 0. Therefore, F(t) 0 and then E(µ µ Q ) ρw(µ,µ Q ) 2 = F(1) 0. Using these ideas and some from Erausquin, M. Ledoux reproved LSI(ρ) and, for the first time, proved HWI(ρ).

43 The circle case Q : T ( π,π] Ris the potential and call V(x)=Q(e ix ). E(µ)= V(x)µ(dx) log e ix e iy µ(dx)µ(dy) There is a measureµ Q such that E Q (µ) is minimized. We define then E(µ µ Q )=E(µ) E(µ Q ). ( I Q (µ)= [Hµ(x) V (x)] 2 µ(dx) V (x)µ(dx) ) 2 where Hµ(x) = cot(x y)µ(dy).

44 If Q(e ix ) ρx 2 is convex onr, then T(ρ) is (ρ+1/4)w(µ,µ Q ) 2 E[µ µ Q ], the LSI(ρ) is and HWI(ρ) is E[µ µ Q ] 1 1+4ρ I(µ µ Q) E Q (µ µ Q ) W(µ,µ Q )I(µ) 1/2 (ρ+1/4)w(µ,µ Q ) 2

45 What Else?

46 What Else? These are not really non-commutative.

47 What Else? These are not really non-commutative. Is there a version of Talagrand s inequality for a tuple of non-commutative random variable with the logarithmic entropy replaced by Voiculescu s free entropy? Partial result was proved using random matrix approximation by Hiai and Ueda for the case E(a 1, a 2,...,a n )= n τ(q i (a i )) χ(a 1, a 2,...,a n ). i=1