Stability results for Logarithmic Sobolev inequality

Transcription

1 Stability results for Logarithmic Sobolev inequality Daesung Kim (joint work with Emanuel Indrei) Department of Mathematics Purdue University September 20, 2017 Daesung Kim (Purdue) Stability for LSI Probability Seminar 1 / 26

2 Logarithmic Sobolev inequality (LSI) The Gaussian measure γ on R n is defined by dγ = (2π) n 2 e x2 2 dx. For dν = fdγ, we define I(ν) = H(ν) = Rn f 2 dγ f f log fdγ R n (Fisher information), (the entropy). The classical logarithmic Sobolev inequality (LSI in short) is 1 I(ν) H(ν). 2 (1) The constant 1 2 is sharp, meaning that there is no constant c less than such that ci(ν) H(ν). 1 2 (2) The constant is dimension free. Daesung Kim (Purdue) Stability for LSI Probability Seminar 2 / 26

3 Logarithmic Sobolev inequality (LSI) Stam (1959): firstly proved the LSI Federbush (1969): the hypercontractivity the LSI Gross (1975): the LSI the hypercontractivity If dν = e bx b2 2 dγ = (2π) n 2 e x b 2 2 dx for b R n, then 1 2I(ν) = H(ν). Carlen (1991) characterized the equality case. That is, he showed that {e bx b2 2 dγ : b R n } are the only optimizers. From the Bechner-Hirschman uncertainty principle, he derived the LSI with remainder term. Daesung Kim (Purdue) Stability for LSI Probability Seminar 3 / 26

4 Stability for LSI The deficit of the LSI is δ LSI (ν) = 1 2I(ν) H(ν). Question: if the deficit δ LSI (ν) is small, how far is the measure ν away from the optimizers? Let X be the set of all admissible, centered probability measures, that is, X := {fdγ : f 0, fdγ = 1, R n xfdγ = 0, f 1 2 R n W 1,2 (R n, dγ)}. Remark (1) The LSI holds for all ν X. (2) Gaussian measure γ is the only optimizer in X. We want to see a distance between ν and γ when the deficit δ LSI (ν) is small. Daesung Kim (Purdue) Stability for LSI Probability Seminar 4 / 26

5 Stability for LSI Let A X, γ A, and d be a metric (or distance) on A. Definition (i) The LSI is stable under (d, A) if δ LSI (ν) 0 implies d(ν, γ) 0. (ii) The LSI is weakly stable under (d, A) if for all {ν k } A satisfying δ LSI (ν k ) 0, there exists a subsequence {ν k(l) } such that d(ν k(l), γ) 0. (iii) The LSI is unstable under (d, A) if there exists {ν k } A such that δ LSI (ν k ) 0 and lim inf k d(ν k, γ) > 0. Daesung Kim (Purdue) Stability for LSI Probability Seminar 5 / 26

6 The Wasserstein distance For p 1, we define the p-th moment of a probability measure µ by m p (µ) = R n x p dµ. Let P be the space of all probability measures and P p (R n ) = {ν P : m p (ν) < }. The Wasserstein distance of order p on P p (R n ) is W p (µ, ν) = = ( inf x y p dπ(x, y) π R n R ( n inf E X Y p) 1 p, where the infimum is taken over all couplings π of µ and ν. W 2 is called the quadratic Wasserstein distance. W 1 is called the Kantorovich Rubinstein distance. ) 1 p, Daesung Kim (Purdue) Stability for LSI Probability Seminar 6 / 26

7 The Wasserstein distance Properties of W p (1) W p defines a metric on P p (R n ). (2) If p 1 < p 2, then W p1 (µ, ν) W p2 (µ, ν). (3) W p (µ, ν k ) 0 if and only if ν k µ weakly and m p (ν k ) m p (µ). (4) (Optimal transport) For µ, ν P 2 (R n ), there exists a map T : R n R n such that ν(a) = µ(t 1 (A)) for all Borel set in R n and W 2 (µ, ν) 2 = T (x) x 2 dµ. R n (5) (Duality for p = 1) { } W 1 (µ, ν) = sup ϕ(dµ dν) : ϕ is 1-Lipschitz function.. R n Daesung Kim (Purdue) Stability for LSI Probability Seminar 7 / 26

8 The total variation distance For probability measures µ and ν, we define the total variation distance d TV (µ, ν) = sup µ(a) ν(a) A { } = sup ϕ(dµ dν) : ϕ 1 R n = inf E[1 {X Y }] π = 1 2 f 1 L 1 (dµ) (if dν = fdµ). If d TV (µ, ν k ) 0, then ν k converges to µ weakly. Daesung Kim (Purdue) Stability for LSI Probability Seminar 8 / 26

9 The entropy Let dν = fdγ. The entropy H(ν) is a distance in the following sense: Pinsker s inequality: for p > 1, 2 f 1 2 L 1 (dγ) H(ν) 2 p 1 f 1 p L p (dγ) + 2 f 1 L p (dγ) Transport (Talagrand) inequality: HWI inequality: W 2 2 (ν, γ) 2H(ν) H(ν) W 2 (ν, γ) I(ν) 1 2 W 2 2 (ν, γ) The entropy H(ν) measures how far the measure ν is away from Gaussian measure γ. Daesung Kim (Purdue) Stability for LSI Probability Seminar 9 / 26

10 Previous results Indrei, Marcon (2014) showed W 2 -stability for a certain class of measures: For M > 0 and ε (0, 1), define F(ε, M) = {e h : ( 1 + ε) D 2 h M}, and A = {fdγ : fdγ = 1, xfdγ = 0, f F(ε, M)}. R n R n Then, there exists C = C(ε, M) > 0 such that δ LSI (ν) CW2 2 (ν, γ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 10 / 26

11 Previous results Fathi, Indrei, Ledoux (2016) considered probability measures that satisfy (2, 2)-Poincaré inequality to obtain W 2 -stability. For λ > 0, let P(λ) be the set of all centered probability measure dν = fdγ such that for all smooth function ϕ : R n R that satisfies R ϕdν = 0, n λ ϕ 2 dν R n ϕ 2 dν. R n Then, there exist C 1 (λ), C 2 (λ) > 0 such that and δ LSI (ν) C 1 (λ)w 2 2 (ν, γ) δ LSI (ν) C 2 (λ) f 1 2 L 1 (dγ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 11 / 26

12 Previous results Feo, Indrei, Posteraro, Roberto (2017) obtained a stability in terms of a certain distance, for general measures. Let A := {fdγ : f > 0, R fdγ = 1, I(fdγ) < }. n Consider only n = 1. Define a distance d on A by 1 T d(µ, ν) = max{1, T } dµ where T : R R pushes forward from µ onto ν. (For F µ (x) = µ((, x]), T = Fµ 1 F ν.) Then, R δ LSI (ν) 1 2 d(ν, γ)2. Daesung Kim (Purdue) Stability for LSI Probability Seminar 12 / 26

13 Previous results Feo, Indrei, Posteraro, Roberto (2017) also proved a convolution type stability using Carlen s LSI with remainder term. Let g(x) = 2 n 4 e π x 2, dm = g 2 dx, and δ c (f) = 1 f 2 dm f 2 log f 2 dm, 2π R n R n which is another equivalent form of LSI deficit (with respect to dm). Then, for θ (0, 1 2 ), we have R n gf g f g g 2 dx cδ c (f) θ ( g(f 1) 2 L q + g(f 1) L 2)2 2θ where f(x) = f( x) and q = 4(1 θ)/(3 2θ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 13 / 26

14 Previous results Bobkov, Gozlan, Roberto, Samson (2014) The LSI is not invariant under scaling. If we scale both Fisher information and the entropy by λ > 0 and optimizing in the factor λ, we get δ LSI (ν) 1 2 ( ) 2 2H(ν) + (m 2 (γ) m 2 (ν)) for ν P 2. In particular, if m 2 (ν) m 2 (γ) = n then δ LSI (ν) ((2H(ν)) 2 + (m 2 (γ) m 2 (ν)) 2) ( W 4 2 (ν, γ) + (m 2 (γ) m 2 (ν)) 2). Talagrand inequality: 2H(ν) W 2 2 (ν, γ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 14 / 26

15 Main results Define P2 M(Rn ) = {ν X : m 2 (ν) M}, the space of probability measures whose second moments are bounded by M. Note that the second moment of γ is n, so that γ P2 M if M n. Theorem (E.Indrei & D.K., 2017+) Let M n and dν = fdγ P2 M(Rn ). Then, we have δ LSI (ν) C n,m min{w 1 (γ, ν), W1 4 (γ, ν)} where C n,m depends only on n and M. Recall that X := {fdγ : f 0, R n fdγ = 1, R n xfdγ = 0, f 1 2 W 1,2 (R n, dγ)}. Daesung Kim (Purdue) Stability for LSI Probability Seminar 15 / 26

16 Main results Remarks 2 1 (1) C n,m = ( (n + M) c 2 (n + M)). So, C n,m 0 if n or M. (2) Since W 1 is weaker than W 2, the stabilities in terms of W 2 implies W 1 results. (3) However, the space P M 2 is quite general. For example, P(λ) that used in [Fathi et al., 2014] is included in P M 2 for some M. In [Bobkov et al., 2014], the stability in terms of W 2 holds for P n 2. P(λ) := {dν = fdγ : λ R n ϕ 2 dν R n ϕ 2 dν for all ϕ satisfying R n ϕdν = 0} Daesung Kim (Purdue) Stability for LSI Probability Seminar 16 / 26

17 Main results Idea of Proof 1. The deficit of Talagrand inequality is δ Tal (ν) = 2H(ν) W 2 2 (ν, γ). From the HWI inequality, we have δ LSI (ν) δ Tal(ν) 16H(ν). 2. H(ν) is bounded by the deficit δ LSI (ν) and the second moment m 2 (ν). 3. Use the stability for Talagrand inequality by Cordero-Erausquin (2017): δ Tal (ν) c min ( W 2 1 (γ, ν), W 1 (γ, ν) ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 17 / 26

18 Main results Theorem (E.Indrei & D.K., 2017+) Let M n and dν j = f j dγ P2 M(Rn ) be centered. If δ LSI (ν j ) 0 as j, then there exists a subsequence {j(k)} k 1 such that f j(k) 1 L 1 (dγ) 0. Equivalent statement For any ε > 0, there exists η > 0 such that if dν = fdγ P2 M and 2H(ν) (1 η)i(ν), then f 1 L 1 (dγ) ε. is centered In [Fathi et al., 2014], the authors gives a stability in terms of L 1 on P(λ), while our result is a weak stability on P2 M which contains P(λ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 18 / 26

19 Main results Idea of Proof 1. I(ν) is bounded by the deficit δ LSI (ν) and the second moment m 2 (ν). So, fγ is bounded in W 1,2 (R n ). 2. Use the Rellich-Kondrasov theorem to see that fγ converges along subsequence. 3. Since the week limit and the strong limit are same, by the previous result, the subsequence converges to 1. Daesung Kim (Purdue) Stability for LSI Probability Seminar 19 / 26

20 Main results As we have seen, stability results in terms of W 2, W 1, and L 1 depend on the space P M 2. Question: is it possible to improve the result by enlarging the admissible space? It turned out that there is a counterexample. Daesung Kim (Purdue) Stability for LSI Probability Seminar 20 / 26

21 Example Observation (1) In the proof of the first result, we have seen that the entropy is bounded by the deficit and the second moment. (2) If δ LSI 0 and H(ν), then m 2 (ν). (3) If m 2 (ν) does not converge to m 2 (γ), then W 2 (ν, γ) does not converge to zero. (4) If H(ν), then f 1 L p (dγ) for p > 1, because H(ν) 2 p 1 f 1 p L p (dγ) + 2 f 1 L p (dγ). Goal: Find a sequence of centered probability measures {ν k } such that δ LSI (ν k ) 0 but H(ν k ). Daesung Kim (Purdue) Stability for LSI Probability Seminar 21 / 26

22 Example We start with a basic example. Let b R n b 2 b x, g b (x) := e 2, and dν b = g b dγ. I(ν b ) = H(ν b ) = m 2 (ν b ) = Thus, δ LSI (ν b ) = 0 for all b. Rn g b 2 dγ = b 2 g b dγ = b 2, g b R n g b log g b dγ = (b (x + b) 12 ) b 2 dγ = 1 R n R n 2 b 2, x 2 dν b = x + b 2 dγ = n + b 2. R n R n Good: H(ν b ), m 2 (ν b ) as b. Bad: the measure ν b is not centered as long as b 0. Daesung Kim (Purdue) Stability for LSI Probability Seminar 22 / 26

23 Example For k N, define f k as follows: cαg b (x) c(1 2α) cαg b (x) k 1 k k k k + 1 k Let dν k = f k dγ. (1) c(k) is a normalization constant (so that ν k is a probability measure). (2) α(k) controls the size of g b (x), and α(k) 0. (3) b(k) is the barycenter of g b, and b(k). Daesung Kim (Purdue) Stability for LSI Probability Seminar 23 / 26

24 Example Competing between α(k) and b(k) (1) If α(k) decays slowly and b(k) grows rapidly, then the entropy and the second moment diverge. (2) If α(k) decays rapidly and b(k) grows slowly, then the deficit tends to zero. Balancing between α(k) and b(k), we obtain the following. Theorem (D.K., 2017+) (i) δ LSI (ν k ) 0 and f k 1 L 1 (dγ) 0 (as a consequence, ν k γ). (ii) the LSI is unstable under (W 2, P2 M ) for any M > n. (iii) the LSI is unstable under (W 1, P 2 ). (iv) the LSI is unstable under (L p, P M 2 ) for any p > 1 and M > n. Daesung Kim (Purdue) Stability for LSI Probability Seminar 24 / 26

25 Summary Metric Space Stability Reference W 2 P2 n stable [Bobkov et al. 2014] P2 M, (M > n) unstable [K ] W 1 P2 M, (M n) stable [Indrei, K ] P 2 unstable [K ] L 1 P(λ) stable [Fathi et al. 2014] P2 M, (M n) weakly stable [Indrei, K ] L p, (p > 1) P2 M, (M > n) unstable [K ] Daesung Kim (Purdue) Stability for LSI Probability Seminar 25 / 26

26 Thank you! Daesung Kim (Purdue) Stability for LSI Probability Seminar 26 / 26