Measure concentration, functional inequalities, and curvature of metric measure spaces

(1)

Measure concentration, functional inequalities, and curvature of metric measure spaces

M. Ledoux, Institut de Math´ematiques de Toulouse, France

Luminy, March 2008

(2)

circleof ideas

between analysis, geometry and probability theory

• concentration of measure phenomenon

• geometric, measure/information theoretic, functional inequalities

• curvature of metric measure spaces

theory of optimal transportation

(3)

circleof ideas

between analysis, geometry and probability theory

(4)

circleof ideas

between analysis, geometry and probability theory

(5)

circleof ideas

between analysis, geometry and probability theory

(6)

circleof ideas

between analysis, geometry and probability theory

(7)

concentration of measure phenomenon

V. Milman 1970

Dvoretzky’s theorem onspherical sections of convex bodies inhigh dimension

for allε >0, there exists δ(ε)>0such that for every convex body K in Rⁿ, there is a subspace F of Rⁿwith dim(F)≥δ(ε) logn and anellipsoidE in F such that

Milman’s idea : find F random using aconcentration property ofspherical measures in high dimension

(8)

concentration of measure phenomenon

V. Milman 1970

(9)

concentration of measure phenomenon

V. Milman 1970

(10)

concentration of measure phenomenon

V. Milman 1970

for allε >0, there exists δ(ε)>0such that for every convex body K in Rⁿ,

there is a subspace F of Rⁿwith dim(F)≥δ(ε) logn and anellipsoidE in F such that

(11)

concentration of measure phenomenon

V. Milman 1970

for allε >0, there exists δ(ε)>0such that for every convex body K in Rⁿ, there is a subspace F of Rⁿwith dim(F)≥δ(ε) logn and anellipsoidE in F

such that

(12)

concentration of measure phenomenon

V. Milman 1970

K ∩F

(13)

concentration of measure phenomenon

V. Milman 1970

(1−ε)E ⊂K ∩F ⊂(1 +ε)E

(14)

concentration of measure phenomenon

V. Milman 1970

(1−ε)E ⊂K ∩F ⊂(1 +ε)E

Milman’s idea : find F random

using aconcentration property ofspherical measures in high dimension

(15)

concentration of measure phenomenon

V. Milman 1970

(1−ε)E ⊂K ∩F ⊂(1 +ε)E

(16)

spherical concentration

isoperimetric inequalityon Sⁿ⊂Rⁿ⁺¹

geodesic ballsare the extremal sets

if vol(A) ≥ vol(B), B geodesic ball

then vol(Ar) ≥ vol(Br), r >0

normalized volume µ(·) = vol(·) vol(Sⁿ)

(17)

(18)

(19)

(20)

(21)

(22)

if µ(A) ≥ 1 2

(B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r) ≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(23)

if µ(A) ≥ 1

2 (B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r) ≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(24)

if µ(A) ≥ 1

2 (B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r)

≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(25)

if µ(A) ≥ 1

2 (B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r) ≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(26)

if µ(A) ≥ 1

2 (B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r) ≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(27)

if µ(A) ≥ 1

2 (B half-sphere)

then, for all r >0,

µ(A_r) ≥ µ(B_r) ≥ 1−e^−(n−1)r²^/2

r∼ 1

√n (n→ ∞)

µ(Ar) ≈ 1

(28)

application byV. Milman

m (normalized) Haar measure on O_n+1

m {T ∈ O_n+1;Tx ∈A}

= µ(A), x ∈Rⁿ⁺¹,A⊂Sⁿ k · k gauge of K

showthat for dim(E)≥δ(ε) logn

mn

T ∈ O_n+1; |z|

1 +ε ≤ kTzk ≤ |z|

1−ε, z ∈Eo

> 0

F =ImT_|E, E=T(B_E)

(29)

application byV. Milman

m (normalized) Haar measure on O_n+1

m {T ∈ O_n+1;Tx ∈A}

= µ(A), x ∈Rⁿ⁺¹,A⊂Sⁿ k · k gauge of K

showthat for dim(E)≥δ(ε) logn

mn

T ∈ O_n+1; |z|

1 +ε ≤ kTzk ≤ |z|

1−ε, z ∈Eo

≈ 1

F =ImT_|E, E=T(B_E)

(30)

framework for measure concentration

metric measure space(X,d, µ)

(X,d) metric space

µ Borel measureonX, µ(X) = 1

concentration function

α(r) =α_(X_,d,µ)(r) = sup

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

µ uniform on Sⁿ⊂Rⁿ⁺¹ : α(r) ≤ e^−(n−1)r²^/2, r >0

(31)

framework for measure concentration

metric measure space(X,d, µ) (X,d) metric space

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

(32)

framework for measure concentration

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

(33)

framework for measure concentration

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

(34)

framework for measure concentration

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

(35)

framework for measure concentration

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r µ uniform on Sⁿ⊂Rⁿ⁺¹ :

α(r) ≤ e^−(n−1)r²^/2, r >0

(36)

framework for measure concentration

1−µ(A_r);µ(A)≥1/2 , r >0

Ar =

x ∈X;d(x,A)<r

(37)

equivalent formulation on functions

F :X →R 1- Lipschitz m=m_F median of F for µ

µ(F ≤m), µ(F ≥m) ≥ 1 2

A={F ≤m} =⇒ A_r ⊂ {F <m+r}

µ

|F −m|<r ≥ 1−2α(r), r >0

(38)

F :X →R 1- Lipschitz

m=m_F median of F for µ µ(F ≤m), µ(F ≥m) ≥ 1

2

A={F ≤m} =⇒ A_r ⊂ {F <m+r}

µ

|F −m|<r ≥ 1−2α(r), r >0

(39)

µ(F ≤m), µ(F ≥m) ≥ 1 2

A={F ≤m} =⇒ A_r ⊂ {F <m+r}

µ

|F −m|<r ≥ 1−2α(r), r >0

(40)

µ(F ≤m), µ(F ≥m) ≥ 1 2

A={F ≤m} =⇒ A_r ⊂ {F <m+r}

µ

|F −m|<r ≥ 1−2α(r), r >0

(41)

µ(F ≤m), µ(F ≥m) ≥ 1 2

A={F ≤m} =⇒ A_r ⊂ {F <m+r}

µ

|F −m|<r ≥ 1−2α(r), r >0

(42)

dual description : observable diameter(M. Gromov)

κ >0 (κ= 10⁻¹⁰)

PartDiamµ(X,d)

= inf

D ≥0,∃A⊂X,Diam(A)≤D, µ(A)≥1−κ

ObsDiam_µ(X,d) = sup

F 1−Lip

PartDiam_µ_F(R)

F :µ→µ_F

(43)

κ >0 (κ= 10⁻¹⁰)

PartDiamµ(X,d)

= inf

F 1−Lip

PartDiam_µ_F(R)

F :µ→µ_F

(44)

κ >0 (κ= 10⁻¹⁰)

PartDiamµ(X,d)

= inf

F 1−Lip

PartDiam_µ_F(R)

F :µ→µ_F

(45)

κ >0 (κ= 10⁻¹⁰)

PartDiamµ(X,d)

= inf

F 1−Lip

PartDiam_µ_F(R)

F :µ→µ_F

(46)

F 1−Lip

PartDiam_µ_F(R)

µ stateon the configuration space (X,d)

F :X →R observable

yielding thetomographic image µ_F on R

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam_µ(X,d) ∼ α⁻¹(κ)

ObsDiam(Sⁿ) = O 1

√n

(47)

F 1−Lip

PartDiam_µ_F(R)

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam(Sⁿ) = O 1

√n

(48)

F 1−Lip

PartDiam_µ_F(R)

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam(Sⁿ) = O 1

√n

(49)

F 1−Lip

PartDiam_µ_F(R)

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam(Sⁿ) = O 1

√n

(50)

F 1−Lip

PartDiam_µ_F(R)

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam(Sⁿ) = O 1

√n

(51)

F 1−Lip

PartDiam_µ_F(R)

µ_F ]m−r,m+r[

=µ

|F −m|<r ≥ 1−2α(r)

ObsDiam(Sⁿ) = O 1

√n

(52)

measure concentration property

less restrictivethanisoperimetry, widely shared

variety of examples and tools

• spectral methods

• combinatorial tools

• product measures(M. Talagrand 1995)

(53)

measure concentration property less restrictivethanisoperimetry, widely shared

(54)

(55)

(56)

(57)

• product measures

(M. Talagrand 1995)

(58)

(59)

(60)

measure concentration property less restrictive than isoperimetry, widely shared

(61)

dimension free inequalities

model : Gaussian measure dγ(x) =e^−|x|²^/2 dx

(2π)^k/2 on (R^k,| · |) concentration property

R ∼√ n

µ uniformon Sⁿ^√_n → γ Gaussian Gauss space : curvature 1 dimension ∞

(62)

dimension free inequalities

(2π)^k/2 on (R^k,| · |)

concentration property

R ∼√ n

(63)

dimension free inequalities

R ∼√ n

(64)

dimension free inequalities

Sⁿ: α(r) ≤ e^−(n−1)r²^/2

R ∼√ n

(65)

dimension free inequalities

SⁿR : α(r) ≤ e^−(n−1)r²^/2R²

R ∼√ n

(66)

dimension free inequalities

SⁿR : α(r) ≤ e^−(n−1)r²^/2R² R ∼√

n

(67)

dimension free inequalities

SⁿR : α(r) ≤ e^−(n−1)r²^/2R² R ∼√

n

µ uniformon Sⁿ^√_n

→ γ Gaussian Gauss space : curvature 1 dimension ∞

(68)

dimension free inequalities

SⁿR : α(r) ≤ e^−(n−1)r²^/2R² R ∼√

n

µ uniformon Sⁿ^√_n → γ Gaussian

Gauss space : curvature 1 dimension ∞

(69)

dimension free inequalities

SⁿR : α(r) ≤ e^−(n−1)r²^/2R² R ∼√

n

(70)

Gaussian concentration α(r) ≤ e^−r²^/2

if A⊂R^k, γ(A) ≥ 1 2 then γ(A_r) ≥ 1−e^−r²^/2, r >0

ObsDiamγ(R^k) = O(1)

independentof the dimension infinite dimensional analysis

(71)

if A⊂R^k, γ(A) ≥ 1 2

then γ(A_r) ≥ 1−e^−r²^/2, r >0

(72)

(73)

(74)

independentof the dimension

infinite dimensional analysis

(75)

(76)

triple description of Gaussian concentration α(r) ≤ e^−r²^/2

• geometric

• measure/information theoretic

• functional

notion of curvaturein metric measure spaces

(PDEand calculus of variations viewpoint)

(77)

• geometric

• functional

(78)

• geometric

• functional

(79)

• geometric

• functional

(80)

• geometric

• functional

(81)

• geometric

• functional

(82)

geometric description : Brunn-Minkowski inequality

Pr´ekopa-Leindler theorem

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

≥ u(x)^θv(y)^1−θ, x,y ∈Rⁿ

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

u =χ_A,v=χ_B,multiplicative form ofBrunn-Minkowski voln θA+ (1−θ)B

≥ voln(A)^θvoln(B)^1−θ

(83)

geometric description : Brunn-Minkowski inequality Pr´ekopa-Leindler theorem

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

(84)

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

(85)

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

(86)

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

(87)

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

u =χ_A,v=χ_B,

multiplicative form ofBrunn-Minkowski voln θA+ (1−θ)B

(88)

θ∈[0,1], u,v,w :Rⁿ→R+

if w θx+ (1−θ)y

then Z

w dx ≥ Z

u dx θ Z

v dx 1−θ

(89)

dx → dγ(x) =e^−|x|²^/2 dx (2π)^n/2

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

choose θ= 1/2, w ≡1, v =χA, u =e^d(·,A)²^/4 Z

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(A_r) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

2 extension : dµ=e^−Vdx, V⁰⁰≥c >0

(90)

dx → dγ(x) =e^−|x|²^/2 dx (2π)^n/2 if w θx+ (1−θ)y

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

choose θ= 1/2, w ≡1, v =χA, u =e^d(·,A)²^/4 Z

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(A_r) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(91)

≥ e−θ(1−θ)|x−y|²/2u(x)^θv(y)^1−θ, x,y ∈Rⁿ

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

choose θ= 1/2, w ≡1, v =χ_A, u =e^d(·,A)²^/4 Z

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(92)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(93)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

choose θ= 1/2, w ≡1, v =χ_A,

u =e^d(·,A)²^/4 Z

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(94)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

choose θ= 1/2, w ≡1, v =χ_A, u =e^d(·,A)²^/4

Z

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(95)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

e^d(·,A)²^/4dγ ≤ 1 γ(A)

γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1 2 extension : dµ=e^−Vdx, V⁰⁰≥c >0

(96)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

2

extension : dµ=e^−Vdx, V⁰⁰≥c >0

(97)

then Z

w dγ ≥ Z

u dγ θ Z

v dγ 1−θ

e^d(·,A)²^/4dγ ≤ 1 γ(A) γ(Ar) ≥ 1−2e^−r²^/4, γ(A) ≥ 1

(98)

measure/information theoretic description : transportation cost inequality

ν probability measure on Rⁿ, ν << γ W₂(ν, γ)² ≤ H(ν|γ)

W₂(ν, γ)² = inf

ν←π→γ

Z Z 1

2|x−y|²dπ(x,y) Wasserstein distance

H(ν|γ) = Z

log dν dγ dν relative entropy M. Talagrand 1996

(99)

ν←π→γ

Z Z 1

H(ν|γ) = Z

(100)

ν←π→γ

Z Z 1

H(ν|γ) = Z

(101)

ν←π→γ

Z Z 1

H(ν|γ) = Z

log dν dγ dν relative entropy

M. Talagrand 1996

(102)

ν←π→γ

Z Z 1

H(ν|γ) = Z

(103)

concentrationvia the transportation cost inequality (K. Marton)

A,B⊂Rⁿ, d(A,B)≥r>0 γ_A = γ(· |A), γ_B = γ(· |B)

W₂(γ_A, γ_B) ≤ log 1

γ(A) 1/2

+ log 1

γ(B) 1/2

√r

2 ≤ W2(γA, γB) ≤ log 1

γ(A) 1/2

+

log 1 γ(B)

1/2

γ(A)≥1/2, B =complementof Ar

γ(A_r) ≥ 1−e^−r²^/4, r ≥r₀

(104)

functional description : logarithmic Sobolev inequality

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Rf logf dγ = H(fdγ|dγ) entropy Z |∇f|²

f dγ Fisher information f →f²:

Z

f²logf²dγ ≤ 2 Z

|∇f|²dγ

A. Stam 1959, L. Gross 1975

(105)

f ≥0 smooth, Z

f dγ = 1

Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Z

|∇f|²dγ

(106)

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Z

|∇f|²dγ

(107)

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Rf logf dγ = H(fdγ|dγ) entropy

Z |∇f|²

Z

|∇f|²dγ

(108)

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

f dγ Fisher information

f →f²: Z

|∇f|²dγ

(109)

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Z

|∇f|²dγ

(110)

f ≥0 smooth, Z

f dγ = 1 Z

f logf dγ ≤ 1 2

Z |∇f|² f dγ

Z

|∇f|²dγ

(111)

concentrationvia the logarithmic Sobolev inequality(I. Herbst)

F :Rⁿ→R 1-Lipschitz, R

F dγ = 0

f =e^λF/R

e^λFdγ, λ∈R differential inequalityon R

e^λFdγ Z

e^λFdγ ≤ e^λ²^/2, λ∈R γ {F <r}

≥ 1−e^−r²^/2, r >0

(112)

Brunn-Minkowski inequality

logarithmic Sobolev inequality

transportation cost inequality

(113)

hierarchy

⇓

dimension free measure concentration

tools toestablish theseinequalities parametrisation methods

(114)

hierarchy

⇓

(115)

hierarchy

⇓

tools toestablish theseinequalities

parametrisation methods

(116)

hierarchy

⇓

(117)

heat kernel parametrisation

logarithmic Sobolev inequality R

f logf dγ ≤ ¹₂R |∇f|² f dγ generator Lf = ∆f −x· ∇f, Pt=e^tL semigroup

γ invariant measure

d dt

Z

Ptf logPtf dγ = −1 2

Z |∇P_tf|² P_tf dγ commutation |∇P_tf| ≤ e^−tP_t(|∇f|)

equivalent to a curvature condition

(118)

heat kernel parametrisation

f logf dγ ≤ ¹₂R |∇f|² f dγ

generator Lf = ∆f −x· ∇f, Pt=e^tL semigroup γ invariant measure

d dt

Z

(119)

heat kernel parametrisation

f logf dγ ≤ ¹₂R |∇f|² f dγ generator Lf = ∆f −x· ∇f,

Pt=e^tL semigroup γ invariant measure

d dt

Z

(120)

heat kernel parametrisation

d dt

Z

(121)

heat kernel parametrisation

d dt

Z

(122)

heat kernel parametrisation

d dt

Z

Z |∇P_tf|² P_tf dγ

commutation |∇P_tf| ≤ e^−tP_t(|∇f|) equivalent to a curvature condition

(123)

heat kernel parametrisation

d dt

Z

(124)

heat kernel parametrisation

d dt

Z

(125)

extensions

dµ=e^−Vdx on Rⁿ, V⁰⁰≥c >0 Riemannian manifolds Ric≥c >0

Bochner’s formula 1

2∆ |∇f|²

− ∇f · ∇(∆f) = kHess(f)k²+Ric(∇f,∇f) ≥ c|∇f|²

equivalent to |∇P_tf| ≤ e^−ctP_t(|∇f|) (Gauss space : curvature 1 )

manifolds withweights dµ=e^−Vdx, Ric+Hess(V)≥c >0

(126)

extensions

dµ=e^−Vdx on Rⁿ, V⁰⁰≥c >0

Riemannian manifolds Ric≥c >0 Bochner’s formula

1

2∆ |∇f|²

(127)

extensions

2∆ |∇f|²

(128)

extensions

2∆ |∇f|²

− ∇f · ∇(∆f) = kHess(f)k²+Ric(∇f,∇f)

≥ c|∇f|²