Approximations of displacement interpolations by entropic interpolations

Transcription

1 Approximations of displacement interpolations by entropic interpolations Christian Léonard Université Paris Ouest Mokaplan 10 décembre 2015

2 Interpolations in P(X ) X : Riemannian manifold (state space) P(X ) : set of all probability measures on X µ 0, µ 1 P(X ) interpolate between µ 0 and µ 1

3 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1

4 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 0

5 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 1

6 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 0

7 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 025

8 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 05

9 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 075

10 Interpolations in P(X ) Standard affine interpolation between µ 0 and µ 1 µ aff t := (1 t)µ 0 + tµ 1 P(X ), 0 t 1 t = 1

11 Interpolations in P(X ) Affine interpolations require mass transference with infinite speed Denial of the geometry of X We need interpolations built upon trans -portation, not tele -portation

12 Interpolations in P(X ) We seek interpolations of this type

13 Interpolations in P(X ) We seek interpolations of this type t = 0

14 Interpolations in P(X ) We seek interpolations of this type t = 025

15 Interpolations in P(X ) We seek interpolations of this type t = 05

16 Interpolations in P(X ) We seek interpolations of this type t = 075

17 Interpolations in P(X ) We seek interpolations of this type t = 1

18 Displacement interpolation µ 0 µ 1

19 Displacement interpolation y = T (x) x y µ 0 µ 1

20 Displacement interpolation geodesics µ 0 µ 1

21 Displacement interpolation geodesics µ 0 µ 1

22 Displacement interpolation

23 Displacement interpolation γ xy t x y

24 Displacement interpolation

25 Displacement interpolation

26 Curvature geodesics and curvature are intimately linked several geodesics give information on the curvature θ δ(t) p δ(t) = ( 2(1 cos θ) t 1 σ p(s) cos 2 ) (θ/2) t 2 + O(t 4 ) 6

27 Displacement interpolation y = T (x) x y µ 0 µ 1

28 Displacement interpolation Respect geometry we have already used geodesics how to choose y = T (x) such that interpolations encrypt curvature as best as possible? no shock perform optimal transport Monge s problem X d 2 (x, T (x)) µ 0 (dx) min; T : T # µ 0 = µ 1 d : Riemannian distance

29 Lazy gas experiment t = 0 0 < t < 1 t = 1 Positive curvature

30 Lazy gas experiment t = 0 0 < t < 1 t = 1 Negative curvature

31 Curvature and displacement interpolations Relative entropy H(p r) := log(dp/dr) dp, p, r : probability measures Convexity of the entropy along displacement interpolations The following assertions are equivalent Ric K along any [µ 0, µ 1 ] disp = (µ t ) 0 t 1, d 2 dt 2 H(µ t vol) KW 2 2 (µ 0, µ 1 ) von Renesse-Sturm (04) W 2 is the Wasserstein distance starting point of the Lott-Sturm-Villani theory

32 Schrödinger s thought experiment Consider a huge collection of non-interacting identical Brownian particles If the density profile of the system at time t = 0 is approximately µ 0 P(R 3 ), you expect it to evolve along the heat flow: { νt = ν 0 e t /2, 0 t 1 ν 0 = µ 0 where is the Laplace operator Suppose that you observe the density profile of the system at time t = 1 to be approximately µ 1 P(R 3 ) with µ 1 different from the expected ν 1 Probability of this rare event exp( CN Avogadro ) Schrödinger s question (1931) Conditionally on this very rare event, what is the most likely path (µ t ) 0 t 1 P(R 3 ) [0,1] of the evolving profile of the particle system?

33 Schrödinger s problem X : Riemannian manifold Ω := {paths} X [0,1] P P(Ω) and (P t ) 0 t 1 P(X ) [0,1] R P(Ω) : Wiener measure (Brownian motion) Schrödinger s problem H(P R) min; P P(Ω) : P 0 = µ 0, P 1 = µ 1 (S dyn ) µ 0, µ 1 P(X ) are the initial and final prescribed profiles Definition R-entropic interpolation [µ 0, µ 1 ] R := (P t ) 0 t 1 with P the unique solution of (S dyn ) It is the answer to Schrödinger s question

34 Lazy gas experiments Lazy gas experiment at zero temperature (Monge) Zero temperature Displacement interpolations Optimal transport Lazy gas experiment at positive temperature (Schrödinger) Positive temperature Entropic interpolations Minimal entropy

35 Lazy gas experiments t = 0 0 < t < 1 t = 1 Negative curvature Zero temperature

36 Lazy gas experiments t = 0 t = 1 Negative curvature Positive temperature

37 Cooling down Aim Drifting from Schrödinger problem to an optimal transport problem To decrease temperature: slow down the particles of the heat bath more generally, decrease fluctuations Slowed down reference measures (B t ) t 0 : Brownian motion on the Riemannian manifold X R : R k : k law of (B t ) 0 t 1 law of (B t/k ) 0 t 1

38 Cooling down k = 1 : γ xy y x R xy t = 0 t = 1 k = 10 : y x R k,xy k = : y x γ xy

39 Cooling down N, k = 1 : the whole particle system performs a rare event to travel from µ 0 to µ 1 cooperative behavior Gibbs conditioning principle (thermodynamical limit: N ) N = 1, k : each individual particle faces a hard task and must travel along an approximate geodesic individual behavior large deviation principle (cooling down limit: k ) Cooling down principle The cooled down sequence (R k ) k 1 encodes some geometry N, k : these two behaviors superpose

40 Results Results 1 displacement interpolations feel curvature entropic interpolations also feel curvature Results 2 entropic interpolations converge to displacement interpolations entropic interpolations regularize displacement interpolations Γ-convergence

41 Results Results 2 (continued) The same kind of results hold in other settings (a) discrete graphs (b) Finsler manifolds (c) interpolations with varying mass (a) graphs: random walk (b) Finsler: jump process in a manifold, (work in progress) (c) varying mass: branching process, (work in progress)

42 Results Results 3 Schrödinger s problem is an analogue of Hamilton s least action principle It allows for dynamical theories of diffusion processes random walks on graphs stochastic Newton equation acceleration is related to curvature

43 Remainder of the talk Let us give some details about Results 2: entropic interpolations converge to displacement interpolations

44 Notation X = {states} Ω = {paths} X [0,1], ω = (ω t ) 0 t 1 Ω X t : ω Ω ω t X, 0 t 1 (canonical process) P P(Ω) P t (dz) := [(X t ) # P](dz) = P(X t dz) P(X )

45 Particle system Ω = {paths} X [0,1] R P(Ω) : R x := R( X 0 = x), reference Markov measure x X (Z i ) 1 i N Ω N, independent dynamical particles Law(Z i ) = R xi, 1 i N N Assume that L N 0 = 1 N N 1 i N δ x i µ 0 P(X ), (t=0) N Empirical measures L N := 1 N N 1 i N δ Z i P(Ω) L N t := 1 N N 1 i N δ Z i t P(X ), 0 t 1

46 Schrödinger s problem Law of large numbers L N (dω) R µ 0 (dω) := N X Rx (dω) µ 0 (dx) (L N t ) 0 t 1 (R µ 0 t ) 0 t 1 P(X ) [0,1] N L N 1 (dy) N Rµ 0 1 (dy) P(X ) Schrödinger s question (1931) N large P(Ω) suppose that you observe: L N 1 µ 1, µ 1 R µ 0 1 (t = 1, rare event) question: conditionally on this rare event, what is the most likely path (L N t ) 0 t 1?

47 Schrödinger s problem Sanov s theorem: Pr(L N A) Relative entropy exp N ( N ) inf H(P R), A P(Ω) P A,P 0 =µ 0 H(p r) := log (dp/dr) dp [0, ] Pr(L N A L N 1 µ 1 ) ( { N }) exp N inf H(P R) inf H(P R) P A,P 0 =µ 0,P 1 =µ 1 P 0 =µ 0,P 1 =µ 1

48 Schrödinger s problem answer to Schrödinger s question: the most likely path is close to the time-marginal flow (P t ) 0 t 1 of the unique solution P of Dynamical Schrödinger problem H(P R) min; P P(Ω) : P 0 = µ 0, P 1 = µ 1 (S dyn ) H Föllmer Definition (entropic interpolation) Let P be the solution of (S dyn ) Then, µ t := P t, 0 t 1 is the R-entropic interpolation between µ 0 and µ 1 in P(X )

49 Schrödinger s problem P 01 = (X 0, X 1 ) # P P(X 2 ) P xy = P( X 0 = x, X 1 = y) P(Ω) : bridge P( ) = X 2 P xy ( ) P 01 (dxdy) P(Ω) Result If it exists, the unique solution P of (S dyn ) satisfies P xy = R xy, x, y P( ) = X 2 R xy ( ) π(dxdy) where P 01 = π P(X 2 ) is the unique solution of (S) below inf(s) = H(P R) = H(P 01 R 01 ) Schrödinger s problem H(π R 01 ) min; π P(X 2 ) : π 0 = µ 0, π 1 = µ 1 (S) H(P R) = H(P 01 R 01 ) + X 2 H(P xy R xy ) P 01 (dxdy)

50 Schrödinger s problem Result Assume: R is m-stationary Markov, m m R 01 m m, H(µ 0 m), H(µ 1 m) <, Then, (S dyn ) and (S) admit a solution long history: Schrödinger, Bernstein, Fortet, Beurling, Csiszár, Rüschendorf & Thomsen, Föllmer & Gantert, L and also: Jamison, Zambrini, Dai Pra, Wakolbinger, Pavon, Mikami, Rœlly, Thieullen,

51 Cooling down (Brownian case) suppose that in addition the heat bath is pretty cold cooling down is (mostly) slowing down the particles R = Law ( (B t ) 0 t 1 ), R k := Law ( (B k 1 t) 0 t 1 ), k H(P R k )/k min; P P(Ω) : P 0 = µ 0, P 1 = µ 1 (S k dyn ) H(π R k 01 )/k min; π P(X 2 ) : π 0 = µ 0, π 1 = µ 1 (S k ) dx t = db t, R-as dx t = 1/k db t, R k -as R01 k (dxdy) = (2π/k) d/2 exp( k y x 2 /2) dxdy with P k solution of (S k dyn ) inf(s k dyn ) = 1 inf(sk ) = X 2 2 y x 2 P01(dxdy) k + o k (1) this suggests that lim k (S k ) = (MK 2 )

52 Cooling down (Brownian case) Cooled down Schrödinger problem H(π R01)/k k min; π P(X 2 ) : π 0 = µ 0, π 1 = µ 1 (S k ) Monge-Kantorovich problem 1 X 2 y x 2 π(dxdy) min; π P(X 2 ) : π 0 = µ 0, π 1 = µ 1 (MK 2 ) Theorem Γ- lim k (S k ) = (MK 2 ) lim k inf(s k ) = inf(mk 2 ) := W 2 2 (µ 0, µ 1 )/2 lim k π k = π : solution of (MK 2 ) Mikami (2004), L (2012)

53 Cooling down (Brownian case) Cooling particles down brings geometry (Brownian case) lim k R k,xy = δ γ xy, γ xy : constant speed geodesic k = 1 : γ xy y k = 10 : x R xy t = 0 t = 1 y x R k,xy k = : y x γ xy

54 Cooling down (Brownian case) Convergence schema P k (dω) = X R k,xy (dω) π k (dxdy) 2 P(dω) = X δ 2 γ xy (dω) π(dxdy) Entropic interpolations converge to displacement interpolations µ k t (dz) = X R k,xy 2 t (dz) π k (dxdy), 0 t 1 µ t (dz) = X δ 2 γ xy (dz) π(dxdy), 0 t 1 t McCann s displacement interpolation [µ 0, µ 1 ] disp := (µ t ) 0 t 1

55 Displacement interpolation µ 0 µ 1

56 Displacement interpolation π(dy x) x y µ 0 µ 1

57 Displacement interpolation geodesics µ 0 µ 1

58 Displacement interpolation geodesics µ 0 µ 1

59 Displacement interpolation

60 Displacement interpolation γ xy t x y

61 Displacement interpolation

62 Displacement interpolation

63 Doubly indexed large deviation principle choose (R k ) k 1 such that it satisfies some LDP LDP for (R k ) k 1 R k,x exp( a k [C + ι {X0 k =x}]), x a k, C : Ω [0, ], coercive {C = 0} is the limiting support of all geodesics Example (slow Brownian motion) a k = k, C(ω) = ω t 2 dt, lim k R k,xy = δ γ xy

64 Γ-convergence Dynamical cooled down Schrödinger problem H(P R k )/a k min; P P(Ω) : P 0 = µ 0, P 1 = µ k 1 (S k dyn ) Dynamical Monge-Kantorovich problem E P C min; P P(Ω) : P 0 = µ 0, P 1 = µ 1 (MK dyn ) Theorem there exists µ k 1 µ 1 such that Γ- lim k (S k dyn ) = (MK dyn) lim k inf(s k dyn ) = inf(mk dyn) lim k P k = P : solution of (MK dyn )

65 Γ-convergence if P k P we get the following schema Convergence schema P k (dω) = X R k,xy (dω) π k (dxdy) 2 P(dω) = X G xy (dω) π(dxdy) 2 Entropic interpolations converge to displacement interpolation µ k t (dz) = X R 2 t k,xy (dz) π k (dxdy), 0 t 1 µ t (dz) = X G xy 2 t (dz) π(dxdy), 0 t 1 Definition (displacement interpolation) [µ 0, µ 1 ] disp := (µ t ) 0 t 1

66 L 2 -type displacement interpolations on a vector space X = R n R k,x : Z k,x t Mogulskii s theorem kt = x + k 1 V j, (V j ) j 1 iid, EV = 0 j=1 C(ω) = 1 0 L V ( ω t ) dt L V (v) = sup p {p v H V (p)}; H V (p) = log E exp(p V ) L V (v) = O v 0 ( v 2 ) Contraction principle c(x, y) = inf{c(ω); ω Ω : ω 0 = x, ω 1 = y} c(x, y) = L V (y x)

67 Interpolations on a discrete graph metric graph (X,, d) x d(x, y) y x y means that (x, y) is an edge length l(ω) := 0 t 1 1 {ω t ω t }d(ω t, ω t ) intrinsic distance d(x, y) = inf {l(ω) : ω Ω, ω 0 = x, ω 1 = y}

68 Interpolations on a discrete graph to recover d: slow down the walk condition at t = 0 and t = 1 reference walk: R P(Ω) with jump kernel Lazy random walks R k Jx k (dy) = J x (dy) = y:y x J x(y) δ y y:y x k d(x,y) J x (y) δ y

69 Interpolations on a discrete graph Geodesics Γ xy := {ω Ω; ω 0 = x, ω 1 = y, l(ω) = d(x, y)} Γ := x,y Γ xy Convergence of bridges G := 1 Γ e 1 0 J Xt (X ) dt R Convergence of the interpolations a k = log k, C = l, c = d lim k inf(s k ) = W 1 (µ 0, µ 1 ) lim k Rk,xy = G xy P(Γ xy )

70 L 1 -type interpolations on a diffuse length space (X, d) : diffuse metric space Definition (diffuse metric space) (X, d) is diffuse if there exists a Borel measure m on X such that sup x m(b 1 x ) < m(b ϵ x) > 0, x X, ϵ > 0 B ϵ x := {y X : d(x, y) < ϵ}

71 L 1 -type interpolations on a diffuse length space Reference processes R J x (dy) = 1 B 1 x (y) m(dy) R k Jx k (dy) = e 1 1 1/k S (y) m(dy) x S ϵ x := B ϵ x \ B ϵ ϵ2 x Convergence of the entropic interpolations a k = k, C = length, c = d H(P R k ) = H(P R) E P log(dr k /dr) log(dr k /dr)/k = #{t : X t X t }/k + O k (1/k) = length(x ) + O k (1/k) work in progress with Luca Tamanini

72 References about entropic interpolations 1 T Mikami Monge s problem with a quadratic cost by the zero-noise 2 limit of h-path processes PTRF, (2004) L From the Schrödinger problem to the Monge-Kantorovich problem 3 JFA, (2012) L A survey of the Schrödinger problem and some of its connections 4 with optimal transport DCDS (A), (2014) L Lazy random walks and optimal transport on graphs AoP, (to appear)

73 Thank you for your attention