Some geometric consequences of the Schrödinger problem

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14299

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This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.

Some geometric consequences of the Schrödinger problem

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application/pdf Some geometric consequences of the Schrödinger problem Christian Leonard
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This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.
Some geometric consequences of the Schrödinger problem

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This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.

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. . . .. . . Some geometric aspects of the Schr¨odinger problem Christian L´eonard Universit´e Paris Ouest GSI’15 ´Ecole Polytechnique. October 28-30, 2015 Interpolations in P(X) X : Riemannian manifold (state space) P(X) : set of all probability measures on X µ0, µ1 ∈ P(X) interpolate between µ0 and µ1 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 0 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 1 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 0 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 0.25 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 0.5 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 0.75 Interpolations in P(X) Standard affine interpolation between µ0 and µ1 µaff t := (1 − t)µ0 + tµ1 ∈ P(X), 0 ≤ t ≤ 1 t = 1 Interpolations in P(X) . . . .. . . Affine interpolations require mass transference with infinite speed 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 Interpolations in P(X) We seek interpolations of this type Interpolations in P(X) We seek interpolations of this type t = 0 Interpolations in P(X) We seek interpolations of this type t = 0.25 Interpolations in P(X) We seek interpolations of this type t = 0.5 Interpolations in P(X) We seek interpolations of this type t = 0.75 Interpolations in P(X) We seek interpolations of this type t = 1 Displacement interpolation µ0 µ1 Displacement interpolation x y µ0 µ1 y = T(x) Displacement interpolation µ0 µ1 geodesics Displacement interpolation µ0 µ1 geodesics Displacement interpolation Displacement interpolation x y γxy t Displacement interpolation Displacement interpolation Curvature geodesics and curvature are intimately linked several geodesics give information on the curvature Curvature geodesics and curvature are intimately linked several geodesics give information on the curvature δ(t) θ p . . . .. . . δ(t) = √ 2(1 − cos θ) t ( 1 − σp(S) cos2(θ/2) 6 t2 + O(t4 ) ) Displacement interpolation x y µ0 µ1 y = T(x) 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 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 d2(x, T(x)) µ0(dx) → min; T : T#µ0 = µ1 d : Riemannian distance Lazy gas experiment t = 0 0 < t < 1 t = 1 Positive curvature Lazy gas experiment t = 0 0 < t < 1 t = 1 Negative curvature 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 [µ0, µ1]disp = (µt)0≤t≤1, d2 dt2 H(µt|vol) ≥ KW 2 2 (µ0, µ1) von Renesse-Sturm (04) W2 is the Wasserstein distance 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 [µ0, µ1]disp = (µt)0≤t≤1, d2 dt2 H(µt|vol) ≥ KW 2 2 (µ0, µ1) von Renesse-Sturm (04) W2 is the Wasserstein distance starting point of the Lott-Sturm-Villani theory Schr¨odinger’s thought experiment Consider a huge collection of non-interacting identical Brownian particles. Schr¨odinger’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 µ0 ∈ P(R3), you expect it to evolve along the heat flow: { νt = ν0et∆/2, 0 ≤ t ≤ 1 ν0 = µ0 where ∆ is the Laplace operator. Schr¨odinger’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 µ0 ∈ P(R3), you expect it to evolve along the heat flow: { νt = ν0et∆/2, 0 ≤ t ≤ 1 ν0 = µ0 where ∆ is the Laplace operator. Suppose that you observe the density profile of the system at time t = 1 to be approximately µ1 ∈ P(R3) with µ1 different from the expected ν1. Probability of this rare event ≃ exp(−CNAvogadro). Schr¨odinger’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 µ0 ∈ P(R3), you expect it to evolve along the heat flow: { νt = ν0et∆/2, 0 ≤ t ≤ 1 ν0 = µ0 where ∆ is the Laplace operator. Suppose that you observe the density profile of the system at time t = 1 to be approximately µ1 ∈ P(R3) with µ1 different from the expected ν1. Probability of this rare event ≃ exp(−CNAvogadro). . Schr¨odinger’s question (1931) .. . . .. . . Conditionally on this very rare event, what is the most likely path (µt)0≤t≤1 ∈ P(R3)[0,1] of the evolving profile of the particle system? Schr¨odinger problem X : compact Riemannian manifold Ω := {paths} ⊂ X[0,1] P ∈ P(Ω) and (Pt)0≤t≤1 ∈ P(X)[0,1] R ∈ P(Ω) : Wiener measure (Brownian motion) Schr¨odinger problem X : compact Riemannian manifold Ω := {paths} ⊂ X[0,1] P ∈ P(Ω) and (Pt)0≤t≤1 ∈ P(X)[0,1] R ∈ P(Ω) : Wiener measure (Brownian motion) . Schr¨odinger problem .. . . .. . . H(P|R) → min; P ∈ P(Ω) : P0 = µ0, P1 = µ1 (S) µ0, µ1 ∈ P(X) are the initial and final prescribed profiles Schr¨odinger problem X : compact Riemannian manifold Ω := {paths} ⊂ X[0,1] P ∈ P(Ω) and (Pt)0≤t≤1 ∈ P(X)[0,1] R ∈ P(Ω) : Wiener measure (Brownian motion) . Schr¨odinger problem .. . . .. . . H(P|R) → min; P ∈ P(Ω) : P0 = µ0, P1 = µ1 (S) µ0, µ1 ∈ P(X) are the initial and final prescribed profiles . Definition. R-entropic interpolation .. . . .. . . [µ0, µ1]R := (Pt)0≤t≤1 with P the unique solution of (S). It is the answer to Schr¨odinger’s question Lazy gas experiments Lazy gas experiment at zero temperature (Monge) Zero temperature Displacement interpolations Optimal transport Lazy gas experiment at positive temperature (Schr¨odinger) Positive temperature Entropic interpolations Minimal entropy Lazy gas experiments t = 0 0 < t < 1 t = 1 Negative curvature Zero temperature Lazy gas experiments t = 0 t = 1 Negative curvature Positive temperature Slowing down . . . .. . . To decrease temperature, slow down the particles of the heat bath Slowing down . . . .. . . To decrease temperature, slow down the particles of the heat bath . Slowed down reference measures .. . . .. . . (Wt)t≥0 : Brownian motion on the Riemannian manifold X R : law of (Wt)0≤t≤1 Rk : law of (Wt/k)0≤t≤1 k → ∞ Slowing down k = 1 : x y γxy Rxy t = 0 t = 1 Slowing down k = 1 : x y γxy Rxy t = 0 t = 1 k = 10 : x y Rk,xy Slowing down k = 1 : x y γxy Rxy t = 0 t = 1 k = 10 : x y Rk,xy k = ∞ : x y γxy Slowing down N → ∞, k = 1 : the whole particle system performs a rare event to travel from µ0 to µ1 cooperative behavior Gibbs conditioning principle (thermodynamical limit: N → ∞) Slowing down N → ∞, k = 1 : the whole particle system performs a rare event to travel from µ0 to µ1 cooperative behavior Gibbs conditioning principle (thermodynamical limit: N → ∞) N = 1, k → ∞ : each individual particle faces a harder task and must travel along an approximate geodesic individual behavior large deviation principle (slowing down limit: k → ∞) Slowing down N → ∞, k = 1 : the whole particle system performs a rare event to travel from µ0 to µ1 cooperative behavior Gibbs conditioning principle (thermodynamical limit: N → ∞) N = 1, k → ∞ : each individual particle faces a harder task and must travel along an approximate geodesic individual behavior large deviation principle (slowing down limit: k → ∞) . Slowing down principle .. . . .. . . The slowed down sequence (Rk)k≥1 encodes some geometry Slowing down N → ∞, k = 1 : the whole particle system performs a rare event to travel from µ0 to µ1 cooperative behavior Gibbs conditioning principle (thermodynamical limit: N → ∞) N = 1, k → ∞ : each individual particle faces a harder task and must travel along an approximate geodesic individual behavior large deviation principle (slowing down limit: k → ∞) . Slowing down principle .. . . .. . . The slowed down sequence (Rk)k≥1 encodes some geometry N → ∞, k → ∞ : these two behaviors superpose Results . Results 1 .. . . .. . . displacement interpolations feel curvature entropic interpolations also feel curvature 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 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 Results . Results 3 .. . . .. . . The same kind of results hold in other settings ...1 discrete graphs ...2 Finsler manifolds ...3 interpolations with varying mass Results . Results 3 .. . . .. . . The same kind of results hold in other settings ...1 discrete graphs ...2 Finsler manifolds ...3 interpolations with varying mass ...1 graphs: random walk Results . Results 3 .. . . .. . . The same kind of results hold in other settings ...1 discrete graphs ...2 Finsler manifolds ...3 interpolations with varying mass ...1 graphs: random walk ...2 Finsler: jump process in a manifold, (work in progress) Results . Results 3 .. . . .. . . The same kind of results hold in other settings ...1 discrete graphs ...2 Finsler manifolds ...3 interpolations with varying mass ...1 graphs: random walk ...2 Finsler: jump process in a manifold, (work in progress) ...3 varying mass: branching process, (work in progress) Results . Results 4 .. . . .. . . Schr¨odinger’s problem is the analogue of Hamilton’s least action principle. It allows for dynamical theories of diffusion processes random walks on graphs Results . Results 4 .. . . .. . . Schr¨odinger’s problem is the 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 References Schr¨odinger (1931) Villani (big yellow book on optimal transport) Zambrini (stochastic deformation of classical mechanics in the diffusion setting) References Schr¨odinger (1931) Villani (big yellow book on optimal transport) Zambrini (stochastic deformation of classical mechanics in the diffusion setting) Conforti + L. (preprint) Mikami (PTRF ’04) L. (JFA ’12, AoP ’15) . . . .. . . Thank you for your attention