Key words. Optimal mass transport, Wasserstein distance, gradient flow, Schrödinger bridge.

Transcription

1 ENTROPIC INTERPOLATION AND GRADIENT FLOWS ON WASSERSTEIN PRODUCT SPACES YONGXIN CHEN, TRYPHON GEORGIOU AND MICHELE PAVON Abstract. We show that the entropic interpolation between two given marginals provie by the Schröinger brige may be characterize as the curve in the Wasserstein space W which minimizes a suitable action. We also stuy the relative entropy evolution between an uncontrolle an a controlle ranom evolution as a graient flow on W W. Key wors. Optimal mass transport, Wasserstein istance, graient flow, Schröinger brige. AMS subject classifications. 47H07, 47H09, 60J5, 34A34, 49J0. Introuction. In the Schröinger brige problem (SBP) 3, one seeks the ranom evolution (a probability measure on path-space) which is closest in the relative entropy sense to a prior Markov iffusion evolution an has certain prescribe initial an final marginals µ an ν. As alreay observe by Schröinger 37, 38, the problem may be reuce to a static problem which, except for the cost, resembles the Kantorovich relaxe formulation of the optimal mass transport problem (OMT). Consiering that since (OMT) also has a ynamic formulation, we have two problems which amit equivalent static an ynamic versions 4. Moreover, in both cases, the solution entails a flow of one-time marginals joining µ an ν. The OMT yiels a isplacement interpolation flow whereas the SBP provies an entropic interpolation flow. Trough the work of Mikami, Mikami-Thieullen an Leonar 6, 7, 8, 3, 4, we know that the OMT may be viewe as a zero-noise limit of SBP when the prior is a sort of uniform measure on path space with vanishing variance. This connection has been extene to more general prior evolutions in 8, 9. Moreover, we know that, thanks to a very useful intuition by Otto 3, the isplacement interpolation flow {µ t ; 0 t } may be viewe as a constant-spee geoesic joining µ an ν in Wasserstein space 40. What can be sai from this geometric viewpoint of the entropic flow? It cannot be a geoesic, but can it be characterize as a curve minimizing a suitable action? In this paper, we show that this is inee the case resorting to a timesymmetric flui ynamic formulation of SBP. This characterization of the Schröinger brige answers as a byprouct a question pose by Carlen 4, pp Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota MN 55455, USA; chen468@umn.eu Department of Electrical an Computer Engineering, University of Minnesota, Minneapolis, Minnesota MN 55455, USA; tryphon@umn.eu Dipartimento i Matematica, Università i Paova, via Trieste 63, 35 Paova, Italy; pavon@math.unip.it Research partially supporte by the NSF, the AFOSR an by the University of Paova Research Project CPDA

2 SBP may be also formulate as a stochastic control problem with atypical bounary constraints. It is therefore interesting to compare the flow associate to the uncontrolle evolution (prior) to the optimal one. In particular, it is interesting to stuy the evolution of the relative entropy on the prouct Wasserstein space. The paper is outline as follows.... Elements of optimal mass transport theory. The literature on this problem is by now so vast an our egree of competence is such that we shall not even attempt here to give a reasonable an/or balance introuction to the various fascinating aspects of this theory. Fortunately, there exist excellent monographs an survey papers on this topic, see 35,, 40,, 4, 33, to which we refer the reaer. We shall only briefly review some concepts an results which are relevant for the topics of this paper... The static problem. Let µ an ν be probability measures on the measurable spaces X an Y, respectively. Let c : X Y 0, + ) be a measurable map with c(x, y) representing the cost of transporting a unit of mass from location x to location y. Let T µν be the family of measurable maps T : X Y such that T #µ = ν, namely such that ν is the push-forwar of µ uner T. Then Monge s optimal mass transport problem (OMT) is (.) inf c(x, T (x))µ(x). T T µν X Y As is well known, this problem may be unfeasible, namely the family T µν may be empty. This is never the case for the relaxe version of the problem stuie by Kantorovich in the 940 s (.) inf π Π(µ,ν) c(x, y)π(x, y) X Y where Π(µ, ν) are couplings of µ an ν, namely probability istributions on X Y with marginals µ an ν. Inee, Π(µ, ν) always contains the prouct measure µ ν. Let us specialize the Monge-Kantorovich problem (.) to the case X = Y = an c(x, y) = x y. Then, if µ oes not give mass to sets of imension n, by Brenier s theorem 40, p.66, there exists a unique optimal transport plan π (Kantorovich) inuce by a µ a.e. unique map T (Monge), T = ϕ, ϕ convex, an we have (.3) π = (I ϕ)#µ, ϕ#µ = ν. Among the extensions of this result, we mention that to strictly convex, superlinear costs c by Gangbo an McCann 5. The optimal transport problem may be use to introuce a useful istance between probability measures. Inee, let P ( ) be the set of probability measures µ on with finite secon moment. For µ, ν P ( ), the Kantorovich-Rubinstein istance, usually calle Wasserstein (Vasershtein) quaratic istance, is efine by ( (.4) W (µ, ν) = inf π Π(µ,ν) ) / x y π(x, y).

3 As is well known 40, Theorem 7.3, W is a bona fie istance. Moreover, it provies a most natural way to metrize weak convergence in P ( ) 40, Theorem 7.,, Proposition 7..5 (the same applies to the case p replacing with p everywhere). The Wasserstein space W is efine as the metric space ( P ( ), W ). It is a Polish space, namely a separable, complete metric space... The ynamic problem. So far, we have ealt with the static optimal transport problem. Nevertheless, in, p.378 it is observe that...a continuum mechanics formulation was alreay implicitly containe in the original problem aresse by Monge... Eliminating the time variable was just a clever way of reucing the imension of the problem. Thus, a ynamic version of the OMT problem was alreay in fieri in Gaspar Monge s 78 mémoire sur la théorie es éblais et es remblais! It was elegantly accomplishe by Benamou an Brenier in by showing that (.5a) W (µ, ν) = inf v(x, t) µ t (x)t, (µ,v) 0 (.5b) µ + (vµ) = 0, t (.5c) µ 0 = µ, µ = ν. Here the flow {µ t ; 0 t } varies over continuous maps from 0, to P ( ) an v over smooth fiels. In 4, Villani states at the beginning of Chapter 7 that two main motivations for the time-epenent version of OMT are a time-epenent moel gives a more complete escription of the transport; the richer mathematical structure will be useful later on. We can a three further reasons: it opens the way to establish a connection with the Schröinger brige problem (see Section 9 below), where the latter appears as a regularization of the former 6, 7, 8, 3, 4, 8, 9; it allows to view the optimal transport problem as an (atypical) optimal control problem 6-9. In some applications such as interpolation of images 0 or spectral morphing, the interpolating flow is essential! Let {µ t ; 0 t } an {v (x, t); (x, t) 0, } be optimal for (.5). Then µ t = ( t)i + t ϕ #µ, with T = ϕ solving Monge s problem, provies, in McCann s language, the isplacement interpolation between µ an ν. Then {µ t ; 0 t } may be viwe as a constant-spee geoesic joining µ an ν in Wasserstein space (Otto). This formally enows W with a pseuo Riemannian structure. McCann iscovere 5 that certain functionals are isplacement convex, namely convex along Wasserstein geoesics. µ k converges weakly to µ if fµ k fµ for every continuous, boune function f. 3

4 This has le to a variety of applications. Following one of Otto s main iscoveries, 3, it turns out that a large class of PDE s may be viewe as graient flows on the Wasserstein space W. This interpretation, because of the isplacement convexity of the functionals, is well suite to establish uniqueness an to stuy energy issipation an convergence to equilibrium. A rigorous setting in which to make sense of the Otto calculus has been evelope by Ambrosio, Gigli an Savaré for a suitable class of functionals. Convexity along geoesics in W also leas to new proofs of various geometric an functional inequalities 5, 40, Chapter 9. Finally, we mention that, when the space is not flat, qualitative properties of optimal transport can be quantifie in terms of how bouns on the Ricci-Curbastro curvature affect the isplacement convexity of certain specific functionals 4, Part II. The tangent space of P ( ) at a probability measure µ, enote by T µ P ( ) may be ientifie with the closure in L µ of the span of { ϕ : ϕ Cc }, where Cc is the family of smooth functions with compact support. It is naturally equippe with the scalar prouct of L µ. 3. The Fokker-Planck equation as a graient flow on Wasserstein space. Let us review the variational formulation of the Fokker-Planck equation as a graient flow on Wasserstein space, 40. Consier a physical system with phase space an with Hamiltonian H : x H(x) = E x. The thermoynamic states of the system are given by the family P( ) of probability istributions P on amitting ensity ρ. On P( ), we efine the internal energy as the expecte value of the Energy observable in state P (3.) U(H, ρ) = E P {H} = H(x) ρ(x)x = H, ρ. Let us also introuce the (ifferential) Gibbs entropy (3.) S(p) = k ln ρ(x)ρ(x)x, where k is Boltzmann s constant. S is strictly concave on P( ). Accoring to the Gibbsian postulate of classical statistical mechanics, the equilibrium state of a microscopic system at constant absolute temperature T an with Hamiltonian function H is necessarily given by the Boltzmann istribution law with ensity (3.3) ρ(x) = Z exp H(x) kt where Z is the partition function. Let us introuce the Free Energy functional F efine by (3.4) F (H, ρ, T ) := U(H, ρ) T S(ρ). Since S is strictly concave on S an U(E, ) is linear, it follows that F is strictly convex on the state space P( ). By Gibbs variational principle, the Boltzmann The letter Z was chosen by Boltzmann to inicate zustänige Summe (pertinent sum- here integral). 4

5 istribution ρ is a minimum point of the free energy F on P( ). Also notice that D(ρ ρ) = log ρ(x) R ρ(x) ρ(x)x N = S(ρ) + log Z + H(x)ρ(x)x = F (H, ρ, T ) + log Z. k kt R kt N Since Z oes not epen on ρ, we conclue that Gibb s principle is a trivial consequence of the fact that ρ minimizes D(ρ ρ) on D( ). Consier now an absolutely continuous curve µ t :, t W. Then, Chapter 8, there exist velocity fiel v t L µ t such that the following continuity equation hols on (0, T ) t µ t + (v t µ t ) = 0. Suppose µ t = x, so that the continuity equation (3.5) ρ t + (vρ) = 0 hols. We want to stuy the free energy functional F (H,, T ) or, equivalently, D( ρ), along the flow { ; t t }. Using (3.5), we get t D( ρ) = + log + R kt H(x) ρt t x N = + log + (3.6) kt H(x) (v )x. Integrating by parts, if the bounary terms at infinity vanish, we get (3.7) t D( ρ) = log + kt H(x) v x = log + kt H(x), v L. Thus, the Wasserstein graient of D( ρ) is The corresponing graient flow is (3.8) t = W D( ρ) = log + kt H(x). ( log + kt H(x) ) = kt H(x) +. But this is precisely the Fokker-Planck equation corresponing to the iffusion process (3.9) X t = kt H(X t)t + W t where W is a stanar n-imensional Wiener process. The process (3.9) has the Boltzmann istribution (3.3) as invariant ensity. Recall that, p.0 F (H,, T ) or, equivalently, D( ρ) are isplacement convex an have therefore a unique minimizer. 5

6 Remark. It seems worthwhile investigating to what extent the funamental assumption of statistical mechanics that the variables with longer relaxation time form a vector Markov process having (3.3) as invariant ensity is equivalent to the requirement that the flow of one-time ensities be a graient flow in Wasserstein space for the free energy. Let us finally plug the steepest escent (3.8) into (3.6). We get, after integrating by parts, the well known formula 6 (3.0) t D( ρ) = = + log + + log + = log ( ρt ρ kt H(x) ρt t x kt H(x) ) x. kt H(x) + x The last integral in (3.0) is sometimes calle the relative Fisher information of with respect to ρ 40, p Relative entropy as a functional on Wasserstein prouct spaces. Consier now two absolutely continuous curves µ t :, t W an µ t :, t W an their velocity fiels v t L µ t an ṽ t L µ t. Then, on (0, T ) (4.) (4.) t µ t + (v t µ t ) = 0, t µ t + (ṽ t µ t ) = 0. Let us suppose that µ t = (x)x an µ t = (x)x, for all t, t. Then (4.)-(4.) become (4.3) ρ + (vρ) = 0, t (4.4) ρ + (ṽ ρ) = 0, t where the fiels v an ṽ satisfy v(x, t) (x)x <, ṽ(x, t) (x)x <. The ifferentiability of the Wasserstein istance W (, ) has been stuie 4, Theorem 3.9. Consier instea the relative entropy functional on W W ( ) ( ) ρt ρ D( ) = h(, )x = log x, h( ρ, ρ) = log ρ. R ρ N t ρ Relative entropy functionals D( γ), where γ is a fixe probability measure (ensity), have been stuie as geoesically convex functionals on P ( ), see, Section 9.4. Our stuy of the evolution of D( ) is motivate by problems on a finite time interval such as the Schröinger brige problem (Section 6) an stochastic control 6

7 problems (Section 8) where it is important to evaluate relative entropy on two flows of marginals. We get t D( ) = (4.5) = h ρ ρ t + h ρ x ρ t ( ( + log log ) ( (ṽ ) + ρ ) t ( (v ) x After an integration by parts, assuming that the bounary terms at infinity vanish, we get ( ) t D( ρ ρt t ) = log ṽ ρ t v x R ρ N t ( ( )) log ρt (ṽ ) ρ ρt (4.6) = t x. ρt v Notice that the last expression looks like ( ( )) log ρt (ṽ ), ρt v L L. Thus, we ientify the graient of the functional D( ρ ρ) on W W as ( ) ( ( )) (4.7) W D( ρ ρ) log ρ ρ =. W D( ρ ρ) ρ Let us now compute the graient flow on W W corresponing to graient (4.7). We get ( ) ) ( ρt (4.8) log ρt ( ) t ρt = 0. Since ( ) ( ) ρt ρt J = log = = J, we observe the remarkable property that in the steepest escent (4.8) on the prouct Wasserstein space the fluxes are opposite an, therefore, ρ t = ρ t. If we plug the steepest escent (4.8) into (4.5), we get what appears to be a new formula ( t D( ) = + log log + ρ ) t ρ x R ρ N t t ( ) ( ) ρt = log ρt + x R ρ N t ( = + ρ ) ( ) t ρt (4.9) log x, 7

8 which shoul be compare to (3.0). Let us return to equation (4.6). By multiplying an iviing by in the last term of the mile expression, we get ( ) (4.0) t D( ρ ρt t ) = log (ṽ v) x which is precisely the expression obtaine in 34, Theorem III.. 5. Elements of Nelson-Föllmer kinematics of finite-energy iffusion processes. Let (Ω, F, P) be a complete probability space. A stochastic process {ξ(t); t t } is calle a finite-energy iffusion with constant iffusion coefficient σ I N if the paths ξ(ω) belong to C (, t ; R ) N (N-imensional continuous functions) an (5.) ξ(t) ξ(s) = t s β(τ)τ + σw + (t) W + (s), s < t t, where β(t) is at each time t a measurable function of the past {ξ(τ); τ t} an W is a stanar N-imensional Wiener process. Moreover, the rift β satisfies the finite energy conition { t } E β τ <. In, Föllmer has shown that a finite-energy iffusion also amits a reverse-time Ito ifferential. Namely, there exists a measurable function γ(t) of the future {ξ(τ); t τ t } calle backwar rift an another Wiener process W such that (5.) ξ(t) ξ(s) = Moreover, γ satisfies t s γ(τ)τ + σw (t) W (s), s < t t. { t } E γ τ <. Let us agree that t always inicate a strictly positive variable. For any function f :, t R let + f(t) = f(t + t) f(t) be the forwar increment at time t an let f(t) = f(t) f(t t) be the backwar increment at time t. For a finite-energy iffusion, Föllmer has also shown in that forwar an backwar rifts may be obtaine as Nelson s conitional erivatives 9 { } + ξ(t) β(t) = lim E ξ(τ), τ t, t 0 t { } ξ(t) γ(t) = lim E ξ(τ), t τ t, t 0 t the limits being taken in L N (Ω, F, P). It was finally shown in that the one-time probability ensity ( ) of ξ(t) (which exists for every t. ) is absolutely continuous on an the following uality relation hols t > 0 (5.3) E {β(t) γ(t) ξ(t)} = σ log ρ(ξ(t), t), a.s.. 8

9 Let us introuce the fiels b + (x, t) = E {β(t) ξ(t) = x}, b (x, t) = E {γ(t) ξ(t) = x}. Then, Ito s rule for the forwar an backwar ifferential of ξ imply that satisfies the two Fokker-Planck equations (5.4) ρ t + (b +ρ) σ ρ = 0, (5.5) ρ t + (b ρ) + σ ρ = 0. Following Nelson, let us introuce the current an osmotic rift of ξ by (5.6) v(t) = β(t) + γ(t), u(t) = β(t) γ(t), respectively. Clearly v is similar to the classical velocity, whereas u is the velocity ue to the noise which tens to zero when σ tens to zero. Let us also introuce v(x, t) = E {v(t) ξ(t) = x} = b +(x, t) + b (x, t). Then, combining (5.4) an (5.5), we get (5.7) ρ + (vρ) = 0, t which has the form of a continuity equation expressing conservation of mass. When ξ is Markovian with β(t) = b + (ξ(t), t) an γ(t) = b (ξ(t), t), (5.3) reuces to Nelson s relation (5.8) b + (x, t) b (x, t) = σ log (x). Then (5.7) hols with (5.9) v(x, t) = b + (x, t) σ log (x). 6. Schröinger briges an entropic interpolation. Let Ω = C(, t ; ) be the space of value continuous functions. Let Wx σ enote Wiener measure on Ω with variance σ I N starting at the point x at time. If, instea of a Dirac measure concentrate at x, we give the volume measure as initial conition, we get the unboune measure on path space W σ = W σ x x. It is a useful tool to introuce the family of istributions P on Ω which are equivalent to it. Let P P represent an a priori ranom evolution an let Q P. Then, we have D(Q P ) = E Q log Q t (6.a) = D(q 0 p 0 ) + E Q P σ βq β P t (6.b) = D(q p ) + E Q t σ γq γ P t Here q 0, q are the marginal ensities of Q at an t, respectively. Similarly, p 0, p are the marginal ensities of P. Let ρ 0 an ρ be two probability ensities. Let 9.

10 P(ρ 0, ρ ) enote the set of istributions in P having the prescribe marginal ensities at an t. Given P P, we consier the following problem: (6.) Minimize D(Q P ) over Q P(ρ 0, ρ ). Conitions for existence an uniqueness for this problem an properties of the minimizing measure have been stuie by many authors, most noticeably by Fortet, Beurlin, Jamison an Föllmer 4, 3, 0, 3. If there is at least one Q in D(ρ 0 ρ ) such that D(Q P ) <, there exists a unique minimizer Q in P(ρ 0 ρ ) calle the Schröinger brige from ρ 0 to ρ over P 3. Existence is guarantee by uner conitions on P, ρ 0 an ρ, see 4, Proposition.5. We shall tacitly assume henceforth that they are satisfie so that Q is well efine. In view of Sanov s theorem 36, solving the maximum entropy problem (6.) is equivalent to a problem of large eviations of the empirical istribution as showe by Föllmer 3 recovering Schröinger s original motivation. Using (5.6) an observing that D(q 0 p 0 ) an D(q p ) are constant for Q varying in P(ρ 0 ρ ), we get that problem (6.) is equivalent to (6.3) t Minimize E Q σ vq v P + σ uq u P t over Q P(ρ 0, ρ ). Suppose now that the prior measure P is Markovian. In this case, the classical results of Jamison 0 imply that the solution of (6.) or, equivalently, of (6.3) is also Markovian. Thus, we can restrict our search to P M (ρ 0, ρ ), namely Markovian measures in P(ρ 0, ρ ). Taking (5.8) into account, we can rewrite (6.3) in the form (6.4) Minimize + σ σ log ρq t ρ P t E Q t (ξ(t), t) t σ vq (ξ(t), t) v P (ξ(t), t) over Q P M (ρ 0 ρ ). Finally, in view of (5.7), we get the following equivalent flui ynamic formulation t (6.5a) Minimize (ρ,v) R σ v(x, t) vp (x, t) N + σ ρ log (x, t) ρ(x, t)xt, 8 ρp (6.5b) ρ + (vρ) = 0, t (6.5c) ρ = 0 0, ρ =. Comparing (6.5) to the OMT with prior formulate an stuie in 8, see also 9 for the Gauss-Markov case, we see that the essential ifference is that there is here an extra term in the action functional which has the form of a relative Fisher information of with respect to the prior one-time ensity ρ P t (Dirichlet form) 40, p.78 integrate over time. Also notice that, in view of (4.7), we have (6.6) log ρ P t = W D( ρ P t ). 0

11 To fin the connection to the classical OMT, let us specialize to the situation where the prior P = W σ. In that case, v P = u P = 0 an, multiplying the criterion by σ, we get the problem t (6.7a) Minimize (ρ,v) R v(x, t) + σ4 8 log (x) (x)xt, N (6.7b) ρ + (vρ) = 0, t (6.7c) ρ = 0 0, ρ =. Again, specializing (6.6) to the case ρ P t (x), we get (6.8) log = W k S(). If σ 0 it appears that in the limit we get the Benamou-Brenier formulation of OMT (.5). This is inee the case, see 6, 7, 8, 3, 4 an 8, 9 for the case with prior. 7. Optimal transport an Nelson s stochastic mechanics. There has been some interest in connecting optimal transport with Nelson s stochastic mechanics 4, 4, p.707 or irectly with the Schröinger equation 4. Consier the case of a free, non-relativistic particle of mass m. Then, a variational principle leaing to the Schröinger equation, can be base on the Guerra-Morato action functional 5 which, in flui ynamic form, is (7.) where A GM (, t ) = = t t m v(x, t) u(x, t) (x)xt m v(x, t) (x)x 8m I() RN ρ (7.) I(ρ) = x ρ t is the Fisher information of ρ since, for the Nelson process, σ = /m. Instea, the Yasue action 44 in flui-ynamic form is (7.3) A Y (, t ) = = t t m v(x, t) + u(x, t) (x)xt m v(x, t) (x)x + 8m I() t. In 4, p.3, Eric Carlen poses the question of minimizing the Yasue action subject to the continuity equation (6.7b) for given initial an final marginals (6.7c) ) stating that...the Euler-Lagrange equations for it are not easy to unerstan. In view of Section 6, we alreay know the solution to this problem: It is provie by the current velocity an the flow of one-time ensities of the Schröinger brige with (6.7c) an stationary Wiener measure as a prior.

12 8. Relative entropy prouction for controlle evolution. Consier on, t a finite-energy Markov process taking values in with forwar Ito ifferential (8.) ξ = b + (ξ(t), t)t + σw +. Let (x) be the probability ensity of ξ(t). Consier also the feeback controlle process ξ u with forwar ifferential (8.) ξ u = b + (ξ u (t), t)t + u(ξ u (t), t)t + σw +. Here the control u is aapte to the past an is such that ξ u is a finite-energy iffusion. Let ρ u t (x) be the probability ensity of ξ u (t). We are intereste in the evolution of D(ρ u t ). By (5.9)-(5.7), the ensities satisfy (8.3) (8.4) ρ u t + (vu ρ u ) = 0, By (4.0), we now get t D(ρu t ) = (8.5) = ρ + (vρ) = 0, t v(x, t) = b +(x, t) σ log (x) v u (x, t) = b + (x, t) + u(x, t) σ log ρu t (x). ( ρ u log t ( ρ u log t ) (v u v) ρ u t x ) ( )) (u σ ρ u log t ρ u t x. Suppose now ρ u t = ρ 0 t is also uncontrolle an iffers from only because of the initial conition at t =. Then (8.5) gives the well known formula generalizing (3.0) ( ) ( ) (8.6) t D(ρ0 t ) = σ ρ 0 log t ρ 0 log t ρ u t x R ρ N t which shows that two solutions of the same Fokker-Plank equation ten to get closer. 9. Schröinger briges as controlle evolution. Consier now the situation where ξ(t) represents a prior evolution on, t an the controlle evolution ξ u = ξ is the solution of the Schröinger brige problem for a pair of initial an final marginals ρ 0 an ρ 3, 43. Then the ifferential of ξ is given by (9.) ξ = b + (ξ (t), t)t + σ log ϕ(ξ (t), t)t + σw + where ϕ is space-time harmonic for the prior evolution, namely it satisfies (9.) ϕ t + b + ϕ + σ ϕ = 0. Let ρ ϕ be the ensity of ξ. Let us single out a special case of the Schröinger brige problem where relative entropy on path space is minimise uner the only constraint that the final marginal ensity be ρ. In such case, we have ρ ϕ t (x) = (x) ϕ(x, t).

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