Extremal curves in Wasserstein space

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We show that known Newton-type laws for Optimal Mass Transport, Schrodinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space. The second order differential equations are accordingly obtained by annihilating the first variation of a suitable action.

Extremal curves in Wasserstein space

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We show that known Newton-type laws for Optimal Mass Transport, Schrodinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space. The second order differential equations are accordingly obtained by annihilating the first variation of a suitable action.
Extremal curves in Wasserstein space

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Extremal curves in Wasserstein space Giovanni Conforti1 and Michele Pavon2 1 Laboratoire Paul Painlevé Université des Sciences et Technologies de Lille 1, 59655 Villeneuve d’Ascq Cedex, France giovanniconfort@gmail.com 2 Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova via Trieste 63, 35121 Padova, Italy pavon@math.unipd.it Abstract. We show that known Newton-type laws for Optimal Mass Transport, Schrödinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space. The second order differential equations are accordingly obtained by annihilating the first variation of a suitable action. Keywords: Kantorovich-Rubinstein metric, calculus of variations, dis- placement interpolation, entropic interpolation, Schrödinger bridge, Madelung fluid. 1 Introduction Continuous random evolutions which are critical for some suitable action occur in many diverse fields of science. We have in mind, in particular, the follow- ing three famous problems: The Benamou-Brenier formulation of the Optimal Mass Transport (OMT) problem with quadratic cost [3], the Schrödinger Bridge Problem (SBP) [45, 46] and the quantum evolution of a nonrelativistic particle in Madelung’s fluid (NSM) [32, 25]. All three problems are considered in their fluid-dynamic form. The flow of one-time marginals {µt; 0 ≤ t ≤ 1} of each solution may be thought of as a curve in Wasserstein space. It is known that SBP may be viewed as a “regularization” of OMT , the latter problem being recovered through a “zero-noise limit” [34–36, 30, 31, 15, 10, 11]. Recently, it was shown by von Renesse [49], for the quantum mechani- cal Madelung fluid [25], and by Conforti [13], for the Schrödinger bridge, that their flows satisfy suitable Newton-like laws in Wasserstein space. In [23], the foundations of a Hamilton-Jacobi theory in Wasserstein space were laid. In this paper, we outline some of the results of [14], where we show that the solution flows of OMT, SBP and NSM may all be seen as extremal curves in Wasserstein space of a suitable action. The actions only differ by the presence or the sign of a (relative) Fisher information functional besides the kinetic energy term. The solution marginals flows correspond to critical points, i.e. annihi- late the first variation, of the respective functionals. The extremality conditions imply indeed the local form of Newton-type second laws in analogy to classi- cal mechanics [26, p.1777]. These are then interpreted in the frame of Otto’s formal Riemannian calculus for optimal transport of probability measures as second-order differential equations in Wasserstein space involving the covariant derivative. Although some of these results are present in some form in the cited literature, our goal here is to develop a coherent framework where an actual cal- culus of variations on Wasserstein space can be developed for various significant problems. The paper is outlined as follows. In Sections 2 and 3, we provide some es- sential background on OMT and SBP, respectively. In Section 4, we obtain the second-order differential equation from an extremality condition for the fluid- dynamic version of the Schrödinger problem. The same is then accomplished for the Madelung fluid - Nelson’s stochastic mechanics in Section 5. 2 Background on optimal mass transport The literature on this problem is by now vast. We refer the reader to the following monographs and survey papers [42, 18, 47, 1, 48, 2, 40, 44]. We shall only briefly review some concepts and results which are relevant for the topics of this paper. The optimal transport problem may be used to introduce a useful distance between probability measures. Indeed, let P2(RN ) be the set of probability mea- sures µ on RN with finite second moment. For ν0, ν1 ∈ P2(RN ), the Kantorovich- Rubinstein (Wasserstein) quadratic distance, is defined by W2(ν0, ν1) =  inf π∈Π(ν0,ν1) Z RN ×RN kx − yk2 dπ(x, y) 1/2 , (1) where Π(ν0, ν1) are “couplings” of ν0 and ν1, namely probability distributions on RN × RN with marginals ν0 and ν1. As is well known [47, Theorem 7.3], W2 is a bona fide distance. Moreover, it provides a most natural way to “metrize” weak convergence3 in P2(RN ) [47, Theorem 7.12], [1, Proposition 7.1.5] (the same applies to the case p ≥ 1 replacing 2 with p everywhere). The Wasserstein space W2 is defined as the metric space P2(RN ), W2  . It is a Polish space, namely a separable, complete metric space. A dynamic version of the OMT problem was elegantly accomplished by Benamou and Brenier in [3] by showing that W2 2 (ν0, ν1) = inf (µ,v) Z 1 0 Z RN kv(x, t)k2 µt(dx)dt, (2a) ∂µ ∂t + ∇ · (vµ) = 0, (2b) µ0 = ν0, µ1 = ν1. (2c) Here the flow {µt; 0 ≤ t ≤ 1} varies over continuous maps from [0, 1] to P2(RN ) and v over smooth fields. In [48, Chapter 7], Villani provides some motivation 3 µk converges weakly to µ if R RN fdµk → R RN fdµ for every continuous, bounded function f. to study the time-dependent version of OMT. Further reasons are the following. It allows to view the optimal transport problem as an (atypical) optimal control problem [7]-[11]. It provides a ground on which the Schrödinger bridge problem appears as a regularization of the former [34–36, 30, 31, 15, 10, 11]. Similarly with the Madelung fluid, see below. In some applications, such as interpolation of images [12] or spectral morphing [28], the interpolating flow is crucial. Let {µ∗ t ; 0 ≤ t ≤ 1} and {v∗ (x, t); (x, t) ∈ RN × [0, 1]} be optimal for (2). Then µ∗ t = [(1 − t)I + t∇ϕ] #ν0, with T = ∇ϕ solving Monge’s problem, provides, in McCann’s language, the dis- placement interpolation between ν0 and ν1 (# denotes “push-forward”). Then {µ∗ t ; 0 ≤ t ≤ 1} may be viewed as a constant-speed geodesic joining ν0 and ν1 in Wasserstein space. This formally endows W2 with a kind of Riemannian struc- ture. McCann discovered [33] that certain functionals are displacement convex, namely convex along Wasserstein geodesics. This has led to a variety of applica- tions. Following one of Otto’s main discoveries [29, 39], it turns out that a large class of PDE’s may be viewed as gradient flows on the Wasserstein space W2. This interpretation, because of the displacement convexity of the functionals, is well suited to establish uniqueness and to study energy dissipation and con- vergence to equilibrium. A rigorous setting in which to make sense of the Otto calculus has been developed by Ambrosio, Gigli and Savaré [1] for a suitable class of functionals. Convexity along geodesics in W2 also leads to new proofs of various geometric and functional inequalities [33], [47, Chapter 9]. The tangent space of P2(RN ) at a probability measure µ, denoted by TµP2(RN ) [1] may be identified with the closure in L2 µ of the span of {∇ϕ : ϕ ∈ C∞ c }, where C∞ c is the family of smooth functions with compact support. It is equipped with the scalar product of L2 µ. 3 Schrödinger bridges and entropic interpolation Let Ω = C([t0, t1]; RN ) be the space of RN valued continuous functions. Let Wσ2 x denote Wiener measure on Ω with variance σ2 IN starting at the point x at time t0. If, instead of a Dirac measure concentrated at x, we give the volume measure as initial condition, we get the unbounded measure on path space Wσ2 = R RN Wσ2 x dx. It is useful to introduce the family of distributions P on Ω which are equivalent to it. Let P(ρ0, ρ1) denote the set of distributions in P having the prescribed marginal marginals densities at t = 0 and t = 1, respectively. Then, the Schrödinger bridge problem (SBP) with Wσ2 as “prior” is the maximum entropy problem Minimize H(P|Wσ2 ) = EP  log dP dWσ2  over P ∈ P(ρ0, ρ1). (3) Conditions for existence and uniqueness for this problem and properties of the minimizing measure (with general Markovian prior) have been studied by many authors, most noticeably by Fortet, Beurlin, Jamison and Föllmer [22, 4, 27, 21, 30],[31, Proposition 2.5]. The solution P∗ is called the Schrödinger bridge from ρ0 to ρ1 over P [21]. We shall tacitly assume henceforth that they are satisfied so that P∗ is well defined. In view of Sanov’s theorem [43], solving the maximum entropy problem is equivalent to a problem of large deviations of the empirical distribution as showed by Föllmer [21] recovering Schrödinger’s original motiva- tion [45, 46]. It has been observed since the early nineties that SBP can be turned, thanks to Girsanov’s theorem, into a stochastic control problem with atypical boundary constraints, see [16, 5, 17, 41, 19]. The latter has a fluid dynamic coun- terpart: When prior is Wσ2 stationary Wiener measure with variance σ2 4 , the solution of the SBP with marginal densities ρ0 and ρ1 can be characterized as the solution of the fluid-dynamic problem [10, p.683], [24, Corollary 5.8]: inf (ρ,v) Z Rn Z 1 0  1 2σ2 kv(x, t)k2 + σ2 8 k∇ log ρ(x, t)k2  ρ(x, t)dtdx, (4a) ∂ρ ∂t + ∇ · (vρ) = 0, (4b) ρ(0, x) = ρ0(x), ρ(1, y) = ρ1(y), (4c) Notice that the only difference from the Benamou-Brenier problem (2) is given by an extra term in the action with the form of a Fisher Information functional I(ρ) = Z Rn k∇ρk2 ρ dx. (5) 4 Variational analysis for the fluid-dynamic SBP Let Pρ0ρ1 be the family of continuous flows of probability densities ρ = {ρ(·, t); 0 ≤ t ≤ 1} satisfying (4c) and let V be the family of continuous feedback control laws v(·, ·). Consider the unconstrained minimization over Pρ0ρ1 ×V of the Lagrangian L(ρ, v; λ) = Z Rn Z 1 0  1 2σ2 kv(x, t)k2 + σ2 8 k∇ log ρ(x, t)k2  ρ(x, t) +λ(x, t)  ∂ρ ∂t + ∇ · (vρ)  dtdx, where λ is a C1 Lagrange multiplier. After integration by parts and discarding the constant boundary terms, we get the problem of minimizing over Pρ0ρ1 × V Z Rn Z 1 0  1 2σ2 kv(x, t)k2 + σ2 8 k∇ log ρ(x, t)k2 +  − ∂λ ∂t − ∇λ · v)  ρ(x, t)dtdx. (6) 4 The case of a general reversible Markovian prior is treated in [14] where all the details of the variational analysis may also be found. Pointwise minimization with respect to v for a fixed flow in Pρ0ρ1 gives v∗ ρ(x, t) = σ2 ∇λ(x, t), (7) which is continuous. Plugging this into (6), we get to minimize over Pρ0ρ1 J(ρ) = − Z Rn Z 1 0  ∂λ ∂t + σ2 2 k∇λk2 − σ2 8 k∇ log ρk2  ρdtdx. (8) Setting the first variation of J in direction δρ equal to zero for all smooth vari- ations vanishing at times t = 0 and t = 1, we get the extremality condition ∂λ ∂t + σ2 2 k∇λk2 + σ2 8 k∇ log ρk2 + σ2 4 ∆ log ρ = 0. (9) By (9), the convective derivative of v∗ in (7) yields the acceleration field a∗ (x, t) =  ∂ ∂t + v∗ · ∇  (v∗ ) (x, t) = −σ2 ∇  σ2 8 k∇ log ρk2 + σ2 4 ∆ log ρ  . (10) The term appearing in the right-hand side of (10) may be viewed as a gradient in Wasserstein space of the entropic part of the Lagrangian (4a), namely −∇  σ2 8 k∇ log ρk2 + σ2 4 ∆ log ρ  = ∇W2 σ2 8 I(ρ). (11) This relation can be found in [49, A.2] and [13]. Indeed, a calculation shows d dt Z Rn k∇ log ρ(x, t)k2 ρ(x, t)dx = − Z Rn ∇  k∇ log ρk2 + 2∆ log ρ  ·v(x, t)ρ(x, t)dx. Finally, since v∗ is a gradient, it follows that the convective derivative a∗ is in fact the covariant derivative ∇W2 µ̇ µ̇ for the smooth curve t → µt = ρ(x, t)dx , see [49, A.3] and [2, Chapter 6]. Thus, (10) takes on the form of a Newton-type law [13] on W2 ∇W2 µ̇ µ̇ = σ4 8 ∇W2 I(µ). (12) In the case when σ2 & 0, we recover in the limit that displacement interpolation provides constant speed geodesics for which ∇W2 µ̇ µ̇ ≡ 0. 5 Optimal transport and Nelson’s stochastic mechanics There has been some interest in connecting optimal transport with Nelson’s stochastic mechanics [6], [48, p.707] or directly with the Madelung fluid [49]. Consider the case of a free non-relativistic particle of mass m (the case of a par- ticle in a force field is treated similarly). Then, a variational principle leading to the Schrödinger equation, can be based on the Guerra-Morato action functional [26] which, in fluid dynamic form, is AGM = Z t1 t0 Z Rn m 2 kv(x, t)k2 ρt(x)dx − ~2 8m I(ρt)  dt. (13) For the Nelson process, we have σ2 = ~/m. It can be derived directly from the classical action [38]. Instead, the Yasue action [50] in fluid-dynamic form is AY (t0, t1) = Z t1 t0 Z RN m 2 kv(x, t)k2 ρt(x)dx + ~2 8m I(ρt)  dt. (14) In [6, p.131], Eric Carlen poses the question of minimizing the Yasue action sub- ject to the continuity equation (4b) for given initial and final marginals (4c). In view of the formulation (4), we already know the solution: It is provided by Nel- son’s current velocity [37] and the flow of one-time densities of the Schrödinger bridge with (4c) and stationary Wiener measure as a prior. Finally observe that the action in (4a) and 1 ~ AGM only differ by the sign in front of the Fisher in- formation functional! Thus the variational analysis can be carried out as in the previous section obtaining eventually the Newton-type law as in [49] ∇W2 µ̇ µ̇ = − ~2 m28 ∇W2 I(µ). (15) 6 Conclusion We have outlined some of the results of [14] where the optimal evolutions for OMT, SBP and NSM are shown to be critical curves for suitable actions. 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