Extremal flows in Wasserstein space

Conforti, Giovanni; Pavon, Michele

doi:10.1063/1.5018402

Cited by 9 publications

(8 citation statements)

References 64 publications

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“…Example 4 (Schrödinger Bridge problem). Consider a Schrödinger system [3] ∂ t η t = 1 2 ∆η t , ∂ t η * t = − 1 2 ∆η * t .…”

Section: It Is Amentioning

confidence: 99%

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Wasserstein Hamiltonian flows

Chow¹,

Li²,

Zhou³

2020

Journal of Differential Equations

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We establish kinetic Hamiltonian flows in density space embedded with the L 2 -Wasserstein metric tensor. We derive the Euler-Lagrange equation in density space, which introduces the associated Hamiltonian flows. We demonstrate that many classical equations, such as Vlasov equation, Schrödinger equation and Schrödinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space.

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“…Example 4 (Schrödinger Bridge problem). Consider a Schrödinger system [3] ∂ t η t = 1 2 ∆η t , ∂ t η * t = − 1 2 ∆η * t .…”

Section: It Is Amentioning

confidence: 99%

“…In literature, the study of Hamiltonian flows in density manifold follows Nelson's stochastic mechanics [1,7,8,9,10]. See related work in [3]. Along with this framework, Lafferty introduces the Riemannian manifold structure of density space.…”

Section: Introductionmentioning

confidence: 99%

Wasserstein Hamiltonian flows

Chow¹,

Li²,

Zhou³

2020

Journal of Differential Equations

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“…It is naturally equipped with the inner product of L 2 \mu . Several recent papers have contributed to the development of second-order calculus in Wasserstein's space [245,69,70,71], building on [164,113,6].…”

Section: The Dynamic Problemmentioning

confidence: 99%

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Chen¹,

Georgiou²,

Pavon³

2021

SIAM Rev.

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Schr\" odinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\" odinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\' ar. The problem, known as the Schr\" odinger bridge problem (SBP) with ``uniform"" prior, was more recently recognized as a regularization of the Monge--Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\" odinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938--1940 specifically for Schr\" odinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\" odinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric.In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\" odinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou--Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview \ast

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“…For the sake of completeness of our discussion, we also reveal the relations among the so-called Schrödinger system [9,11,3] and our derived systems ( 13) and ( 16). All three PDE systems are derived from the SBP.…”

Section: Discrete Sbp Based On Relative Entropy and Reference Markov ...mentioning

confidence: 99%

“…Remark 4.1. In approach (A), the Hamiltonian systems ( (13), ( 16) and ( 21)) are corresponding to the control problem (11), which is derived from discretizing the relative entropy H(P |Q) in (27); In approach (B), the Hamiltonian systems ( (24) and ( 25)) are corresponding to the control problem (23), which is derived via discretizing the Fisher information I(ρ) in (28). It worth mentioning that under continuous cases, ( 27) and (28) are equivalent under the transform (29) and their corresponding Hamiltonian systems are also equivalent.…”

Section: Discrete Sbp Based On Minimum Action With Fisher Informationmentioning

confidence: 99%

What is a stochastic Hamiltonian process on finite graph? An optimal transport answer

Cui¹,

Shu²,

Zhou³

2021

Preprint

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We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrödinger equation and Schrödinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.

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Extremal flows in Wasserstein space

Cited by 9 publications

References 64 publications

Wasserstein Hamiltonian flows

Wasserstein Hamiltonian flows

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

What is a stochastic Hamiltonian process on finite graph? An optimal transport answer

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