Wasserstein Hamiltonian flows

Chow, Shui-Nee; Li, Wuchen; Zhou, Haomin

doi:10.1016/j.jde.2019.08.046

Cited by 16 publications

(27 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In short, the Euler-Lagrange equation is from the primal coordinates (ρ t , ∂ t ρ t ) and the Hamiltonian flow is from the dual coordinates (ρ t , Φ t ). Similar interpretations can be found in [8].…”

Section: ∂G(ρsupporting

confidence: 87%

See 1 more Smart Citation

Accelerated Information Gradient Flow

Wang

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.

show abstract

Section: ∂G(ρsupporting

confidence: 87%

“…It is identical to the Wasserstein Hamiltonian flow introduced by [8]. The derivation simply comes from that…”

Section: Example 16 (Fisher-rao Hamiltonian Flow) the Fisher-rao Hamiltonian Flow Followsmentioning

confidence: 99%

Accelerated Information Gradient Flow

Wang

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many well-known equations, such as Schrödinger equation, Schrödinger bridge problem and Vlasov equation, can be written as Hamiltonian systems on the density manifold. In this sense, they can be considered as members of the so-called Wasserstein Hamiltonian flows ( [41,3,22,13,11,12,16]). The study of Wasserstein Hamiltonian flow can be traced back to Nelson's mechanics ( [35,36,37,38]).…”

Section: Introductionmentioning

confidence: 99%

“…with a Hamiltonian H 0 on the density manifold and δ δS , δ δρ being the variational derivatives, which is proposed by only imposing randomness on the initial position of the phase space [12]. This is different from the Hamiltonian flows considered in [3], where the authors consider and construct the solutions of the ODEs in the measure space of even dimensional phase variables equipped with the Wasserstein metric.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Wasserstein Hamiltonian Flows

Cui¹,

Liu²,

Zhou³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with L 2 -Wasserstein metric tensor, via the Wong-Zakai approximation. We begin our investigation by showing that the stochastic Euler-Lagrange equation, regardless it is deduced from either variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wassersetin Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schrödinger equation, Schrödinger equation with random dispersion, and Schrödinger bridge problem with common noise.

show abstract

“…0, subject to boundary conditions ρ(0) = µ, ρ(1) = ν. This is the well-known geodesic equation between two densities µ and ν on the Wasserstein manifold [27], and can also be viewed as a Wasserstein Hamiltonian flow with Hamiltonian H(ρ, S) = 1 2 R d |∇S| 2 ρdx when C(t) = 0, [8]. If S 0 = S t=0 is known, the optimal value g W (µ, ν), the L 2 -Wasserstein distance between µ and ν, equals 2H(µ, S 0 ).…”

Section: Introductionmentioning

confidence: 99%

A continuation multiple shooting method for Wasserstein geodesic equation

Cui¹,

Dieci²,

Zhou³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we propose a numerical method to solve the classic L 2 -optimal transport problem. Our algorithm is based on use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. We exploit the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, and our method is designed to retain the underlying Hamiltonian structure. Several numerical examples are presented to illustrate the performance of the method.

show abstract

Wasserstein Hamiltonian flows

Cited by 16 publications

References 12 publications

Accelerated Information Gradient Flow

Accelerated Information Gradient Flow

Stochastic Wasserstein Hamiltonian Flows

A continuation multiple shooting method for Wasserstein geodesic equation

Contact Info

Product

Resources

About