Parametric Fokker-Planck Equation

Li, Wuchen; Shu, Liu; Zha, Hongyuan; Zhou, Haomin

doi:10.1007/978-3-030-26980-7_74

Cited by 10 publications

(7 citation statements)

References 20 publications

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“…Here the generative models are powerful in machine learning [15]. It applies the reparameterization trick (known as push-forward relation) to do efficient sampling.…”

Section: And the Wasserstein Information Matrix Satisfiesmentioning

confidence: 99%

Wasserstein information matrix

Li¹,

Zhao²

2019

Preprint

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We study information matrices for statistical models by the L 2 -Wasserstein metric. We call it Wasserstein information matrix (WIM), which is an analog of the classical Fisher information matrix. Based on this matrix, we introduce Wasserstein score functions and study covariance operators in statistical models. Using them, we establish a Wasserstein-Cramer-Rao bound for estimation. Also, by a ratio of the Wasserstein and Fisher information matrices, we prove various functional inequalities within statistical models, including both the Log-Sobolev and Poincaré inequalities. These inequalities relate to a new efficiency property named Poincaré efficiency, introduced via a Wasserstein natural gradient of maximal likelihood estimation. Furthermore, online efficiency for Wasserstein natural gradient methods is also established. Several analytical examples and approximations of the WIM are presented, including location-scale families, independent families, Gaussian mixture models, and rectified linear unit (ReLU) generative models.

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“…Here the generative models are powerful in machine learning [15]. It applies the reparameterization trick (known as push-forward relation) to do efficient sampling.…”

Section: And the Wasserstein Information Matrix Satisfiesmentioning

confidence: 99%

Wasserstein information matrix

Li¹,

Zhao²

2019

Preprint

Self Cite

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“…We expect that this joint study would be useful in developing transport estimation theory of quantum information theory, and designing AI-driven quantum computing algorithms for quantum systems. In the future, we will continue this line of study following transport information geometry [26,30].…”

Section: Discussionmentioning

confidence: 98%

“…In particular, the parameter estimation problem of probability measures by using parameterized Wasserstein gradient flows on either Kullback-Leibler (KL) divergence, also referred to as relative entropy, or L 2 -Wasserstein distance has been addressed by the second author [10,28,29]. This leads to a joint study between optimal transport [42] and information geometry [1,2], namely transport information geometry [26,30]. Here, the natural gradient induced by optimal transport is first applied for statistical learning problems.…”

Section: Introductionmentioning

confidence: 99%

Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Becker

2021

J Stat Phys

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In this article, we introduce a new approach towards the statistical learning problem $$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$ argmin ρ ( θ ) ∈ P θ W Q 2 ( ρ ⋆ , ρ ( θ ) ) to approximate a target quantum state $$\rho _{\star }$$ ρ ⋆ by a set of parametrized quantum states $$\rho (\theta )$$ ρ ( θ ) in a quantum $$L^2$$ L 2 -Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $$C^*$$ C ∗ algebras. For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold with quantum Wasserstein information matrix. Using a quantum analogue of the Benamou–Brenier formula, we derive a natural gradient flow on the parameter space. We also discuss certain continuous-variable quantum states by studying the transport of the associated Wigner probability distributions.

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“…When dimension d = 1, according to Corollary 5.1, G(θ) has explicit solution. Thus, push-forward approximation of 1D Fokker-Planck equation can be directly computed by solving the ODE system (27) with forward-Euler scheme [25]. In this section, we will mainly focus on numerical methods for (27) with dimension d ≥ 2.…”

Section: Methodsmentioning

confidence: 99%

Neural Parametric Fokker-Planck Equations

Liu,

Li,

Zha

et al. 2020

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In this paper, we develop and analyze numerical methods for high dimensional Fokker-Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs neural parametric Fokker-Planck equation. The fact that the Fokker-Planck equation can be viewed as the L 2 -Wasserstein gradient flow of Kullback-Leibler (KL) divergence allows us to derive the ODEs as the constrained L 2 -Wasserstein gradient flow of KL divergence on the set of probability densities generated by neural networks. For numerical computation, we design a variational semi-implicit scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle Fokker-Planck equations in higher dimensional spaces. Moreover, we also establish bounds for the asymptotic convergence analysis of the neural parametric Fokker-Planck equation as well as its error analysis for both the continuous and discrete (forward-Euler time discretization) versions.Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis.

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Parametric Fokker-Planck Equation

Abstract: We derive the Fokker-Planck equation on the parametric space. It is the Wasserstein gradient flow of relative entropy on the statistical manifold. We pull back the PDE to a finite dimensional ODE on parameter space. Some analytical example and numerical examples are presented.

Cited by 10 publications

References 20 publications

Wasserstein information matrix

Wasserstein information matrix

Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Neural Parametric Fokker-Planck Equations

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