SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1633-1658, January 2020.
We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.

中文翻译：

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贝叶斯推断的仿射不变相互作用朗格文动力学

SIAM应用动力系统杂志，第19卷第3期，第1633-1658页，2020年1月。
我们提出了一种计算方法（缩写为ALDI），用于基于一阶（过阻尼）Langevin动力学从给定目标分布进行采样，该方法满足仿射不变性的性质。ALDI的中心思想是运行具有经验协方差的粒子集合，以作为其潜在的Langevin动力学的前提。ALDI不需要取经验协方差矩阵的逆或平方根，从而可以应用于高维采样问题。从非简并性和遍历性的角度研究了ALDI的理论特性。此外，我们研究了它与黎曼流形和Wasserstein梯度流上的扩散的关系。贝叶斯推理是ALDI的主要应用领域。如果正向问题存在加性高斯测量误差，ALDI允许整体卡尔曼滤波器的精髓实现无梯度近似。针对PDE约束的贝叶斯逆问题，提供了无梯度和基于梯度的ALDI之间的计算比较。