Abstract
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.
Highlights
We propose an efficient sampling method for Bayesian inference which is based on first-order Langevin dynamics [33] and which satisfies the property of affine invariance [13]
We have proposed a finite ensemble size implementation of the Kalman-Wasserstein gradient flow formalism put forward in [11], which requires the inclusion of a correction term (3.16) due to the multiplicative nature of the noise in the Langevin equations (3.10) [30]
Further computational savings can be achieved through the gradient-free implementation (3.14) for Bayesian inverse problem (BIP) as introduced in Example 2.1
Summary
We propose an efficient sampling method for Bayesian inference which is based on first-order (overdamped) Langevin dynamics [33] and which satisfies the property of affine invariance [13]. \delta \\wpidiet ilt Building upon the affine invariance property of the nonlinear Fokker--Planck equation (2.22), we demonstrate how to obtain stochastic evolution equations of the form (2.10) which satisfy all three properties (i)--(iii) from above.
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