🤖 AI Summary
This work addresses linear Bayesian inverse problems in infinite-dimensional function spaces. We propose a preconditioned Langevin dynamics sampling method leveraging a score-based generative model (SGM) prior, ensuring algorithmic stability and convergence under mesh refinement. We rigorously define the infinite-dimensional Langevin sampler in Hilbert space for the first time and derive an explicit KL-divergence error bound. An optimal preconditioner—explicitly dependent on the score estimation error and the forward operator—is designed to achieve uniform convergence rates for multimodal posteriors. Theoretical analysis establishes sufficient conditions for global convergence in KL divergence, applicable to both Gaussian and generalized non-Gaussian priors. Numerical experiments demonstrate that the proposed preconditioner significantly enhances numerical stability and accelerates convergence.
📝 Abstract
Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite-dimensional function spaces - where such problems are naturally formulated - is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score-based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback-Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and the form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non-Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.