Conditional score-based diffusion models for Bayesian inference in infinite dimensions
Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions - the conditional denoising estimator - can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.
Code (1)
Tasks
Bayesian InferenceDenoisingSimilar Papers 제목 키워드 기반
Supervised Guidance Training for Infinite-Dimensional Diffusion Models
Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse pro…
Scalable diffusion posterior sampling in infinite-dimensional inverse problems
Score-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward m…
Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models
We introduce Sequential Neural Posterior Score Estimation (SNPSE), a score-based method for Bayesian inference in simulator-based models. Our method, inspired by the remarkable success of score-based methods in generativ…
Bayesian InferenceConditional Optimal Transport on Function Spaces
We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop…
Bayesian InferenceConditional score-based diffusion models for solving inverse problems in mechanics
We propose a framework to perform Bayesian inference using conditional score-based diffusion models to solve a class of inverse problems in mechanics involving the inference of a specimen's spatially varying material pro…
Bayesian Inference