Score-based diffusion models in Bayesian inverse problems

Abstract

This thesis investigates the use of score-based diffusion models for Bayesian inverse problems, with a focus on high-dimensional imaging applications. After introducing relevant applications such as inpainting, computed tomography (CT), deblurring and diffuse optical tomography (DOT) and reviewing the mathematical framework of Bayesian inverse problems, the thesis surveys recent developments in diffusion-based generative modelling, including conditional and unconditional score functions, infinite-dimensional formulations, and neural operator-based approximations.

This work develops a principled framework for interpolating between two score functions, such as a data-driven and a model-based score. It is demonstrated that, in the small diffusion-time regime, the proposed interpolation approximates the score of a geometric mixture distribution. A tunable mixing parameter enables the adaptive weighting of the two score components without retraining, and the theoretical findings are supported by numerical experiments in Gaussian settings.

This dissertation introduces UCoS, a formulation expressing conditional score functions as affine transformations of intermediate task-dependent scores for linear inverse problems. The approach is derived in the infinite-dimensional Hilbert-space setting and leads to a novel framework for posterior sampling that avoids repeated evaluations of the forward operator during inference. A modified score matching objective is introduced to enable training, and numerical errors are theoretically quantified. Numerical experiments demonstrate the performance of the proposed methods across several imaging problems. In inpainting with Gaussian priors, the proposed and conditional methods accurately reproduce posterior statistics, while unconditional approaches exhibit varying deviations. In large-scale CT imaging, UCoS achieves significant computational speed-ups and improved performance in low-parameter regimes. In high-dimensional deblurring, UCoS produces more realistic reconstructions than competing methods while requiring less computation. In a DOT problem with experimental and simulated data, the proposed methods show strong performance, with regularised variants improving reconstruction quality and coverage in challenging settings.

Overall, the results highlight the potential of score-based diffusion models for scalable Bayesian inference and demonstrate how learned generative models can be integrated with a classical inverse problem structure. The thesis further suggests that hybrid and task-dependent approaches offer promising directions for improving both computational efficiency and reconstruction quality in large-scale scientific imaging problems.