An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems
This addresses the problem of computational efficiency for researchers and practitioners in fields like medical imaging and image processing, offering a scalable solution for large-scale inverse problems.
The paper tackles the computational cost of sampling from posterior distributions in linear inverse problems by proposing an unconditional representation of the conditional score function (UCoS), which avoids forward model evaluations during sampling and is shown to be exact and discretization-invariant, with validation in high-dimensional CT and image deblurring experiments.
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 mapping to generate a single sample, resulting in significant computational costs for large-scale inverse problems. To address this, we propose an unconditional representation of the conditional score function (UCoS) tailored to linear inverse problems, which avoids forward model evaluations during sampling by shifting computational effort to an offline training phase. In this phase, a \emph{task-dependent} score function is learned based on the linear forward operator. Crucially, we show that the conditional score can be derived \emph{exactly} from a trained (unconditional) score using affine transformations, eliminating the need for conditional score approximations. Our approach is formulated in infinite-dimensional function spaces, making it inherently discretization-invariant. We support this formulation with a rigorous convergence analysis that justifies UCoS beyond any specific discretization. Finally we validate UCoS through high-dimensional computed tomography (CT) and image deblurring experiments, demonstrating both scalability and accuracy.