An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems

TL;DR

This paper introduces an offline-trained, discretization-invariant method for efficiently sampling from posterior distributions in linear inverse problems using score-based diffusion models, reducing computational costs.

Contribution

It presents an unconditional score representation that enables exact derivation of the conditional score from an offline-trained unconditional score, avoiding multiple forward model evaluations during sampling.

Findings

The method is exact and discretization-invariant.

It reduces computational costs in high-dimensional inverse problems.

Validated on CT and image deblurring tasks with scalable and accurate results.

Abstract

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.