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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. 

The inference stage of diffusion models involves running a reverse-time diffusion stochastic differential equation, transforming samples from a Gaussian latent distribution into samples from a target distribution on a low-dimensional manifold. The intermediate values can be interpreted as noisy images, with the amount of noise determined by the forward diffusion process noise schedule. Boomerang is an approach for local sampling of image manifolds, which involves adding noise to an input image, moving it closer to the latent space, and mapping it back to the image manifold through a partial reverse diffusion process. Boomerang can be used with any pretrained diffusion model without adjustments to the reverse diffusion process, and we present three applications: constructing privacy-preserving datasets with controllable anonymity, increasing generalization performance with Boomerang for data augmentation, and enhancing resolution with a perceptual image enhancement framework.