will study conditional sampling with diffusion models under linear constraints, with a focus on understanding how a pre-trained unconditional diffusion model can be used to sample from a conditional distribution. I will present a normal–tangent decomposition of the conditional score that separates the effect of the observed constraints from the remaining uncertainty in the distribution. This decomposition provides a way to characterize the discrepancy between the unconditional and conditional diffusion dynamics, and to relate this discrepancy to information-theoretic quantities.

Based on this perspective, I will introduce a sampling method that combines projected Langevin initialization on the constraint set with guided reverse diffusion. I will discuss theoretical and information-theoretic guarantees for the resulting sampler, as well as numerical results illustrating its behavior on linear inverse problems. At the end, I will also briefly discuss related results showing how the information content and structural properties of a target distribution can reduce the dependence of diffusion sampling on the ambient dimension.