Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees

2026-05-06

AI Summary

This work addresses the conditional distribution bias in pre-trained diffusion models when applied to zero-shot linear inverse problems—such as image inpainting and super-resolution—where enforcing only observation consistency leads to suboptimal solutions. The authors propose a novel sampling method based on a normal–tangential score decomposition, leveraging the key insight that the score component along the observation direction is precisely determined by measurements, while the tangential component can be approximated using the unconditional score. By deriving a dimension-independent upper bound on the conditional mutual information, they quantify the mismatch error between initialization and path-wise scores, and accordingly design a projected Langevin initialization together with a guided reverse denoising strategy. The resulting approach consistently outperforms existing projection-based sampling baselines across multiple tasks, yielding high-quality samples that better align with the true conditional data distribution.

Abstract

We study zero-shot conditional sampling with pretrained diffusion models for linear inverse problems, including inpainting and super-resolution. In these problems, the observation determines only part of the unknown signal. The remaining degrees of freedom must be sampled according to the correct conditional data distribution. Existing projection-based samplers enforce measurement consistency by correcting the observed component during reverse diffusion. However, measurement consistency alone does not determine how probability mass should be distributed along the feasible set, and this can lead to biased conditional samples. We analyze this issue through a normal--tangent decomposition of the score function. For Gaussian noising, the observed-direction score is exactly determined by the measurement; only the tangent conditional score is unknown. We prove that the error from replacing this score by the unconditional tangent score is upper bounded by a dimension-free conditional mutual information between observed and unobserved components. This gives an information-theoretic decomposition into initialization and pathwise score-mismatch errors. Motivated by the theory, we propose a projected-Langevin initialization followed by guided reverse denoising, which outperforms a strong projection-based baseline in inpainting and super-resolution experiments.

Problem

• conditional diffusion
• linear inverse problems
• measurement consistency
• conditional sampling
• score function

Innovation

• conditional diffusion
• linear inverse problems
• score decomposition
• information-theoretic guarantees
• projected Langevin