SNaP: One-Step Posterior Sampling for Noisy Inverse Problems

2026-09-28

AI Summary

This study addresses the slow sampling of diffusion models in solving inverse problems and the difficulty of MeanFlow in ensuring measurement consistency. To this end, we propose SNaP, a one-step posterior sampler. This method innovatively designs an anisotropic Gaussian source distribution determined by the measurement operator, which precisely anchors measurable directions while preserving weakly informative variations. By integrating MeanFlow, conditional flow matching, and Gaussian noise modeling, SNaP enables efficient solutions to linear inverse problems. Requiring only a single network evaluation to generate high-quality samples, the proposed approach achieves a 30- to 2250-fold speedup over conventional iterative methods.

Abstract

Diffusion and flow-matching models can produce high-quality posterior samples for inverse problems, but typically require tens to thousands of network evaluations per draw. MeanFlow enables one-step generation, yet applying it to inverse problems leaves no intermediate steps at which to enforce measurement consistency. We introduce SNaP, a one-step MeanFlow posterior sampler for linear inverse problems with Gaussian noise. Its central innovation is a measurement-adapted source: a Gaussian distribution whose mean and anisotropic covariance are determined by the measurement operator, observation, and noise level. The source anchors well-measured directions while preserving variation where the measurements are weak or uninformative. We show that the exact conditional flow transports this source to the true posterior. Across natural-image restoration and multi-coil MRI, SNaP produces diverse, high-quality samples with one network evaluation per draw, 30 to 2250 $\times$ faster than iterative samplers.

Problem

• inverse problems
• posterior sampling
• one-step generation
• diffusion models
• measurement consistency

Innovation

• One-Step Posterior Sampling
• MeanFlow
• Inverse Problems
• Measurement-Adapted Source
• Conditional Flow