Towards Unbiased Diffusion Variational Inversion via Principled Posterior Matching

Diffusion models provide expressive priors for imaging inverse problems, but scalable variational inference with an implicit, non-degenerate posterior remains challenging because the prior KL term depends on the score of the variational distribution. We propose Principled Posterior Matching (PPM), a score-based variational framework that supports both particle-based optimization for individual observations and amortized inference through a reconstruction network for rapid, single-pass reconstruction. For two distributions evolved under the same forward diffusion process, we use the KL-dissipation relation to express their finite-time discrepancy in terms of an integrated Fisher divergence and a terminal KL term. We further derive a pathwise gradient representation using a score-projection identity under explicit differentiability, integrability, and reparameterization assumptions. When the prior and variational scores are exact, Monte Carlo sampling yields an unbiased estimator of the resulting gradient. In practice, these scores are calculated by pretrained and auxiliary models, and its performance is therefore affected only by score-estimation and optimization errors. We evaluate PPM on a range of computational imaging problems, including natural-image inpainting, super-resolution, and deblurring, fluorescence-microscopy reconstruction, and synthetic radio-interferometric black-hole imaging. Across these tasks, PPM produces more diverse reconstruction samples while maintaining competitive reconstruction fidelity relative to existing approaches. These results demonstrate that PPM provides a unified approach to sample-based variational inference and efficient amortized reconstruction across diverse imaging inverse problems.