Analyzing and Guiding Zero-Shot Posterior Sampling in Diffusion Models

TL;DR

This paper provides a rigorous spectral analysis of zero-shot diffusion-based inverse problem solvers, introducing a principled parameter design framework that improves reconstruction quality.

Contribution

It offers a closed-form spectral analysis of diffusion-based posterior samplers under Gaussian prior assumptions and proposes a new parameter tuning framework.

Findings

Closed-form spectral representations of diffusion samplers and ideal posteriors.

A new parameter design framework that balances perceptual quality and fidelity.

Spectral recommendations that differ from standard heuristics and adapt with diffusion steps.

Abstract

Recovering a signal from its degraded measurements is a long standing challenge in science and engineering. Recently, zero-shot diffusion based methods have been proposed for such inverse problems, offering a posterior sampling based solution that leverages prior knowledge. Such algorithms incorporate the observations through inference, often leaning on manual tuning and heuristics. In this work we propose a rigorous analysis of these approximate posterior samplers, relying on a Gaussianity assumption of the prior. Under this regime, we show that both the ideal posterior sampler and diffusion-based reconstruction algorithms can be expressed in closed-form, enabling their thorough analysis and comparisons in the spectral domain. Building on these representations, we introduce a principled framework for parameter design, replacing heuristic selection strategies used to date. The proposed approach is method-agnostic and yields tailored parameter choices that jointly account for the characteristics of the prior, the degraded signal, and the diffusion dynamics. We show that our spectral recommendations differ structurally from standard heuristics and vary with the diffusion step size, resulting in a consistent balance between perceptual quality and signal fidelity.