A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm

• URL: http://arxiv.org/abs/2509.20511v1
• Date: Wed, 24 Sep 2025 19:35:23 GMT
• Title: A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm
• Authors: Oscar Leong, Yann Traonmilin,
• Abstract summary: High-dimensional corrupted measurements are a central challenge in inverse problems.<n>Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors.
• Score: 5.71305698739856
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recovering high-dimensional signals from corrupted measurements is a central challenge in inverse problems. Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors, but rigorous recovery guarantees remain limited. In this work, we develop a theoretical framework for analyzing deterministic diffusion-based algorithms for inverse problems, focusing on a deterministic version of the algorithm proposed by Kadkhodaie \& Simoncelli \cite{kadkhodaie2021stochastic}. First, we show that when the underlying data distribution concentrates on a low-dimensional model set, the associated noise-convolved scores can be interpreted as time-varying projections onto such a set. This leads to interpreting previous algorithms using diffusion priors for inverse problems as generalized projected gradient descent methods with varying projections. When the sensing matrix satisfies a restricted isometry property over the model set, we can derive quantitative convergence rates that depend explicitly on the noise schedule. We apply our framework to two instructive data distributions: uniform distributions over low-dimensional compact, convex sets and low-rank Gaussian mixture models. In the latter setting, we can establish global convergence guarantees despite the nonconvexity of the underlying model set.

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