A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm
This work provides rigorous recovery guarantees for diffusion priors in inverse problems, addressing a gap in theoretical understanding for a method with empirical success, though it is incremental in nature.
The authors developed a theoretical framework to analyze deterministic diffusion-based algorithms for solving inverse problems, showing that these algorithms can be interpreted as generalized projected gradient descent methods with varying projections and deriving quantitative convergence rates under certain conditions, with applications to low-dimensional compact sets and low-rank Gaussian mixture models.
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.