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

2025-09-24

AI Summary

This paper addresses the inverse problem of recovering high-dimensional signals from corrupted measurements. We propose a deterministic algorithmic framework grounded in diffusion priors: modeling the noise-convolved score function as a time-varying projection operator onto a low-dimensional model set, and designing a generalized projected gradient descent method. We establish, for the first time, a deterministic convergence theory for diffusion-prior-based algorithms—achieving global convergence without convexity or strong convexity assumptions, and providing quantitative convergence rates dependent on the noise schedule. Our analysis integrates properties of the score function, low-dimensional structural priors, and the restricted isometry property (RIP). Empirical validation on uniform distributions and low-rank Gaussian mixture models demonstrates substantial improvements in reconstruction accuracy and robustness. The core contribution lies in uncovering the intrinsic connection between diffusion priors and projection-based optimization, thereby bridging a critical gap in the theoretical foundations of data-driven inverse problem solvers.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Develops theoretical guarantees for diffusion-based signal recovery algorithms
• Analyzes deterministic diffusion priors for solving inverse problems
• Establishes convergence rates for nonconvex low-dimensional model sets

Innovation

• Deterministic diffusion prior algorithm analysis
• Interpreting diffusion scores as time-varying projections
• Deriving convergence rates via restricted isometry property