为扩散模型求解反问题提供理论分析，揭示其本质是时变投影梯度下降。
A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm
- 将扩散模型的得分函数视为对低维数据集的时变投影
- 在受限等距条件下，给出收敛速度与噪声调度的显式关系
- 适用于低维凸集和低秩高斯混合模型，支持全局收敛
从受损观测中恢复高维信号是反问题的核心挑战。近年来生成式扩散模型在提供数据驱动先验方面展现出显著的实证成功，但严格的恢复保证仍有限。本文建立了一个理论框架，用于分析基于扩散的确定性算法在反问题中的应用，聚焦于Kadkhodaie & Simoncelli提出的算法的确定性版本。首先，我们证明当底层数据分布集中在低维模型集上时，对应的噪声卷积得分可解释为对该集合的时变投影。这使得使用扩散先验的算法可被理解为具有可变投影的广义投影梯度下降法。当传感矩阵在模型集上满足受限等距性质时，可推导出依赖于噪声调度的定量收敛速率。我们将该框架应用于两种典型数据分布：低维紧凸集上的均匀分布和低秩高斯混合模型。在后者情形下，尽管模型集非凸，仍可建立全局收敛保证。
原文摘要 · Abstract (English)
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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