为去噪器求解MAP估计提供理论依据，证明其可收敛到最优解。
MAP Estimation with Denoisers: Convergence Rates and Guarantees
- 基于对数凹性假设，证明去噪器可逼近后验的近似梯度算子。
- 提出算法在平滑目标函数上进行梯度下降，实现收敛。
- 适用于使用预训练去噪器解决逆问题的研究者。
去噪器模型已成为求解逆问题的强大工具，能够利用预训练网络近似平滑先验分布的得分函数。这些模型常用于启发式迭代算法中，以求解最大后验（MAP）优化问题，其中负对数先验的近邻算子起核心作用。然而该算子通常不可计算，实践中常以预训练去噪器作为替代，但缺乏普遍的理论支持。本文表明，在先验 $p$ 满足对数凹性假设下，一种与实际中广泛使用的算法密切相关的简单算法，可严格收敛于近邻算子。我们进一步将该算法解释为在平滑近邻目标函数上的梯度下降。因此，本分析为一类虽经验有效但此前仅具启发性的方法提供了理论基础。
原文摘要 · Abstract (English)
Denoiser models have become powerful tools for inverse problems, enabling the use of pretrained networks to approximate the score of a smoothed prior distribution. These models are often used in heuristic iterative schemes aimed at solving Maximum a Posteriori (MAP) optimisation problems, where the proximal operator of the negative log-prior plays a central role. In practice, this operator is intractable, and practitioners plug in a pretrained denoiser as a surrogate-despite the lack of general theoretical justification for this substitution. In this work, we show that a simple algorithm, closely related to several used in practice, provably converges to the proximal operator under a log-concavity assumption on the prior $p$. We show that this algorithm can be interpreted as a gradient descent on smoothed proximal objectives. Our analysis thus provides a theoretical foundation for a class of empirically successful but previously heuristic methods.
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