研究前向后向算法中用迭代去噪替代近端算子的收敛性，发现无论迭代次数多少结果一致。
Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms
- 用字典框架构建分析与合成型高斯去噪器，通过反向传播优化
- 证明单次或无限次子迭代下，算法最终解相同，理论严谨
- 适合图像恢复与压缩感知领域研究人员参考
本文研究在前向后向（FB）算法中，以子迭代方式近似高斯去噪器替代近端算子时的性质，采用插件式（PnP）范式。特别地，在字典框架下分别通过双前向后向迭代和前向后向迭代获得分析型与合成型高斯去噪器。我们分析了对应的最小化问题及迭代的渐近行为。结果显示，合成型高斯去噪问题可视为一个近端算子；对于分析与合成两种情形，无论使用一次还是无限次子迭代求解去噪问题，FB-PnP算法均收敛到同一目标。为此，我们证明在热启动策略下，单次子迭代可被解释为原-对偶算法。此外，还给出了任意子迭代数下全局问题的Moreau-Yosida光滑化结果。最后，通过数值实验验证理论：包括一个简单的压缩感知示例，以及基于深度字典框架的图像恢复问题。
原文摘要 · Abstract (English)
In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework.
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