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.

翻译：本文研究了在前向后向算法中，以即插即用方式用子迭代过程近似高斯降噪器替代邻近算子时的算法行为。特别地，我们在字典框架下分别通过展开对偶前向后向迭代或前向后向迭代，获得了分析与合成两类高斯降噪器。我们分析了相关的极小化问题以及所得前向后向即插即用迭代的渐近行为。特别地，我们证明了合成高斯降噪问题可视为邻近算子。对于分析与合成两种情形，我们证明了无论每次迭代中使用一次还是无限次子迭代求解降噪问题，前向后向即插即用算法均求解同一问题。为此，我们证明了前向后向即插即用中每种"单次子迭代"策略在使用热重启策略时可解释为原始-对偶算法。我们进一步展示了在任意子迭代次数下，对全局问题采用莫罗-吉田光滑化时的类似结果。最后，我们通过数值模拟验证理论结果，包括一个压缩感知的示例性实验以及在深度字典框架下的图像复原问题。