Abstract
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.
Abstract (translated)
在这项工作中，我们研究了在插件式（PnP）框架下用子迭代过程近似高斯去噪器时前向后向（FB）算法的行为。特别地，在字典框架内，我们将分析和综合高斯去噪器分别通过展开对偶-FB迭代或FB迭代获得。我们分析了相关的最小化问题以及由此产生的FB-PnP迭代的渐进行为。特别是，我们展示了合成高斯降噪问题可以被视为邻近算子。对于每种情况，无论是分析还是综合，我们都证明了无论是在每次迭代中只使用一次还是无限次子迭代来解决去噪问题，FB-PnP算法都会求解相同的问题。为此，我们展示了在使用热启动策略时，FB-PnP中的“单次子迭代”策略可以被解释为一种原始对偶算法。此外，当使用任意数量的子迭代进行全局问题的Moreau-Yosida平滑处理时，我们也提供了类似的结果。最后，我们通过数值模拟来说明我们的理论结果。特别地，首先考虑了一个玩具压缩感知示例以及深度字典框架下的图像恢复问题。
URL
https://arxiv.org/abs/2411.13276
