Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as non-expansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate significantly faster convergence as compared to other provable PnP methods with similar convergence results.

翻译：即插即用（PnP）方法是一类高效的迭代方法，旨在通过经典优化算法（如ISTA或ADMM）将数据保真项与深度去噪器相结合。可证明的PnP方法是具有收敛保证（如不动点收敛或收敛至某个能量函数的临界点）的PnP方法的子类。许多现有可证明的PnP方法对去噪器或保真函数施加了严格限制，分别要求去噪器非扩张性或保真函数严格凸性。在本工作中，我们提出了一种新颖的算法框架，将拟牛顿步长融入基于邻近去噪器的可证明PnP框架中，从而在保持对去噪器较弱假设的同时显著加速收敛。通过将去噪器表征为弱凸函数的邻近算子，我们证明所提出的拟牛顿PnP算法的不动点是弱凸函数的临界点。在图像去模糊和超分辨率任务上的数值实验表明，与具有类似收敛结果的其他可证明PnP方法相比，该方法具有显著更快的收敛速度。