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, with applications in inverse problems and imaging. 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 2--8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.

翻译：即插即用（PnP）方法是一类高效的迭代方法，旨在利用经典优化算法（如ISTA或ADMM）将数据保真项与深度降噪器相结合，广泛应用于逆问题与成像领域。可证明PnP方法是具有收敛保证的PnP方法子类，例如不动点收敛或收敛至某能量函数的临界点。现有多种可证明PnP方法对降噪器或保真函数施加了严格限制，例如要求降噪器非扩张或保真函数强凸。本文提出一种新颖的算法方法，将拟牛顿步骤融入基于邻近降噪器的可证明PnP框架，在保持降噪器轻量假设的同时大幅加速收敛。通过将降噪器表征为弱凸函数的邻近算子，我们证明所提拟牛顿PnP算法的不动点即为弱凸函数的临界点。针对图像去模糊和超分辨率的数值实验表明，与其它具有相似重建质量的可证明PnP方法相比，本文方法收敛速度提升2至8倍。