  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.
