Trust Region Plug-and-Play With Provable Convergence for Image Restoration

Abstract

The Plug-and-Play (PnP) method is an optimization-based splitting algorithm that solves image inversion problems by combining data fidelity with a deep learning-based denoiser as an implicit regularization term. While many theoretical studies have been conducted to develop convergent PnP methods, most frameworks remain restricted to first-order descent or quasi-Newton methods. This study proposes the first true Newton trust-region method designed directly on the Douglas-Rachford Envelope as a differentiable surrogate function for PnP. The trust-region subproblem is then solved using Steihaug-CG as the inner solver, enabling it to handle negative curvature in the optimization landscape. Meanwhile, the Hessian-vector product required for subproblem quadratic model is computed matrix-free via JVP on the denoiser. Global convergence is proven via the trust-region mechanism by leveraging non-expansive proximal denoiser, and local Q-quadratic rate further proven under the Kurdyka-Łojasiewicz conditions. Numerical experiments on image deblurring and super-resolution tasks indicate that the proposed method achieves competitive restoration capabilities compared with other state-of-the-art convergent PnP methods. Despite a higher per-iteration cost, it requires 5–10x fewer iterations, yielding faster convergence than first-order PnP methods.