In plug-and-play (PnP) regularization, the proximal operator in algorithms
such as ISTA and ADMM is replaced by a powerful denoiser. This formal
substitution works surprisingly well in practice. In fact, PnP has been shown
to give state-of-the-art results for various imaging applications. The
empirical success of PnP has motivated researchers to understand its
theoretical underpinnings and, in particular, its convergence. It was shown in
prior work that for kernel denoisers such as the nonlocal means, PnP-ISTA
provably converges under some strong assumptions on the forward model. The
present work is motivated by the following questions: Can we relax the
assumptions on the forward model? Can the convergence analysis be extended to
PnP-ADMM? Can we estimate the convergence rate? In this letter, we resolve
these questions using the contraction mapping theorem: (i) for symmetric
denoisers, we show that (under mild conditions) PnP-ISTA and PnP-ADMM exhibit
linear convergence; and (ii) for kernel denoisers, we show that PnP-ISTA and
PnP-ADMM converge linearly for image inpainting. We validate our theoretical
findings using reconstruction experiments.

通过收缩映射定理，我们解决了关于 PnP-ISTA 和 PnP-ADMM 在对称去噪器和核去噪器上的线性收敛性的问题，并在图像修复上验证了我们的理论发现。