  It's well-known that inverse problems are ill-posed and to solve them
meaningfully, one has to employ regularization methods. Traditionally, popular
regularization methods are the penalized Variational approaches. In recent
years, the classical regularization approaches have been outclassed by the
so-called plug-and-play (PnP) algorithms, which copy the proximal gradient
minimization processes, such as ADMM or FISTA, but with any general denoiser.
However, unlike the traditional proximal gradient methods, the theoretical
underpinnings, convergence, and stability results have been insufficient for
these PnP-algorithms. Hence, the results obtained from these algorithms, though
empirically outstanding, can't always be completely trusted, as they may
contain certain instabilities or (hallucinated) features arising from the
denoiser, especially when using a pre-trained learned denoiser. In fact, in
this paper, we show that a PnP-algorithm can induce hallucinated features, when
using a pre-trained deep-learning-based (DnCNN) denoiser. We show that such
instabilities are quite different than the instabilities inherent to an
ill-posed problem. We also present methods to subdue these instabilities and
significantly improve the recoveries. We compare the advantages and
disadvantages of a learned denoiser over a classical denoiser (here, BM3D), as
well as, the effectiveness of the FISTA-PnP algorithm vs. the ADMM-PnP
algorithm. In addition, we also provide an algorithm to combine these two
denoisers, the learned and the classical, in a weighted fashion to produce even
better results. We conclude with numerical results which validate the developed
theories.
