  There is a growing interest in deep model-based architectures (DMBAs) for
solving imaging inverse problems by combining physical measurement models and
learned image priors specified using convolutional neural nets (CNNs). For
example, well-known frameworks for systematically designing DMBAs include
plug-and-play priors (PnP), deep unfolding (DU), and deep equilibrium models
(DEQ). While the empirical performance and theoretical properties of DMBAs have
been widely investigated, the existing work in the area has primarily focused
on their performance when the desired image prior is known exactly. This work
addresses the gap in the prior work by providing new theoretical and numerical
insights into DMBAs under mismatched CNN priors. Mismatched priors arise
naturally when there is a distribution shift between training and testing data,
for example, due to test images being from a different distribution than images
used for training the CNN prior. They also arise when the CNN prior used for
inference is an approximation of some desired statistical estimator (MAP or
MMSE). Our theoretical analysis provides explicit error bounds on the solution
due to the mismatched CNN priors under a set of clearly specified assumptions.
Our numerical results compare the empirical performance of DMBAs under
realistic distribution shifts and approximate statistical estimators.
