  Imaging is a standard example of an inverse problem, where the task of
reconstructing a ground truth from a noisy measurement is ill-posed. Recent
state-of-the-art approaches for imaging use deep learning, spearheaded by
unrolled and end-to-end models and trained on various image datasets. However,
many such methods require the availability of ground truth data, which may be
unavailable or expensive, leading to a fundamental barrier that can not be
bypassed by choice of architecture. Unsupervised learning presents an
alternative paradigm that bypasses this requirement, as they can be learned
directly on noisy data and do not require any ground truths. A principled
Bayesian approach to unsupervised learning is to maximize the marginal
likelihood with respect to the given noisy measurements, which is intrinsically
linked to classical variational regularization. We propose an unsupervised
approach using maximum marginal likelihood estimation to train a convex neural
network-based image regularization term directly on noisy measurements,
improving upon previous work in both model expressiveness and dataset size.
Experiments demonstrate that the proposed method produces priors that are near
competitive when compared to the analogous supervised training method for
various image corruption operators, maintaining significantly better
generalization properties when compared to end-to-end methods. Moreover, we
provide a detailed theoretical analysis of the convergence properties of our
proposed algorithm.
