Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation

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.