  There has been significant recent interest in the use of deep learning for
regularizing imaging inverse problems. Most work in the area has focused on
regularization imposed implicitly by convolutional neural networks (CNNs)
pre-trained for image reconstruction. In this work, we follow an alternative
line of work based on learning explicit regularization functionals that promote
preferred solutions. We develop the Explicit Learned Deep Equilibrium
Regularizer (ELDER) method for learning explicit regularizers that minimize a
mean-squared error (MSE) metric. ELDER is based on a regularization functional
parameterized by a CNN and a deep equilibrium learning (DEQ) method for
training the functional to be MSE-optimal at the fixed points of the
reconstruction algorithm. The explicit regularizer enables ELDER to directly
inherit fundamental convergence results from optimization theory. On the other
hand, DEQ training enables ELDER to improve over existing explicit regularizers
without prohibitive memory complexity during training. We use ELDER to train
several approaches to parameterizing explicit regularizers and test their
performance on three distinct imaging inverse problems. Our results show that
ELDER can greatly improve the quality of explicit regularizers compared to
existing methods, and show that learning explicit regularizers does not
compromise performance relative to methods based on implicit regularization.
