  The recovery of magnetic resonance (MR) images from undersampled measurements
is a key problem that has seen extensive research in recent years. Unrolled
approaches, which rely on end-to-end training of convolutional neural network
(CNN) blocks within iterative reconstruction algorithms, offer state-of-the-art
performance. These algorithms require a large amount of memory during training,
making them difficult to employ in high-dimensional applications. Deep
equilibrium (DEQ) models and the recent monotone operator learning (MOL)
approach were introduced to eliminate the need for unrolling, thus reducing the
memory demand during training. Both approaches require a Lipschitz constraint
on the network to ensure that the forward and backpropagation iterations
converge. Unfortunately, the constraint often results in reduced performance
compared to unrolled methods. The main focus of this work is to relax the
constraint on the CNN block in two different ways. Inspired by
convex-non-convex regularization strategies, we now impose the monotone
constraint on the sum of the gradient of the data term and the CNN block,
rather than constrain the CNN itself to be a monotone operator. This approach
enables the CNN to learn possibly non-monotone score functions, which can
translate to improved performance. In addition, we only restrict the operator
to be monotone in a local neighborhood around the image manifold. Our
theoretical results show that the proposed algorithm is guaranteed to converge
to the fixed point and that the solution is robust to input perturbations,
provided that it is initialized close to the true solution. Our empirical
results show that the relaxed constraints translate to improved performance and
that the approach enjoys robustness to input perturbations similar to MOL.
