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The recovery of magnetic resonance (MR) images from undersampled measurementsis a key problem that has seen extensive research in recent years. Unrolledapproaches, which rely on end-to-end training of convolutional neural network(CNN) blocks within iterative reconstruction algorithms, offer state-of-the-artperformance. These algorithms require a large amount of memory during training,making them difficult to employ in high-dimensional applications. Deepequilibrium (DEQ) models and the recent monotone operator learning (MOL)approach were introduced to eliminate the need for unrolling, thus reducing thememory demand during training. Both approaches require a Lipschitz constrainton the network to ensure that the forward and backpropagation iterationsconverge. Unfortunately, the constraint often results in reduced performancecompared to unrolled methods. The main focus of this work is to relax theconstraint on the CNN block in two different ways. Inspired byconvex-non-convex regularization strategies, we now impose the monotoneconstraint 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 approachenables the CNN to learn possibly non-monotone score functions, which cantranslate to improved performance. In addition, we only restrict the operatorto be monotone in a local neighborhood around the image manifold. Ourtheoretical results show that the proposed algorithm is guaranteed to convergeto the fixed point and that the solution is robust to input perturbations,provided that it is initialized close to the true solution. Our empiricalresults show that the relaxed constraints translate to improved performance andthat the approach enjoys robustness to input perturbations similar to MOL.

Local monotone operator learning using non-monotone operators: MnM-MOL