Local Monotone Operator Learning Using Non-Monotone Operators: MNM-MOL

Model-based deep learning methods, which train convolutional neural network blocks within iterative algorithms in an end-to-end fashion, have been shown to provide state-of-the-art performance in several inverse problems. Deep equilibrium (DEQ) models and monotone operator learning (MOL) were introduced to reduce the memory demand of unrolled algorithms. However, the Lipschitz constraint or monotone constraint in DEQ methods often results in a decrease in performance compared to unrolled methods. This work relaxes the monotone constraint in two ways. We impose the monotone constraint on the sum of the gradient of the data term and the CNN block, allowing the CNN to learn possibly non-monotone score functions. Additionally, the operator is only restricted to being monotone in a local neighborhood around the image manifold. We present theoretical results demonstrating that the proposed scheme is guaranteed to converge to the fixed point and that the solution is robust to input perturbations. Empirical results show that the relaxed constraint leads to improved performance, and that the approach has robustness to input perturbations similar to that of MOL.