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Computational imaging has been revolutionized by compressed sensingalgorithms, which offer guaranteed uniqueness, convergence, and stabilityproperties. Model-based deep learning methods that combine imaging physics withlearned regularization priors have emerged as more powerful alternatives forimage recovery. The main focus of this paper is to introduce a memory efficientmodel-based algorithm with similar theoretical guarantees as CS methods. Theproposed iterative algorithm alternates between a gradient descent involvingthe score function and a conjugate gradient algorithm to encourage dataconsistency. The score function is modeled as a monotone convolutional neuralnetwork. Our analysis shows that the monotone constraint is necessary andsufficient to enforce the uniqueness of the fixed point in arbitrary inverseproblems. In addition, it also guarantees the convergence to a fixed point,which is robust to input perturbations. We introduce two implementations of theproposed MOL framework, which differ in the way the monotone property isimposed. The first approach enforces a strict monotone constraint, while thesecond one relies on an approximation. The guarantees are not valid for thesecond approach in the strict sense. However, our empirical studies show thatthe convergence and robustness of both approaches are comparable, while theless constrained approximate implementation offers better performance. Theproposed deep equilibrium formulation is significantly more memory efficientthan unrolled methods, which allows us to apply it to 3D or 2D+time problemsthat current unrolled algorithms cannot handle.

Memory-efficient model-based deep learning with convergence and robustness guarantees