  Inverse problems consist of recovering a signal from a collection of noisy
measurements. These problems can often be cast as feasibility problems;
however, additional regularization is typically necessary to ensure accurate
and stable recovery with respect to data perturbations. Hand-chosen analytic
regularization can yield desirable theoretical guarantees, but such approaches
have limited effectiveness recovering signals due to their inability to
leverage large amounts of available data. To this end, this work fuses
data-driven regularization and convex feasibility in a theoretically sound
manner. This is accomplished using feasibility-based fixed point networks
(F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of
which is the composition of a projection-based operator and a data-driven
regularization operator. Fixed point iteration is used to compute fixed points
of these operators, and weights of the operators are tuned so that the fixed
points closely represent available data. Numerical examples demonstrate
performance increases by F-FPNs when compared to standard TV-based recovery
methods for CT reconstruction and a comparable neural network based on
algorithm unrolling.
