  Unrolled neural networks emerged recently as an effective model for learning
inverse maps appearing in image restoration tasks. However, their
generalization risk (i.e., test mean-squared-error) and its link to network
design and train sample size remains mysterious. Leveraging the Stein's
Unbiased Risk Estimator (SURE), this paper analyzes the generalization risk
with its bias and variance components for recurrent unrolled networks. We
particularly investigate the degrees-of-freedom (DOF) component of SURE, trace
of the end-to-end network Jacobian, to quantify the prediction variance. We
prove that DOF is well-approximated by the weighted \textit{path sparsity} of
the network under incoherence conditions on the trained weights. Empirically,
we examine the SURE components as a function of train sample size for both
recurrent and non-recurrent (with many more parameters) unrolled networks. Our
key observations indicate that: 1) DOF increases with train sample size and
converges to the generalization risk for both recurrent and non-recurrent
schemes; 2) recurrent network converges significantly faster (with less train
samples) compared with non-recurrent scheme, hence recurrence serves as a
regularization for low sample size regimes.
