  The James-Stein (JS) shrinkage estimator is a biased estimator that captures
the mean of Gaussian random vectors.While it has a desirable statistical
property of dominance over the maximum likelihood estimator (MLE) in terms of
mean squared error (MSE), not much progress has been made on extending the
estimator onto manifold-valued data.
  We propose C-SURE, a novel Stein's unbiased risk estimate (SURE) of the JS
estimator on the manifold of complex-valued data with a theoretically proven
optimum over MLE. Adapting the architecture of the complex-valued SurReal
classifier, we further incorporate C-SURE into a prototype convolutional neural
network (CNN) classifier. We compare C-SURE with SurReal and a real-valued
baseline on complex-valued MSTAR and RadioML datasets.
  C-SURE is more accurate and robust than SurReal, and the shrinkage estimator
is always better than MLE for the same prototype classifier. Like SurReal,
C-SURE is much smaller, outperforming the real-valued baseline on MSTAR
(RadioML) with less than 1 percent (3 percent) of the baseline size
