  The James-Stein estimator is an estimator of the multivariate normal mean and
dominates the maximum likelihood estimator (MLE) under squared error loss. The
original work inspired great interest in developing shrinkage estimators for a
variety of problems. Nonetheless, research on shrinkage estimation for
manifold-valued data is scarce. In this paper, we propose shrinkage estimators
for the parameters of the Log-Normal distribution defined on the manifold of $N
\times N$ symmetric positive-definite matrices. For this manifold, we choose
the Log-Euclidean metric as its Riemannian metric since it is easy to compute
and is widely used in applications. By using the Log-Euclidean distance in the
loss function, we derive a shrinkage estimator in an analytic form and show
that it is asymptotically optimal within a large class of estimators including
the MLE, which is the sample Fr\'echet mean of the data. We demonstrate the
performance of the proposed shrinkage estimator via several simulated data
experiments. Furthermore, we apply the shrinkage estimator to perform
statistical inference in diffusion magnetic resonance imaging problems.
