An Empirical Bayes Approach to Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices

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