Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices with Applications to Neuroimaging

Abstract

The James-Stein shrinkage estimator was proposed in the field of Statistics as an estimator of the mean for samples drawn from a Gaussian distribution and shown to dominate the maximum likelihood estimator (MLE) in terms of the risk. This seminal work lead to a flurry of activity in the field of shrinkage estimation. However, there has been very little work on shrinkage estimation for data samples that reside on manifolds. In this paper, we present a novel shrinkage estimator of the Fréchet Mean (FM) of manifold-valued data for the manifold, $P_n$, of symmetric positive definite matrices of size ‘n’. We choose to endow $P_n$ with the well known Log-Euclidean metric for its simplicity and ease of computation. With this choice of the metric, we show that the shrinkage estimator can be derived in an analytic form. Further, we prove that the shrinkage estimate of FM dominates the MLE of the FM in terms of the risk. We present several synthetic data examples with noise along with performance comparisons to estimated FM using other non-shrinkage estimators. As an application of shrinkage FM-estimation to real data, we compute the average motor sensory area (M1) tract from diffusion MR brain scans of controls and patients with Parkinson Disease (PD). We first show the dominance of the shrinkage FM estimator over the MLE of FM in this setting and then perform group testing to show differences between PD and controls based on the M1 tracts.