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

TL;DR

This paper develops a new shrinkage estimator for symmetric positive-definite matrices on the Log-Euclidean manifold, demonstrating its asymptotic optimality and effectiveness in diffusion MRI applications.

Contribution

It introduces an analytic shrinkage estimator for manifold-valued data, specifically on the SPD matrices with Log-Euclidean metric, extending shrinkage estimation to this domain.

Findings

Estimator is asymptotically optimal among a large class of estimators.

Performs well in simulated experiments.

Effective in diffusion MRI statistical inference.

Abstract

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.