FRECHET MEANS OF RIEMANNIAN DISTANCES: EVALUATIONS AND APPLICATIONS

Abstract

The space of symmetric positive definite matrices forms a manifold with an ambient is Euclidean space. In order to measure the distances between the objects on this manifold several metrics have been proposed. In this work we study the concept of averaging over the elements of the manifold by using the notion of Frechet mean. The main advantageous of this method is its connection to the metrics as a result of which we can utilize the Reimannian distances to obtain the mean of positive definite matrices. We consider three Reimannian metrics which have been developed on the manifold of symmetric positive definite matrices. The methods of obtaining the Frechet mean in the case of each metric will be discussed. The performance of each estimator will be demonstrated by using models based on matrix Cholesky factor, matrix square root and matrix logarithm. The deviation from the nominal covariance in each case will be evaluated using loss function, Euclidean distance and root Euclidean distance. We will see that depending on the model under investigation, Frechet mean of Reimannian distances performs better in most of the cases.

In terms of application, we analyse the performance of each Frechet mean estimator in a classification task. For this purpose we evaluate the method of distance to the center of mass using the simulated data. This method will also be applied on the high content cell image data set in order to classify the cells with respect to the type of treatment that has been used.