Sample Mean Versus Sample Fréchet Mean for Combining Complex Wishart Matrices: A Statistical Study

The space of covariance matrices is a non-Euclidean space. The matrices form a manifold which if equipped with a Riemannian metric becomes a Riemannian manifold, and recently this idea has been used for comparison and clustering of complex-valued spectral matrices, which at a given frequency are typically modeled as complex Wishart-distributed random matrices. Identically distributed sample complex Wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator. However, using the Riemannian geometry, their so-called sample Fréchet mean can also be found. We derive the expected value of the determinant of the sample Fréchet mean and the expected value of the sample Fréchet mean itself. The population Fréchet mean is shown to be a scaled version of the true covariance matrix. The risk under convex loss functions for the standard sample mean is never larger than for the Fréchet mean. In simulations, the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix, namely partial coherence.