  Many statistical settings call for estimating a population parameter, most
typically the population mean, based on a sample of matrices. The most natural
estimate of the population mean is the arithmetic mean, but there are many
other matrix means that may behave differently, especially in high dimensions.
Here we consider the matrix harmonic mean as an alternative to the arithmetic
matrix mean. We show that in certain high-dimensional regimes, the harmonic
mean yields an improvement over the arithmetic mean in estimation error as
measured by the operator norm. Counter-intuitively, studying the asymptotic
behavior of these two matrix means in a spiked covariance estimation problem,
we find that this improvement in operator norm error does not imply better
recovery of the leading eigenvector. We also show that a Rao-Blackwellized
version of the harmonic mean is equivalent to a linear shrinkage estimator
studied previously in the high-dimensional covariance estimation literature,
while applying a similar Rao-Blackwellization to regularized sample covariance
matrices yields a novel nonlinear shrinkage estimator. Simulations complement
the theoretical results, illustrating the conditions under which the harmonic
matrix mean yields an empirically better estimate.
