Intrinsic and Extrinsic Means on the Circle - A Maximum Likelihood Interpretation

For data samples in Rⁿ, the mean is a well known estimator. When the data set belongs to an embedded manifold M in Rⁿ, e.g. the unit circle in R², the definition of a mean can be extended and constrained to M by choosing either the intrinsic Riemannian metric of the manifold or the extrinsic metric of the embedding space. A common view has been that extrinsic means are approximate solutions to the intrinsic mean problem. This paper study both means on the unit circle and reveal how they are related to the ML estimate of independent samples generated from a Brownian distribution. The conclusion is that on the circle, intrinsic and extrinsic means are maximum likelihood estimators in the limits of high SNR and low SNR respectively.