  The asymptotic concentration of the Fr{\'e}chet mean of IID random variables
on a Rieman-nian manifold was established with a central limit theorem by
Bhattacharya \& Patrangenaru (BP-CLT) [6]. This asymptotic result shows that
the Fr{\'e}chet mean behaves almost as the usual Euclidean case for
sufficiently concentrated distributions. However, the asymptotic covariance
matrix of the empirical mean is modified by the expected Hessian of the squared
distance. This Hessian matrix was explicitly computed in [5] for constant
curvature spaces in order to relate it to the sectional curvature. Although
explicit, the formula remains quite difficult to interpret, and the intuitive
effect of the curvature on the asymptotic convergence remains unclear.
Moreover, we are most often interested in the mean of a finite sample of small
size in practice. In this work, we aim at understanding the effect of the
manifold curvature in this small sample regime. Last but not least, one would
like computable and interpretable approximations that can be extended from the
empirical Fr{\'e}chet mean in Rie-mannian manifolds to the empirical
exponential barycenters in affine connection manifolds. For distributions that
are highly concentrated around their mean, and for any finite number of
samples, we establish explicit Taylor expansions on the first and second moment
of the empirical mean thanks to a new Taylor expansion of the Riemannian
log-map in affine connection spaces. This shows that the empirical mean has a
bias in 1/n proportional to the gradient of the curvature tensor contracted
twice with the covariance matrix, and a modulation of the convergence rate of
the covariance matrix proportional to the covariance-curvature tensor. We show
that our non-asymptotic high concentration expansion is consistent with the
asymptotic expansion of the BP-CLT. Experiments on constant curvature spaces
demonstrate that both expansions are very accurate in their domain of validity.
Moreover, the modulation of the convergence rate of the empirical mean's
covariance matrix is explicitly encoded using a scalar multiplicative factor
that gives an intuitive vision of the impact of the curvature: the variance of
the empirical mean decreases faster than in the Euclidean case in negatively
curved space forms, with an infinite speed for an infinite negative curvature.
This suggests potential links with the stickiness of the Fr{\'e}chet mean
described in stratified spaces. On the contrary, the variance of the empirical
mean decreases more slowly than in the Euclidean case in positive curvature
space forms, with divergence when we approach the limits of the Karcher \&
Kendall concentration conditions with a uniform distribution on the equator of
the sphere, for which the Fr{\'e}chet mean is not a single point any more.
