Title: Inconsistency of Template Estimation with the Fr{é}chet mean in Quotient Space,
Abstract: We tackle the problem of template estimation when data have been randomly
transformed under an isometric group action in the presence of noise. In order
to estimate the template, one often minimizes the variance when the influence
of the transformations have been removed (computation of the Fr{é}chet mean
in quotient space). The consistency bias is defined as the distance (possibly
zero) between the orbit of the template and the orbit of one element which
minimizes the variance. In this article we establish an asymptotic behavior of
the consistency bias with respect to the noise level. This behavior is linear
with respect to the noise level. As a result the inconsistency is unavoidable
as soon as the noise is large enough. In practice, the template estimation with
a finite sample is often done with an algorithm called max-max. We show the
convergence of this algorithm to an empirical Karcher mean. Finally, our
numerical experiments show that the bias observed in practice cannot be
attributed to the small sample size or to a convergence problem but is indeed
due to the previously studied inconsistency.