Title: Inconsistency of Template Estimation by Minimizing of the Variance/Pre-Variance in the Quotient Space,
Abstract: We tackle the problem of template estimation when data have been randomly
deformed under a 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 the
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 the first part, we restrict ourselves to isometric
group action, in this case the Hilbertian distance is invariant under the group
action. We establish an asymptotic behavior of the consistency bias which is
linear with respect to the noise level. As a result the inconsistency is
unavoidable as soon as the noise is enough. In practice, template estimation
with a finite sample is often done with an algorithm called "max-max". In the
second part, also in the case of isometric group finite, we show the
convergence of this algorithm to an empirical Karcher mean. Our numerical
experiments show that the bias observed in practice can not be attributed to
the small sample size or to a convergence problem but is indeed due to the
previously studied inconsistency. In a third part, we also present some
insights of the case of a non invariant distance with respect to the group
action. We will see that the inconsistency still holds as soon as the noise
level is large enough. Moreover we prove the inconsistency even when a
regularization term is added.