Template estimation in computational anatomy: Fréchet means in top and quotient spaces are not consistent

In this article, we study the consistency of the template estimation with the Fréchet mean in quotient spaces. The Fréchet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases. Keyword : Template, Fréchet mean, group action, quotient space, inconsistency, consistency bias, empirical Fréchet mean, Hilbert space, manifold ∗Université Côte d’Azur, Inria, France, [email protected] †CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ‡CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France §Université Côte d’Azur, Inria, France