# Inconsistency of Template Estimation with the Fr{\'e}chet mean in Quotient Space

## TL;DR

This paper investigates the inconsistency in template estimation using the Fréchet mean in quotient spaces, showing that noise causes unavoidable bias that grows linearly with noise level, affecting practical algorithms.

## Contribution

It establishes the asymptotic linear behavior of the consistency bias due to noise and proves the convergence of the max-max algorithm to an empirical Karcher mean.

## Key findings

- Bias increases linearly with noise level
- Inconsistency is unavoidable at high noise levels
- Practical bias is due to theoretical inconsistency, not sample size or convergence issues

## 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{\'e}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.