Estimating the Template in the Total Space with the Fréchet Mean on Quotient Spaces may have a Bias: a Case Study on Vector Spaces Quotiented by the Group of Translations
International audienceWhen we have a deformation group acting on a vector space of observations, these data are not anymore elements of our space but rather orbits for the group action we consider. If the data are generated from an unknown template with noise, to estimate this template, one may want to minimize the variance in the quotient set. In this article we study statistics on a particular quotient space. We prove that the expected value of a random variable in our vector space mapped in the quotient space is different from the Fréchet mean in the quotient space when the observations are noisy
