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

Stéphanie Allassonnière 1 Loïc Devilliers 1 Xavier Pennec 2
1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : When 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.
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Communication dans un congrès
Mathematical Foundations of Computational Anatomy (MFCA'15), Oct 2015, Munich, Germany. pp.131-142, 2015, Proceedings of the fifth international workshop on Mathematical Foundation sof Computational Anatomy (MFCA'15). 〈http://www-sop.inria.fr/asclepios/events/MFCA15/〉
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Stéphanie Allassonnière, Loïc Devilliers, Xavier Pennec. 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. Mathematical Foundations of Computational Anatomy (MFCA'15), Oct 2015, Munich, Germany. pp.131-142, 2015, Proceedings of the fifth international workshop on Mathematical Foundation sof Computational Anatomy (MFCA'15). 〈http://www-sop.inria.fr/asclepios/events/MFCA15/〉. 〈hal-01203816〉

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