Minimax properties of Fréchet means of discretely sampled curves
Résumé
We study the problem of estimating a mean pattern from a set of similar curves in the setting where the variability in the data is due to random geometric deformations and additive noise. This problem requires to define non-Euclidean distances by using the action of a Lie group on an infinite dimensional space of curves. This approach leads to the construction of estimators based on the notion of Fréchet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. A recent research direction in nonparametric statistics is the study of the properties of the Fréchet mean in deformable models, and the development of consistent estimators of a mean pattern. Using such models, we show the links that exist between minimax theory in nonparametric statistics and the problem of estimating a mean pattern from a sequence of curves.
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