Minimax properties of Fréchet means of discretely sampled curves
The Annals of Statistics · 2013 · Volume 41 · Issue 2
Abstract
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. We propose an estimator based on the notion of Fréchet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. We derive a minimax rate for this estimation problem, and we show that our estimator achieves this optimal rate under the asymptotics where both the number of curves and the number of sampling points go to infinity.
Keywords
62G08, 62G20, Curve registration, deformable models, Fréchet mean, Functional data analysis, lie group action, minimax rate of convergence, non-Euclidean metric, Sobolev balls
Enable JavaScript for citation tools and additional paper interactions.