Minimax properties of Fréchet means of discretely sampled curves

Jérémie Bigot (1) , Xavier Gendre (2)

1 DMIA - Département de Mathématiques, Informatique, Automatique
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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 Frechet 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.

Mots clés: Fréchet mean; Non-Euclidean metric; Deformable models; Lie group action; Curve registration; Minimax rate of convergence; Sobolev balls; Functional data analysis

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jérémie Bigot, Xavier Gendre. Minimax properties of Fréchet means of discretely sampled curves. Annals of Statistics, 2013, 41 (2), pp.923-956. ⟨10.1214/13-AOS1104⟩. ⟨hal-00737560v3⟩