Minimax properties of Fr\'echet means of discretely sampled curves

TL;DR

This paper introduces a Fréchet mean-based estimator for averaging discretely sampled curves affected by geometric deformations and noise, achieving minimax optimal rates as data size increases.

Contribution

It proposes a novel estimator using Fréchet means for curve data and establishes its minimax optimality in a high-sample regime.

Findings

Estimator achieves minimax rate under asymptotics

Optimal rate derived for the estimation problem

Method handles geometric deformations and noise effectively

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 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.