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2019 Convergence rates for the generalized Fréchet mean via the quadruple inequality
Christof Schötz
Electron. J. Statist. 13(2): 4280-4345 (2019). DOI: 10.1214/19-EJS1618


For sets $\mathcal{Q}$ and $\mathcal{Y}$, the generalized Fréchet mean $m\in \mathcal{Q}$ of a random variable $Y$, which has values in $\mathcal{Y}$, is any minimizer of $q\mapsto \mathbb{E}[\mathfrak{c}(q,Y)]$, where $\mathfrak{c}\colon \mathcal{Q}\times \mathcal{Y}\to \mathbb{R}$ is a cost function. There are little restrictions to $\mathcal{Q}$ and $\mathcal{Y}$. In particular, $\mathcal{Q}$ can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fréchet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fréchet means, we do not require a finite diameter of the $\mathcal{Q}$ or $\mathcal{Y}$. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.


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Christof Schötz. "Convergence rates for the generalized Fréchet mean via the quadruple inequality." Electron. J. Statist. 13 (2) 4280 - 4345, 2019.


Received: 1 December 2018; Published: 2019
First available in Project Euclid: 28 October 2019

zbMATH: 07136617
MathSciNet: MR4023955
Digital Object Identifier: 10.1214/19-EJS1618

Primary: 62G05
Secondary: 62G20

Keywords: Barycenter , Fréchet mean , Hadamard space , power inequality , quadruple inequality , rate of convergence


Vol.13 • No. 2 • 2019
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