Convergence rates for the generalized Fréchet mean via the quadruple inequality

By Christof Schötz. Electronic Journal of Statistics, 2018

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For sets $\mathcal Q$ and $\mathcal Y$, the generalized Frechet 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 Frechet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Frechet 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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b50d636bb89dc295bbe484ce567b7df757ba4f00