# Convergence Rates for the Generalized Fr\'echet Mean via the Quadruple Inequality

## TL;DR

This paper establishes convergence rates for the empirical generalized Fréchet mean in broad metric spaces using a novel quadruple inequality, extending previous results that required finite diameters.

## Contribution

It introduces the quadruple inequality as a key condition, enabling convergence rate analysis without finite diameter restrictions in general metric spaces.

## Key findings

- Provides convergence rates in probability and expectation
- Generalizes Fréchet mean analysis to non-Euclidean spaces
- Applies to Hadamard spaces and certain metric powers

## Abstract

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