Weighted Frechet Means as Convex Combinations in Metric Spaces: Properties and Generalized Median Inequalities

TL;DR

This paper explores the properties of weighted Frechet means in metric spaces, establishing their algebraic characteristics and deriving median inequalities applicable to various spaces, highlighting limitations for certain weights.

Contribution

It introduces a comprehensive analysis of weighted Frechet means, including their properties and median inequalities, extending understanding in abstract metric spaces.

Findings

Weighted Frechet mean is commutative, non-associative, idempotent.

Median inequalities are established for negative and positive Alexandrov spaces.

Inequalities for weights less than one cannot be generally derived.

Abstract

In this short note, we study the properties of the weighted Frechet mean as a convex combination operator on an arbitrary metric space, (Y,d). We show that this binary operator is commutative, non-associative, idempotent, invariant to multiplication by a constant weight and possesses an identity element. We also treat the properties of the weighted cumulative Frechet mean. These tools allow us to derive several types of median inequalities for abstract metric spaces that hold for both negative and positive Alexandrov spaces. In particular, we show through an example that these bounds cannot be improved upon in general metric spaces. For weighted Frechet means, however, such inequalities can solely be derived for weights equal or greater than one. This latter limitation highlights the inherent difficulties associated with working with abstract-valued random variables.