Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm

TL;DR

This paper investigates the properties of the sample mean in graph spaces, addresses fundamental open problems, and introduces the Majorize-Minimize-Mean (MMM) Algorithm, which outperforms existing methods in approximating graph means.

Contribution

The paper provides conditions to resolve key issues with graph sample means and proposes the MMM-Algorithm, a novel method with improved approximation capabilities.

Findings

MMM-Algorithm outperforms six other mean algorithms in experiments

Conditions established to ensure existence and uniqueness of graph means

Experimental results on image and molecule datasets demonstrate effectiveness

Abstract

One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show that the MMM-Algorithm best approximates a sample mean of graphs compared to six other mean algorithms.