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

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.

The Mean Partition Theorem in consensus clustering

“…The Mean Partition Theorem is a special case of the same theorem for the mean of a sample of attributed graphs [14]. Any partition can be regarded as an attributed graph without edges.…”