Bures-Wasserstein Means of Graphs

TL;DR

This paper introduces a novel Wasserstein-based framework for defining and computing a mean of graphs, enabling structural preservation and improved performance in graph-related machine learning tasks.

Contribution

It proposes a new graph mean concept via Wasserstein embeddings, with proven existence, uniqueness, and an iterative algorithm for practical computation.

Findings

Outperforms baseline methods in graph clustering and classification tasks.

Achieves consistent and improved performance across various graph datasets.

Provides a theoretically sound and computationally feasible approach for graph averaging.

Abstract

Finding the mean of sampled data is a fundamental task in machine learning and statistics. However, in cases where the data samples are graph objects, defining a mean is an inherently difficult task. We propose a novel framework for defining a graph mean via embeddings in the space of smooth graph signal distributions, where graph similarity can be measured using the Wasserstein metric. By finding a mean in this embedding space, we can recover a mean graph that preserves structural information. We establish the existence and uniqueness of the novel graph mean, and provide an iterative algorithm for computing it. To highlight the potential of our framework as a valuable tool for practical applications in machine learning, it is evaluated on various tasks, including k-means clustering of structured aligned graphs, classification of functional brain networks, and semi-supervised node classification in multi-layer graphs. Our experimental results demonstrate that our approach achieves consistent performance, outperforms existing baseline approaches, and improves the performance of state-of-the-art methods.