Bures-Wasserstein Means of Graphs

TL;DR

This work introduces the Bures-Wasserstein mean of graphs by embedding each graph as a zero-mean Gaussian with covariance $L^†$ (where $L$ is the graph Laplacian) and formulating the graph mean as a BW barycenter in this embedding. The core result is that the BW mean corresponds to a Laplacian $L = S^†$ with $S$ solving $S = \sum_j \lambda_j (S^{1/2} L_j^† S^{1/2})^{1/2}$ subject to range$(S)= (\mathrm{span}\{\mathbf{1}_N\})^{\perp}$, and it can be computed via a convergent fixed-point iteration using a PD transform. The theory extends to graph filters and geodesic interpolation for two graphs, and empirical evaluations on graph fusion, k-means clustering, brain-network classification, and multi-layer learning show robust, competitive improvements over baselines. These results demonstrate the practical value of a principled OT-based graph mean that preserves both local and global structural information through smooth graph signal embeddings, and point to future work on unaligned graphs and generative modeling.

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.

Figures (5)

• Figure 1: Illustration of different means for two sets of graphs. The edge widths are proportional to the absolute value of edge weights, and red edges correspond to negative edge weights. Our method is compared with the arithmetic mean, harmonic mean, and the geometric (Karcher) mean of the given graph Laplacians.
• Figure 2: Normalized mutual information of graph classification with k-means clustering for different values of number of classes (clusters), and inter-community edge probability (connectivity).
• Figure 3: Test error of node classification for different percentage of observed node labels using several types of graph means and distances. Here, BW low pass and BW high pass correspond to the filter distance with $g(L)= L^\dag$ and $g(L)= L^{1/2}$, respectively.
• Figure 4: Some data samples of the k-means clustering example in Section \ref{['subsec:kmeans']} with $k=N_C=5$ clusters and inter-community edge probability $p=0.2$.
• Figure 5: Evolution of the centroids for one trial of the k-means clustering example with data as illustrated in Figure \ref{['fig:kmeans_samples']}.

Theorems & Definitions (17)

• Definition 3.2
• Theorem 3.3
• proof : Proof sketch:
• Proposition 3.4
• proof
• Theorem 3.5
• proof
• Definition 7.1
• Corollary 7.2
• proof