A generative model for graph matching and embedding

This paper shows how to construct a generative model for graph structure through the embedding of the nodes of the graph in a vector space. We commence from a sample of graphs where the correspondences between nodes are unknown ab initio. We also work with graphs where there may be structural differences present, i.e. variations in the number of nodes in each graph and their edge structure. We characterise the graphs using the heat-kernel, and this is obtained by exponentiating the Laplacian eigensystem with time. The idea underpinning the method is to embed the nodes of the graphs into a vector space by performing a Young-Householder decomposition of the heat-kernel into an inner product of node co-ordinate matrices. The co-ordinates of the nodes are determined by the eigenvalues and eigenvectors of the Laplacian matrix, together with a time-parameter which can be used to scale the embedding. Node correspondences are located by applying Scott and Longuet-Higgins algorithm to the embedded nodes. We capture variations in graph structure using the covariance matrix for corresponding embedded point positions. We construct a point-distribution model for the embedded node positions using the eigenvalues and eigenvectors of the covariance matrix. We show how to use this model to both project individual graphs into the eigenspace of the point position covariance matrix and how to fit the model to potentially noisy graphs to reconstruct the Laplacian matrix. We illustrate the utility of the resulting method for shape analysis using data from the Caltech–Oxford and COIL databases.