Spectral embedding of graphs

In this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern-space. We adopt a graph-spectral approach. We use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix. For each eigenmode, we compute vectors of spectral properties. These include the eigenmode perimeter, eigenmode volume, Cheeger number, inter-mode adjacency matrices and intermode edge-distance. We embed these vectors in a pattern-space using two contrasting approaches. The first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors. The second approach involves performing multidimensional scaling on the L2 norm for pairs of pattern vectors. We illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2D images of 3D polyhedral objects. Two problems are investigated. The first of these is the clustering of graphs representing distinct objects viewed from different directions. The second is the identification of characteristic views of single objects. These two studies reveal that both embedding methods result in well-structured view spaces for graph-data extracted from 2D views of 3D objects.