A Spectral Approach to Learning Structural Variations in Graphs

Abstract

This paper shows how to construct a linear deformable model for graph structure by performing principal components analysis (PCA) on the vectorised adjacency matrix. We commence by using correspondence information to place the nodes of each of a set of graphs in a standard reference order. Using the correspondences order, we convert the adjacency matrices to long-vectors and compute the long-vector covariance matrix. By projecting the vectorised adjacency matrices onto the leading eigenvectors of the covariance matrix, we embed the graphs in a pattern-space. We illustrate the utility of the resulting method for shape-analysis.

Two new graphs kernels in chemoinformatics

“…A vectorial embedding of a graph may, for example, be defined by associating to each graph its distance to a set of graph prototypes (Emms et al, 2007). Another graph embedding approach is based on the spectral analysis of graphs (Caelli and Kosinov, 2004;Luo et al, 2006). In this last case, the embedding is deduced from the analysis of the eigenvectors and eigenvalues of the adjacency matrix.…”

Network Sampling Algorithms and Applications

“…Spectral embedding methods [28,6,27] base the vectorial representation of a graph on different characteristics of the spectrum of the graph laplacian matrix. The basic idea of this family of methods is that the spectrum of the graph laplacian matrix is insensitive to any rotation of the adjacency matrix and may thus be considered as a characteristic of the graph rather than a characteristic of its adjacency matrix.…”

Automatic segmentation of colon glands using object-graphs

“…The graph constructed in this step is an example of an attributed graph, in which the nodes represent image primitives and the edges represent the relations between these primitives. Attributed graphs have been shown to be effective in representing structural knowledge and have been used in many computer vision applications including structural matching (Christmas et al, 1995), similarity searching (Petrakis and Faloutsos, 1997), object recognition (Sanfeliu et al, 2002;Ahmadyfard and Kittler, 2003), object tracking (Tang and Tao, 2008), and face recognition (Luo et al, 2006). These applications mainly rely on comparing the attributed graphs of different structures with a graph matching algorithm, which has high computational complexity (Jain and Wysotzki, 2004).…”