Bai Xiao

Graph characteristics from the heat kernel trace

Graph structures have been proved important in high level-vision since they can be used to represent structural and relational arrangements of objects in a scene. One of the problems that arises in the analysis of structural abstractions of objects is graph clustering. In this paper, we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering. The heat kernel is the solution of the heat equation and is a compact representation of the path-length distribution on a graph. The trace of the heat kernel is given by the sum of the Laplacian eigenvalues exponentiated with time. We explore three different approaches to characterizing the heat kernel trace as a function of time. Our first characterization is based on the zeta function, which from the Mellin transform is the moment generating function of the heat kernel trace. Our second characterization is unary and is found by computing the derivative of the zeta function at the origin. The third characterization is derived from the heat content, i.e. the sum of the elements of the heat kernel. We show how the heat content can be expanded as a power series in time, and the coefficients of the series can be computed using the Laplacian spectrum. We explore the use of these characterizations as a means of representing graph structure for the purposes of clustering, and compare them with the use of the Laplacian spectrum. Experiments with the synthetic and real-world databases reveal that each of the three proposed invariants is effective and outperforms the traditional Laplacian spectrum. Moreover, the heat-content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real-world object recognition problems.

Geometric characterization and clustering of graphs using heat kernel embeddings

In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph in a vector space. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat kernel of the graph is found by exponentiating the Laplacian eigensystem over time. In this paper, we explore how graphs can be characterized in a geometric manner using embeddings into a vector space obtained from the heat kernel. We explore two different embedding strategies. The first of these is a direct method in which the matrix of embedding co-ordinates is obtained by performing a Young-Householder decomposition on the heat kernel. The second method is indirect and involves performing a low-distortion embedding by applying multidimensional scaling to the geodesic distances between nodes. We show how the required geodesic distances can be computed using parametrix expansion of the heat kernel. Once the nodes of the graph are embedded using one of the two alternative methods, we can characterize them in a geometric manner using the distribution of the node co-ordinates. We investigate several alternative methods of characterization, including spatial moments for the embedded points, the Laplacian spectrum for the Euclidean distance matrix and scalar curvatures computed from the difference in geodesic and Euclidean distances. We experiment with the resulting algorithms on the COIL database.

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.

Geometric characterisation of graphs

In this paper, we explore whether the geometric properties of the point distribution obtained by embedding the nodes of a graph on a manifold can be used for the purposes of graph clustering. The embedding is performed using the heat-kernel of the graph, computed by exponentiating the Laplacian eigen-system. By equating the spectral heat kernel and its Gaussian form we are able to approximate the Euclidean distance between nodes on the manifold. The difference between the geodesic and Euclidean distances can be used to compute the sectional curvatures associated with the edges of the graph. To characterise the manifold on which the graph resides, we use the normalised histogram of sectional curvatures. By performing PCA on long-vectors representing the histogram bin-contents, we construct a pattern space for sets of graphs. We apply the technique to images from the COIL database, and demonstrate that it leads to well defined graph clusters.

Isotree: Tree clustering via metric embedding

One of the problems that hinders the spectral analysis of trees is that they have a strong tendency to be co-spectral. As a result, structurally distinct trees possess degenerate graph-spectra, and spectral methods can be reliably used to neither compute distances between trees nor to cluster trees. The aim of this paper is to describe a method that can be used to alleviate this problem. We use the ISOMAP algorithm to embed the trees in a Euclidean space using the pattern of shortest distances between nodes. From the arrangement of nodes in this space, we compute a weighted proximity matrix, and from the proximity matrix a Laplacian matrix is computed. By transforming the graphs in this way we lift the co-spectrality of the trees. The spectrum of the Laplacian matrix for the embedded graphs may be used for purposes of comparing trees and for clustering them. Experiments on sets of shock graphs reveal the utility of the method on real-world data.

Manifold embedding for shape analysis

Shape analysis played important role in computer vision based tasks. The importance of shape information relies that it usually contains perceptual information, and thus can be used for high level visual information analysis. Currently, there are many ways that shapes can be represented as a structural manner using graphs. Hence shapes can be analyzed by using graph methods. This paper describes how graph-spectral methods can be used to transform the node correspondence problem into one of point-sets alignment. We commence by using the ISOMAP algorithm to embed the nodes of a graph in a low-dimensional Euclidean space. With the nodes in the graph transformed to points in a metric space, we can recast the problem of graph-matching into that of aligning the point-sets. Here we use semidefinite programming to develop a robust point-sets correspondences algorithm. Variations in graph structure using the covariance matrix for corresponding embedded point-positions is captured. We construct a statistical 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 project individual graph, i.e. shape into the eigenspace of the point position covariance matrix. We illustrate the utility of the resulting method for shape analysis and recognition on COIL and MPEG-7 databases.

Learning invariant structure for object identification by using graph methods

The problem of learning the class identity of visual objects has received considerable attention recently. With rare exception, all of the work to date assumes low variation in appearance, which limits them to a single depictive style usually photographic. The same object depicted in other styles - as a drawing, perhaps - cannot be identified reliably. Yet humans are able to name the object no matter how it is depicted, and even recognize a real object having previously seen only a drawing. This paper describes a classifier which is unique in being able to learn class identity no matter how the class instances are depicted. The key to this is our proposition that topological structure is a class invariant. Practically, we depend on spectral graph analysis of a hierarchical description of an image to construct a feature vector of fixed dimension. Hence structure is transformed to a feature vector, which can be classified using standard methods. We demonstrate the classifier on several diverse classes.

A spectral generative model for graph structure

This paper shows how to construct a generative model for graph structure. 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 the edge-structure. The idea underpinning the method is to embed the nodes of the graphs into a vector space by performing kernel PCA on the heat kernel. 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 COIL database.

Trace formula analysis of graphs

In this paper, we explore how the trace of the heat kernel can be used to characterise graphs for the purposes of measuring similarity and clustering. The heat-kernel is the solution of the heat-equation and may be computed by exponentiating the Laplacian eigensystem with time. To characterise the shape of the heat-kernel trace we use the zeta-function, which is found by exponentiating and summing the reciprocals of the Laplacian eigenvalues. From the Mellin transform, it follows that the zeta-function is the moment generating function of the heat-kernel trace. We explore the use of the heat-kernel moments as a means of characterising graph structure for the purposes of clustering. Experiments with the COIL and Oxford-Caltech databases reveal the effectiveness of the representation.

Graph Matching Using Manifold Embedding

This paper describes how graph-spectral methods can be used to transform the node correspondence problem into one of point-set alignment. We commence by using a heat kernel analysis to compute geodesic distances between nodes in the graphs. With geodesic distances to hand, we use the ISOMAP algorithm to embed the nodes of a graph in a low-dimensional Euclidean space. With the nodes in the graph transformed to points in a metric space, we can recast the problem of graph-matching into that of aligning the points. Here we use a variant of the Scott and Longuet-Higgins algorithm to find point correspondences. We experiment with the resulting algorithm on a number of real-world problems.