Showing papers on "Spectral graph theory published in 2007"

Spectral Graph Theory and its Applications

Abstract: Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.

Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes

Abstract: This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators (ii) A specific instantiation of this approach where global basis functions called proto-value functions (PVFs) are formed using the eigenvectors of the graph Laplacian on an undirected graph formed from state transitions induced by the MDP (iii) A three-phased procedure called representation policy iteration comprising of a sample collection phase, a representation learning phase that constructs basis functions from samples, and a final parameter estimation phase that determines an (approximately) optimal policy within the (linear) subspace spanned by the (current) basis functions. (iv) A specific instantiation of the RPI framework using least-squares policy iteration (LSPI) as the parameter estimation method (v) Several strategies for scaling the proposed approach to large discrete and continuous state spaces, including the Nystrom extension for out-of-sample interpolation of eigenfunctions, and the use of Kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored MDPs (vi) Finally, a series of illustrative discrete and continuous control tasks, which both illustrate the concepts and provide a benchmark for evaluating the proposed approach. Many challenges remain to be addressed in scaling the proposed framework to large MDPs, and several elaboration of the proposed framework are briefly summarized at the end.

Clustering and Embedding Using Commute Times

Abstract: This paper exploits the properties of the commute time between nodes of a graph for the purposes of clustering and embedding and explores its applications to image segmentation and multibody motion tracking. Our starting point is the lazy random walk on the graph, which is determined by the heat kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterize the random walk using the commute time (that is, the expected time taken for a random walk to travel between two nodes and return) and show how this quantity may be computed from the Laplacian spectrum using the discrete Green's function. Our motivation is that the commute time can be anticipated to be a more robust measure of the proximity of data than the raw proximity matrix. In this paper, we explore two applications of the commute time. The first is to develop a method for image segmentation using the eigenvector corresponding to the smallest eigenvalue of the commute time matrix. We show that our commute time segmentation method has the property of enhancing the intragroup coherence while weakening intergroup coherence and is superior to the normalized cut. The second application is to develop a robust multibody motion tracking method using an embedding based on the commute time. Our embedding procedure preserves commute time and is closely akin to kernel PCA, the Laplacian eigenmap, and the diffusion map. We illustrate the results on both synthetic image sequences and real-world video sequences and compare our results with several alternative methods.

Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems

Abstract: Graph Laplacians.- Eigenfunctions and Nodal Domains.- Nodal Domain Theorems for Special Graph Classes.- Computational Experiments.- Faber-Krahn Type Inequalities.

The main eigenvalues of a graph: A survey

Abstract: We survey results relating main eigenvalues and main angles to the structure of a graph. We provide a number of short proofs, and note the connection with star partitions. We discuss graphs with just two main eigenvalues in the context of measures of irregularity, and in the context of harmonic graphs.

Laplacian eigenvectors and eigenvalues and almost equitable partitions

Abstract: Relations between Laplacian eigenvectors and eigenvalues and the existence of almost equitable partitions (which are generalizations of equitable partitions) are presented. Furthermore, on the basis of some properties of the adjacency eigenvectors of a graph, a necessary and sufficient condition for the graph to be primitive strongly regular is introduced.

On the Effectiveness of Laplacian Normalization for Graph Semi-supervised Learning

Abstract: This paper investigates the effect of Laplacian normalization in graph-based semi-supervised learning. To this end, we consider multi-class transductive learning on graphs with Laplacian regularization. Generalization bounds are derived using geometric properties of the graph. Specifically, by introducing a definition of graph cut from learning theory, we obtain generalization bounds that depend on the Laplacian regularizer. We then use this analysis to better understand the role of graph Laplacian matrix normalization. Under assumptions that the cut is small, we derive near-optimal normalization factors by approximately minimizing the generalization bounds. The analysis reveals the limitations of the standard degree-based normalization method in that the resulting normalization factors can vary significantly within each connected component with the same class label, which may cause inferior generalization performance. Our theory also suggests a remedy that does not suffer from this problem. Experiments confirm the superiority of the normalization scheme motivated by learning theory on artificial and real-world data sets.

Four proofs for the Cheeger inequality and graph partition algorithms

Abstract: We will give four proofs of the Cheeger inequality which relates the eigenvalues of a graph with various isoperimetric variations of the Cheeger constant. The rst is a simplied proof of the classical Cheeger inequality using eigenvectors. The second is based on a rapid mixing result for random walks by Lov asz and Simonovits. The third uses PageRank, a quantitative ranking of the vertices introduced by Brin and Page. The fourth proof is by an improved notion of the heat kernel pagerank. The four proofs lead to further improvements of graph partition algorithms and in particular the local partition algorithms with cost proportional to its output instead of in terms of the total size of the graph.

Interlacing for weighted graphs using the normalized Laplacian

Abstract: The problem of relating the eigenvalues of the normalized Laplacian for a weighted graph G and GH ,f orH a subgraph of G is considered. It is shown that these eigenvalues interlace and that the tightness of the interlacing is dependent on the number of nonisolated vertices of H. Weak coverings of a weighted graph are also defined and interlacing results for the normalized Laplacian for such a covering are given. In addition there is a discussion about interlacing for the Laplacian of directed graphs.

The Laplacian spectral radius of a graph under perturbation

Abstract: In this paper, we investigate how the Laplacian spectral radius changes when one graph is transferred to another graph obtained from the original graph by adding some edges, or subdivision, or removing some edges from one vertex to another.

On the third largest Laplacian eigenvalue of a graph

Abstract: In this article, a sharp lower bound for the third largest Laplacian eigenvalue of a graph is given in terms of the third largest degree of the graph.

Dynamic Coverage Verification in Mobile Sensor Networks Via Switched Higher Order Laplacians

Abstract: In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on time-evolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about the coverage properties of the network. Our main result states that the asymptotic stability the switched linear system implies that every point of the domain covered by the mobile sensor nodes is visited infinitely often, hence verifying dynamic coverage. The enabling mathematical technique for this result is the theory of higher order Laplacian operators, which is a generalization of the graph Laplacian used in spectral graph theory and continuous-time consensus problems.

On eigenvectors of mixed graphs with exactly one nonsingular cycle

Abstract: Let G be a mixed graph. The eigenvalues and eigenvectors of G are respectively defined to be those of its Laplacian matrix. If G is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of G corresponding to its second smallest eigenvalue (also called the algebraic connectivity of G). For G being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of G corresponding to its smallest eigenvalue.

Asymptotic spectral analysis of growing regular graphs

Abstract: We propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph. We prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states. The condition for the growth is described in terms of simple statistics arising from the stratification of the graph. The asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction.

Multiple Testing Correction in Medical Image Analysis

Abstract: We present a novel method for correcting the significance level of hypothesis testing that requires multiple comparisons. It is based on the spectral graph theory, in which the variables are seen as the vertices of a complete undirected graph and the correlation matrix as the adjacency matrix that weights its edges. The method increases the statistical power of the analysis by refuting the assumption of independence among variables, while keeping the probability of false positives low. By computing the eigenvalues of the correlation matrix, it is possible to obtain valuable information about the dependence levels among the variables of the problem, so that the effective number of independent variables can be estimated. The method is compared to other available models and its effectiveness illustrated in case studies involving high-dimensional sets of variables.

Sharp edge, vertex, and mixed Cheeger type inequalities for finite Markov kernels

Abstract: We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.

Learning on Graph with Laplacian Regularization

Abstract: We consider a general form of transductive learning on graphs with Laplacian regularization, and derive margin-based generalization bounds using appropriate geometric properties of the graph. We use this analysis to obtain a better understanding of the role of normalization of the graph Laplacian matrix as well as the effect of dimension reduction. The results suggest a limitation of the standard degree-based normalization. We propose a remedy from our analysis and demonstrate empirically that the remedy leads to improved classification performance.

Shape Indexing through Laplacian Spectra

Abstract: With ever growing databases containing multimedia data, indexing has become a necessity to avoid a linear search. We propose a novel technique for indexing multimedia databases, whose entries can be represented as graph structures. In our method, the topological structure of a graph as well as that of its subgraphs are represented as vectors in which the components correspond to the sorted Laplacian eigenvalues of the graph or subgraphs. We draw from recently-developed techniques in the field of spectral integral variation to overcome the problem of computing the Laplacian spectrum for every subgraph individually. By doing a nearest neighbor search around the query spectra, similar but not necessarily isomorphic graphs are retrieved. The novelties of the proposed method come from the powerful representation of the graph topology and successfully adopting the concept of spectral integral variation in an indexing algorithm. Our experiments, consisting of recognition trials in the domain of 2D and 3D object recognition, including a comparison with a competing indexing method, demonstrate both the robustness and efficacy of the approach.

Bounds on the Expansion Properties of Tanner Graphs

Abstract: This work focuses on the expansion properties of a Tanner Graph because they are known to be related to the performance of associated iterative message-passing algorithms over various channels. By analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a Tanner Graph, lower bounds on these expansion properties are derived. Specifically, for the binary erasure channel, these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference-set codes (type-I 2-D projective geometry low-density parity-check (LDPC) codes).

Classification by Cheeger Constant Regularization

Abstract: This paper develops a classification algorithm in the framework of spectral graph theory where the underlying manifold of a high dimensional data set is described by a graph. The classification on the data is performed on the graph. The classifier optimizes an objective functional that combines prior information with the Cheeger constant. We interpret this approach as a regularized version of the Cheeger constant based classifier that we introduced recently. Our derivation shows that Cheeger regularization removes noise like a Laplacian based classifier but preserves better sharp boundaries needed for class separation. Experimental results show good performance of our proposed approach for classification applications.

Graph spectral image smoothing

Abstract: A new method for smoothing both gray-scale and color images is presented that relies on the heat diffusion equation on a graph. We represent the image pixel lattice using a weighted undirected graph. The edge weights of the graph are determined by the Gaussian weighted distances between local neighbouring windows. We then compute the associated Laplacian matrix (the degree matrix minus the adjacency matrix). Anisotropic diffusion across this weighted graph-structure with time is captured by the heat equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigen-system with time. Image smoothing is accomplished by convolving the heat kernel with the image, and its numerical implementation is realized by using the Krylov subspace technique. The method has the effect of smoothing within regions, but does not blur region boundaries. We also demonstrate the relationship between our method, standard diffusion-based PDEs, Fourier domain signal processing and spectral clustering. Experiments and comparisons on standard images illustrate the effectiveness of the method.

How sensor graph topology affects localization accuracy

Abstract: We characterize the accuracy of a cooperative localization algorithm based on Kalman Filtering, as expressed by the trace of the covariance matrix, in terms of the algebraic graph theoretic properties of the sensing graph. In particular, we discover a weighted Laplacian in the expression that yields the constant, steady state value of the covariance matrix. We show how one can reduce the localization uncertainty by manipulating the eigenvalues of the weighted Laplacian. We thus provide insight to recent optimization results which indicate that increased connectivity implies higher accuracy. We offer an analysis method that could lead to more efficient ways of achieving the desired accuracy by controlling the sensing network.