Showing papers on "Spectral graph theory published in 2004"
TL;DR: The contribution of this paper is a method that substantially reduces the computational requirements of grouping algorithms based on spectral partitioning making it feasible to apply them to very large grouping problems.
Abstract: Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation. However, due to the computational demands of these approaches, applications to large problems such as spatiotemporal data and high resolution imagery have been slow to appear. The contribution of this paper is a method that substantially reduces the computational requirements of grouping algorithms based on spectral partitioning making it feasible to apply them to very large grouping problems. Our approach is based on a technique for the numerical solution of eigenfunction problems known as the Nystrom method. This method allows one to extrapolate the complete grouping solution using only a small number of samples. In doing so, we leverage the fact that there are far fewer coherent groups in a scene than pixels.
1,420 citations
Proceedings Article•
01 Jan 2004
382 citations
06 Oct 2004
TL;DR: An algorithm is developed that favors segmentation along concave regions, which is inspired by human perception and theoretically sound, efficient, simple to implement, and able to achieve high-quality segmentation results on 3D meshes.
Abstract: We formulate and apply spectral clustering to 3D mesh segmentation for the first time and report our preliminary findings Given a set of mesh faces, an affinity matrix which encodes the likelihood of each pair of faces belonging to the same group is first constructed Spectral methods then use selected eigenvectors of the affinity matrix or its closely related graph Laplacian to obtain data representations that can be more easily clustered We develop an algorithm that favors segmentation along concave regions, which is inspired by human perception Our algorithm is theoretically sound, efficient, simple to implement, andean achieve high-quality segmentation results on 3D meshes
268 citations
TL;DR: In this paper, the Laplacian matrix of a simple graph G = (V, E) is defined as L(G) = D (G) - A(G).
Abstract: Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertexdegrees and by A(G) its adjacency matrix. Then, the Laplacian matrix of G is L(G) = D(G) - A(G). The first and second section of this paper contains introduction and some known results, respectively. The third section is devoted to properties of Laplacian spectrum. The fourth section contains characterization of graphs. The fifth section relates the Laplacian eigenvalues with the graph structure.
126 citations
01 Jul 2004
TL;DR: It is shown that spectral clustering usually converges to an intuitively appealing limit partition of the data space and argues that in case of the unnormalized graph Laplacian, equally strong convergence results are difficult to obtain.
Abstract: Given a set of n randomly drawn sample points, spectral clustering in its simplest form uses the second eigenvector of the graph Laplacian matrix, constructed on the similarity graph between the sample points, to obtain a partition of the sample. We are interested in the question how spectral clustering behaves for growing sample size n. In case one uses the normalized graph Laplacian, we show that spectral clustering usually converges to an intuitively appealing limit partition of the data space. We argue that in case of the unnormalized graph Laplacian, equally strong convergence results are difficult to obtain.
62 citations
01 Apr 2004
TL;DR: iFind is described, a system for clustering and searching WWW images, which is less sensitive to noisy links than previous methods like PageRank, HITS, and PicASHOW, and hence the image graph can better reflect the semantic relationship between images.
Abstract: Due to the rapid growth of the number of digital images on the Web, there is an increasing demand for effective and efficient method for organizing and retrieving the images available. This paper describes ImageSeer, a system for clustering and searching WWW images. By using a vision-based page segmentation algorithm, a web page is partitioned into blocks, and the textual and link information of an image can be accurately extracted within the block containing that image. The textual information is used for image representation. By extracting the page-to-block, blockto-image, block-to-page relationships through link structure and page layout analysis, we construct an image graph. Our method is less sensitive to noisy links than previous methods like PicASHOW, and hence the image graph can better reflect the semantic relationship between images. With the graph models, we use techniques from spectral graph theory and Markov Chain theory for image ranking, clustering and embedding. Some experimental results are given in the paper.
59 citations
TL;DR: In this paper, the eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigen values of its generators.
Abstract: In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd.
57 citations
27 Jun 2004
TL;DR: A method for clustering and embedding WWW images by using a vision-based page segmentation algorithm, and the textual and link information of an image can be accurately extracted from the block containing that image.
Abstract: Due to the rapid growth of the number of digital images on the Web, there is an increasing demand for an effective and efficient method of organizing and retrieving the images available. This paper describes a method for clustering and embedding WWW images. By using a vision-based page segmentation algorithm, a Web page is partitioned into blocks, and the textual and link information of an image can be accurately extracted from the block containing that image. By extracting the page-to-block, block-to-image, block-to-page relationships through a link structure and page layout analysis, we construct an image graph. With the image graph model, we use techniques from spectral graph theory for image clustering and embedding. Some experimental results are given in the paper.
51 citations
TL;DR: In this article, a general method for obtaining a vacuum spectral distribution of the adjacency matrix of a star graph is established within the framework of quantum probability theory, and the spectral distribution tends asymptotically to the Bernoulli distribution.
Abstract: A general method for obtaining a vacuum spectral distribution of the adjacency matrix of a star graph is established within the framework of quantum probability theory. The spectral distribution tends asymptotically to the Bernoulli distribution as the number of leaves of a star graph tends to the infinity.
49 citations
TL;DR: In this article, the connection between the standard inverse eigenvalue problem and describing all the possible associated ordered multiplicity lists, along with determining the minimum number of distinct eigenvalues for a symmetric matrix with graph G, is made.
Abstract: For a given acyclic graph G, an important problem is to characterize all of the eigenvalues over all symmetric matrices with graph G. Of particular interest is the connection between this standard inverse eigenvalue problem and describing all the possible associated ordered multiplicity lists, along with determining the minimum number of distinct eigenvalues for a symmetric matrix with graph G. In this note two important open questions along these lines are resolved, both in the negative.
TL;DR: Efficient methods are developed for decomposing the graphs into subgraphs and healing the sub graphs to maintain the information corresponding to the eigenvalues and eigenvectors of the original graph for calculating the natural frequency of symmetric structures.
23 Aug 2004
TL;DR: This work uses the Levenshtein distance to compare spectral representations under graph edit operations which add or delete vertices and uses the concept of the string-edit distance to allow for the missing eigenmodes and compare the correct modes to each other.
Abstract: Graph structures play a critical role in computer vision, but they are inconvenient to use in pattern recognition tasks because of their combinatorial nature and the consequent difficulty in constructing feature vectors. Spectral representations have been used for this task which are based on the eigensystem of the graph Laplacian matrix. However, graphs of different sizes produce eigensystems of different sizes where not all eigenmodes are present in both graphs. We use the Levenshtein distance to compare spectral representations under graph edit operations which add or delete vertices. The spectral representations are therefore of different sizes. We use the concept of the string-edit distance to allow for the missing eigenmodes and compare the correct modes to each other. We evaluate the method by first using generated graphs to compare the effect of vertex deletion operations. We then examine the performance of the method on graphs from a shape database.
29 Sep 2004
TL;DR: The min Chi indicator is introduced which helps in selecting the number of clusters and confirming the existence of a partition of the data and gives a non-probabilistic alternative to statistical mixture-models.
Abstract: The problem of clustering data can be formulated as a graph partitioning problem. Spectral methods for obtaining optimal solutions have reveceived a lot of attention recently. We describe Perron Cluster Cluster Analysis (PCCA) and, for the first time, establish a connection to spectral graph partitioning. We show that in our approach a clustering can be efficiently computed using a simple linear map of the eigenvector data. To deal with the prevalent problem of noisy and possibly overlapping data we introduce the min Chi indicator which helps in selecting the number of clusters and confirming the existence of a partition of the data. This gives a non-probabilistic alternative to statistical mixture-models. We close with showing favorable results on the analysis of gene expressi on data for two different cancer types.
TL;DR: In this article, the authors apply non-negative matrix theory to the matrix K = D+A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ 1(G) of G and the spectral radius ρ(K) of K. And then by using this relation they present two upper bounds for λ1(G), and determine the extrernal graphs which achieve the upper bounds.
Abstract: We first apply non-negative matrix theory to the matrix K=D+A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1(G) of G and the spectral radius ρ(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extrernal graphs which achieve the upper bounds.
TL;DR: In this article, a formula for the characteristic polynomial of any graph covering is described, and a generalization of this formula to regular or irregular graph covers is given for graphs.
Abstract: In this note, a formula for the characteristic polynomial of any (regular or irregular) graph covering is described.
TL;DR: In this article, lower bounds for the largest eigenvalue, the second largest value, and the sum of the two largest values of the Laplacian matrix of a graph are presented.
Abstract: In this article, we present lower bounds for the largest eigenvalue, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph.
TL;DR: In this article, an efficient method for calculating the eigenvalues of space structures with regular topologies is presented, where the topology of a structure is formed as the Cartesian product of its generators.
Journal Article•
TL;DR: In this article, the asymptotic behaviour of the neighborhood of the graph of a type of rapidly oscillating continuous functions is studied and necessary and sucient conditions for rapid oscillations of solutions of the main equation are given.
Abstract: We study the asymptotic behaviour of "-neighbourhood of the graph of a type of rapidly oscillating continuous functions. Next, we estate necessary and sucient conditions for rapid oscillations of solutions of the main equation. This enables us to verify some new singular properties of bounded continuous solutions of a class of nonlinear p-Laplacian by calculating lower and upper bounds for the Minkowski content and the s-dimensional density of the graph of each solution and its derivative.
TL;DR: It is reported that weighting the original quadratic cost function results in a notable improvement of the matching performance, even in medium and high noise conditions.
Abstract: In this paper we propose a simple way of significantly improving the performance of the Softassign graph-matching algorithm of Gold and Rangarajan. Exploiting recent theoretical results in spectral graph theory we use diffusion kernels to transform a matching problem between unweighted graphs into a matching between weighted ones in which the weights rely on the entropies of the probability distributions associated to the vertices after kernel computation. In our experiments, we report that weighting the original quadratic cost function results in a notable improvement of the matching performance, even in medium and high noise conditions.
06 Jun 2004
TL;DR: A k-way graph partitioning algorithm based on clustering using recursive spectral bisection that generates partitions with 83.8~ 108.4% cutsets compared to the strict spectral bisections or multi-level partitions.
Abstract: The recursive spectral bisection for the k-way graph partition has been underestimated because it tries to balance the bipartition strictly. However, by loosening the balancing constraint, the spectral bisection can identify clusters efficiently. We propose a k-way graph partitioning algorithm based on clustering using recursive spectral bisection. After a graph is divided into a partition, the partition is adjusted in order to meet the balancing constraint. Experimental results show that the clustering based k-way partitioning generates partitions with 83.8~ 108.4% cutsets compared to the strict recursive spectral bisections or multi-level partitions.
01 Jan 2004
TL;DR: In this paper, the authors apply non-negative matrix theory to the matrix K = D + A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ 1 (G) of G and the spectral radius p(K) of K. And then by using this relation they present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.
Abstract: We first apply non-negative matrix theory to the matrix K = D + A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.
01 Jun 2004
TL;DR: In this paper, a web page is partitioned into blocks, and the textual and link information of an image can be accurately extracted from the block containing that image by extracting the page-to-block, block-toimage, blockto-page relationships through link structure and page layout analysis, and using techniques from spectral graph theory for image clustering and embedding.
Abstract: Due to the rapid growth of the number of digital images on the Web, there is an increasing demand for effective and efficient method for organizing and retrieving the images available This paper describes a method for clustering and embedding WWW images By using a vision-based page segmentation algorithm, a web page is partitioned into blocks, and the textual and link information of an image can be accurately extracted from the block containing that image By extracting the page-to-block, block-to-image, block-to-page relationships through link structure and page layout analysis, we construct an image graph With the image graph model, we use techniques from spectral graph theory for image clustering and embedding Some experimental results are given in the paper
TL;DR: In this paper, the mass spectrum of a model constructed in a theory space is expressed in terms of eigenvalues of the Laplacian on the graph structure of the theory space.
Abstract: The mass spectrum of a model constructed in a theory space is expressed in terms of eigenvalues of the Laplacian on the graph structure of the theory space. The nature of the one-loop UV divergence in the vacuum energy is then determined by only the degree matrix of the graph. Using these facts, we construct models of induced gravity that do not exhibit quadratic divergences at the one-loop level.
Journal Article•
TL;DR: In this article, the maximum spectral radius for the Laplacian matrix of a graph with e edges and n vertices was determined, where e is the number of vertices in the graph.
Abstract: This note determines the maximum spectral radius for the Laplacian matrix of a graph with e edges and n vertices.
22 Nov 2004
TL;DR: In this paper, the spectral curve of the Lax representation becomes the graph of integrable systems, such as the open Toda molecule, and generalizations for which a function lives on a cylinder, torus or a Riemann surface of higher genus.
Abstract: For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a function $A(\lambda)$. Those integrable systems provide an interesting ``toy model'' of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which $A(\lambda)$ lives on a cylinder, a torus or a Riemann surface of higher genus.
14 May 2004
TL;DR: This paper describes a technique to compare two data partitions of different data sets by means of matrices called Graph Adjacency Matrices which represent the data sets and returns an estimation of the level of similarity between the data set.
Abstract: A frequently recurring problem in several applications is to compare two or more data sets and evaluate the level of similarity In this paper we describe a technique to compare two data partitions of different data sets The comparison is obtained by means of matrices called Graph Adjacency Matrices which represent the data sets Then, a match coefficient returns an estimation of the level of similarity between the data sets