Showing papers on "Spectral graph theory published in 2003"

Semi-supervised learning using Gaussian fields and harmonic functions

Abstract: An approach to semi-supervised learning is proposed that is based on a Gaussian random field model. Labeled and unlabeled data are represented as vertices in a weighted graph, with edge weights encoding the similarity between instances. The learning problem is then formulated in terms of a Gaussian random field on this graph, where the mean of the field is characterized in terms of harmonic functions, and is efficiently obtained using matrix methods or belief propagation. The resulting learning algorithms have intimate connections with random walks, electric networks, and spectral graph theory. We discuss methods to incorporate class priors and the predictions of classifiers obtained by supervised learning. We also propose a method of parameter learning by entropy minimization, and show the algorithm's ability to perform feature selection. Promising experimental results are presented for synthetic data, digit classification, and text classification tasks.

Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

Abstract: First we recall a Faber-Krahn type inequality and an estimate forp() in terms of the so-called Cheeger constant. Then we prove that the eigenvaluep() converges to the Cheeger constant h() as p → 1. The associated eigenfunction up converges to the characteristic function of the Cheeger set, i.e. a subset of which minimizes the ratio |@D|/|D| among all simply connected D ⊂⊂ . As a byproduct we prove that for convex the Cheeger set ! is also convex.

Binary partitioning, perceptual grouping, and restoration with semidefinite programming

Abstract: We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints. The approach is (tuning) parameter-free and computes high-quality combinatorial solutions using interior-point methods (convex programming) and a randomized hyperplane technique. Apart from a symmetry condition, no assumptions (such as metric pairwise interactions) are made with respect to the objective criterion. As a consequence, the approach can be applied to a wide range of problems. Applications to unsupervised partitioning, figure-ground discrimination, and binary restoration are presented along with extensive ground-truth experiments. From the viewpoint of relaxation of the underlying combinatorial problem, we show the superiority of our approach to relaxations based on spectral graph theory and prove performance bounds.

On the Multiplicities of Graph Eigenvalues

Abstract: Star complements and associated quadratic functions are used to obtain a sharp upper bound for the order of a graph with an eigenspace of prescribed codimension. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph G if and only if G is an extremal strongly regular graph. 2000 Mathematics Subject Classification 05C50.

On graph partitioning, spectral analysis, and digital mesh processing

Abstract: Partitioning is a fundamental operation on graphs. In this paper we briefly review the basic concepts of graph partitioning and its relationship to digital mesh processing. We also elaborate on the connection between graph partitioning and spectral graph theory. Applications in computer graphics are described.

Generalized Graph Matrix, Graph Geometry, Quantum Chemistry, and Optimal Description of Physicochemical Properties

Abstract: The generalized graph matrix Γ(x,v) (Estrada, E. Chem. Phys. Lett. 2001, 336, 247) is shown to encompass several of the applications of graph theory in physical chemistry in a more compact and effective way. It defines several n-Euclidean graph metrics, which simulate a graph defolding by changing the exponent v from 0 to 0.5 in a continuous way. This matrix is included in the formalism of the Huckel molecular orbital approach by considering that the resonance integrals between nonneighbor atoms are a function of the topological distance in terms of β. In doing so, the isospectrality between graphs disappears by changing the x parameter in this matrix as a consequence of considering the interactions between nonneighbor atoms. The Γ(x,v) matrix permits several of the “classical” topological indices to be (re)defined using only one graph invariant. These indices include the connectivity index, Balaban J index, Zagreb indices, Wiener index, and Harary indices, which are represented in an 8-dimensional space ...

Solving Symmetric Diagonally-Dominant Systems by Preconditioning

Abstract: In this paper we design support-tree preconditioners for n × n matrices with m nonzeros that are symmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix). This reduces to designing such a preconditioner for a Laplacian matrix, A, which can be interpreted as an undirected nonnegatively-weighted graph, G with n vertices and m edges. Preconditioners accelerate the convergence of iterative methods for solving linear systems, and our preconditioner allows us to analyze the convergence of a particular algorithm, due to Gremban and Miller, called support-tree conjugate gradient (STCG). An advantage of support-tree preconditioners is that STCG parallelizes well. We show that STCG equipped with our preconditioner requires O(m log n · √ dilexp(G)) work and O(m) space to solve the system Ax = b, where dilexp(G) is an edge-expansion-based upper bound on the diameter of G. Existing bounds depend only on the size of the matrix (graph), hence our bound is incomparable. For instance, if G is a bounded-degree expander graph with uniform edge weights, dilexp(G) = O(log 2 n), and the work is O(n log n). This is currently the best known bound for Laplacians of expander graphs. We show that dilexp(G) is always at most n, hence our bound is at most O(m √ n log n) for any Laplacian (or SDD) matrix. For sufficiently dense systems, when m = Ω(n), this bound offers the best known work guarantee of any linear-space method. The main technical contributions of this paper include (i) adapting a recent result of Racke to designing support-tree preconditioners, (ii) extending a power dissipation approach for bounding support numbers of preconditioners, and (iii) applying the methods used in Leighton and Rao’s approximate max-flow min-cut theorem to the “asymmetric” product flows the arise in Racke’s construction.

Computational models of perceptual organization

Abstract: Perceptual organization refers to the process of organizing sensory input into coherent and interpretable perceptual structures. This process is challenging due to the chicken-and-egg nature between the various sub-processes such as image segmentation, figure-ground segregation and object recognition. Low-level processing requires the guidance of high-level knowledge to overcome noise; while high-level processing relies on low-level processes to reduce the computational complexity. Neither process can be sufficient on its own. Consequently, any system that carries out these processes in a sequence is bound to be brittle. An alternative system is one in which all processes interact with each other simultaneously. In this thesis, we develop a set of simple yet realistic interactive processing models for perceptual organization. We model the processing in the framework of spectral graph theory, with a criterion encoding the overall goodness of perceptual organization. We derive fast solutions for near-global optima of the criterion, and demonstrate the efficacy of the models on segmenting a wide range of real images. Through these models, we are able to capture a variety of perceptual phenomena: a unified treatment of various grouping, figure-ground and depth cues to produce popout, region segmentation and depth segregation in one step; and a unified framework for integrating bottom-up and top-down information to produce an object segmentation from spatial and object attention. We achieve these goals by empowering current spectral graph methods with a principled solution for multiclass spectral graph partitioning; expanded repertoire of grouping cues to include similarity, dissimilarity and ordering relationships; a theory for integrating sparse grouping cues; and a model for representing and integrating higher-order relationships. These computational tools are also useful more generally in other domains where data need to be organized effectively.

The Cheeger Constant, the Heat Kernel, and the Green Kernel of an Infinite Graph

Abstract: This paper gives a new approach to estimate the Cheeger constant, the heat kernel, and the Green kernel of the combinatorial Laplacian for an infinite graph.

Quantum query complexity of graph connectivity

Abstract: Harry Buhrman et al gave an Omega(sqrt n) lower bound for monotone graph properties in the adjacency matrix query model. Their proof is based on the polynomial method. However for some properties stronger lower bounds exist. We give an Omega(n^{3/2}) bound for Graph Connectivity using Andris Ambainis' method, and an O(n^{3/2} log n) upper bound based on Grover's search algorithm. In addition we study the adjacency list query model, where we have almost matching lower and upper bounds for Strong Connectivity of directed graphs.

On the optimality of J. Cheeger and P. Buser inequalities

Abstract: We study the relationship between the first eigenvalue of the Laplacian and Cheeger constant when the Cheeger constant converges to zero, in the case of compact Riemannian manifolds and of finite graphs.

Pattern spaces from graph polynomials

Abstract: Although graph structures have proved useful in high level vision for object recognition and matching, they can prove computationally cumbersome because of the need to establish reliable correspondences between nodes. Hence, standard pattern recognition techniques cannot be easily applied to graphs since feature vectors are not easily constructed. To overcome this problem, we turn to the spectral matrix. We show how the elements of this matrix can be used to construct symmetric polynomials that are permutation invariant. The coefficients of these polynomials can be used as graph-features which can be encoded in a vectorial manner. Hence, the symmetric polynomials lead to a representation which is invariant under node permutations and so represents the graph structure without the need for labelling or correspondence operations. We demonstrate that these features are complete and continuous for 'simple' graphs (those without repeated eigenvalues in their spectrum). The notions of stability and discrimination are discussed, and we present experimental evaluation of these properties. Finally, we show that these graph characterizations can be used to cluster graphs from real datasets.

Some properties of Laplacian eigenvectors.

Abstract: Let G be a graph on n vertices, G its complement and Kn the complete graph on n vertices. We show that if G is connected, then any Laplacian eigenvector of G is also a Laplacian eigenvector of G and of Kn . This result holds, with a slight modification, also for disconnected graphs. We establish also some other results, all showing that the structural information contained in the Laplacian eigenvectors is rather limited. An analogy between the theories of Laplacian and ordinary graph spectra is pointed out.

Two shorter proofs in spectral graph theory

Abstract: Let G be a simple graph with V (E) as its vertex (respectively, edge) set. The spectrum of G is the spectrum of its adjacency matrix. For all other definitions (or notation not given here), especially those related to graph spectra, see [1]. The purpose of this note is to provide shorter proofs of two inequalities already known in spectral graph theory. The first bounds vertex eccentricities of a connected graph in terms of some spectral quantities, while the second bounds the spectral radius of a graph in terms of vertex degrees. The first inequality is a slight generalization of a bound due to C.D. Godsil, while the second, also known as the Runge-Hofmeister conjecture, was first proved by A.J. Hoffman et al. (The proofs given here were obtained by the first and second author, respectively.)

Cheeger Inequality for Infinite Graphs

Abstract: In this paper we prove the Cheeger inequality for infinite weighted graphs endowed with 'corresponding' measure. This measure has already been developed in the study of tree lattices. Our graphs have finite volumes. A similar theory has already been developed for manifolds of finite volumes.

Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems

Abstract: The spectrum of a graph is the set of all eigenvalues of the Laplacian matrix of the graph. There is a closed relationship between the Laplacian spectrum of graphs and some properties of graphs such as connectivity. In the recent years Laplacian spectrum of graphs has been widely applied in many fields. The application of Laplacian spectrum of graphs to circuit partitioning problems is reviewed in this paper. A new criterion of circuit partitioning is proposed and the bounds of the partition ratio for weighted graphs are also presented. Moreover, the deficiency of graph-partitioning algorithms by Laplacian eigenvectors is addressed and an algorithm by means of the minimal spanning tree of a graph is proposed. By virtue of taking the graph structure into consideration this algorithm can fulfill general requirements of circuit partitioning.