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180
Normalized Cuts and Image Segmentation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2000
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Consensus Problems in Networks of Agents with Switching Topology and TimeDelays
, 2003
"... In this paper, we discuss consensus problems for a network of dynamic agents with fixed and switching topologies. We analyze three cases: i) networks with switching topology and no timedelays, ii) networks with fixed topology and communication timedelays, and iii) maxconsensus problems (or leader ..."
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Cited by 435 (13 self)
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In this paper, we discuss consensus problems for a network of dynamic agents with fixed and switching topologies. We analyze three cases: i) networks with switching topology and no timedelays, ii) networks with fixed topology and communication timedelays, and iii) maxconsensus problems (or leader determination) for groups of discretetime agents. In each case, we introduce a linear/nonlinear consensus protocol and provide convergence analysis for the proposed distributed algorithm. Moreover, we establish a connection between the Fiedler eigenvalue of the information flow in a network (i.e. algebraic connectivity of the network) and the negotiation speed (or performance) of the corresponding agreement protocol. It turns out that balanced digraphs play an important role in addressing averageconsensus problems. We introduce disagreement functions that play the role of Lyapunov functions in convergence analysis of consensus protocols. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
A Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems
 Experience
, 1994
"... Unstructured meshes are used in many largescale scientific and engineering problems, including finitevolume methods for computational fluid dynamics and finiteelement methods for structural analysis. If unstructured problems such as these are to be solved on distributedmemory parallel computers, ..."
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Cited by 284 (7 self)
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Unstructured meshes are used in many largescale scientific and engineering problems, including finitevolume methods for computational fluid dynamics and finiteelement methods for structural analysis. If unstructured problems such as these are to be solved on distributedmemory parallel computers, their data structures must be partitioned and distributed across processors; if they are to be solved efficiently, the partitioning must maximize load balance and minimize interprocessor communication. Recently the recursive spectral bisection method (RSB) has been shown to be very effective for such partitioning problems compared to alternative methods. Unfortunately, RSB in its simplest form is rather expensive. In this report we shall describe a multilevel implementation of RSB that can attain about an orderofmagnitude improvement in run time on typical examples. Keywords: graph partitioning, domain decomposition, MIMD machines, multilevel algorithm, spectral bisection, sp...
Survey of clustering data mining techniques
, 2002
"... Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in math ..."
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Cited by 247 (0 self)
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Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in mathematics, statistics, and numerical analysis. From a machine learning perspective clusters correspond to hidden patterns, the search for clusters is unsupervised learning, and the resulting system represents a data concept. From a practical perspective clustering plays an outstanding role in data mining applications such as scientific data exploration, information retrieval and text mining, spatial database applications, Web analysis, CRM, marketing, medical diagnostics, computational biology, and many others. Clustering is the subject of active research in several fields such as statistics, pattern recognition, and machine learning. This survey focuses on clustering in data mining. Data mining adds to clustering the complications of very large datasets with very many attributes of different types. This imposes unique
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 151 (1 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
A Minmax Cut Algorithm for Graph Partitioning and Data Clustering
, 2001
"... An important application of graph partitioning is data clustering using a graph model  the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. Here we propose a new algorithm for graph partition with an object ..."
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Cited by 150 (12 self)
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An important application of graph partitioning is data clustering using a graph model  the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. Here we propose a new algorithm for graph partition with an objective function that follows the minmax clustering principle. The relaxed version of the optimization of the minmax cut objective function leads to the Fiedler vector in spectral graph partition. Theoretical analyses of minmax cut indicate that it leads to balanced partitions, and lower bonds are derived. The minmax cut algorithm is tested on newsgroup datasets and is found to outperform other current popular partitioning/clustering methods. The linkagebased re nements in the algorithm further improve the quality of clustering substantially. We also demonstrate that the linearized search order based on linkage di erential is better than that based on the Fiedler vector, providing another e ective partition method.
Motion Segmentation and Tracking Using Normalized Cuts
, 1998
"... We propose a motion segmentation algorithm that aims to break a scene into its most prominent moving groups. A weighted graph is constructed on the ira. age sequence by connecting pixels that arc in the spatiotemporal neighborhood of each other. At each pizel, we define motion profile vectors which ..."
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Cited by 145 (5 self)
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We propose a motion segmentation algorithm that aims to break a scene into its most prominent moving groups. A weighted graph is constructed on the ira. age sequence by connecting pixels that arc in the spatiotemporal neighborhood of each other. At each pizel, we define motion profile vectors which capture the probability distribution of the image veloczty. The distance between motion profiles is used to assign a weight on the graph edges. 5rsmg normalized cuts we find the most salient partitions of the spatiotemporaI graph formed by the image sequence. For swmenting long image sequences,' we have developed a recursire update procedure that incorporates knowledge of segmentation in previous frames for efficiently finding the group correspondence in the new frame.
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 144 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Randomwalk computation of similarities between nodes of a graph, with application to collaborative recommendation
 IEEE Transactions on Knowledge and Data Engineering
, 2006
"... Abstract—This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markovchain model of random walk through the database. More precisely, we compute quantities (the average comm ..."
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Cited by 116 (14 self)
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Abstract—This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markovchain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the “length ” of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commutetime distance. This graph PCA provides a nice interpretation to the “Fiedler vector, ” widely used for graph partitioning. The model is evaluated on a collaborativerecommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacianbased similarities perform well in comparison with other methods. The model, which nicely fits into the socalled “statistical relational learning ” framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machinelearning and patternrecognition tasks involving a relational database. Index Terms—Graph analysis, graph and database mining, collaborative recommendation, graph kernels, spectral clustering, Fiedler vector, proximity measures, statistical relational learning. 1
Fundamentals of Spherical Parameterization for 3D Meshes
 PROCEEDINGS OF THE 2006 SYMPOSIUM ON INTERACTIVE 3D GRAPHICS AND GAMES, MARCH 1417, 2006
, 2003
"... Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the ..."
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Cited by 104 (26 self)
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Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. Satisfying the nonoverlapping requirement is the most difficult and critical component of this process. We present a generalization of the method of barycentric coordinates for planar parametrization which solves the spherical parametrization problem, prove its correctness by establishing a connection to spectral graph theory and describe efficient numerical methods for computing these parametrizations.