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On Spectral Clustering: Analysis and an algorithm
 ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2001
"... Despite many empirical successes of spectral clustering methods  algorithms that cluster points using eigenvectors of matrices derived from the distances between the points  there are several unresolved issues. First, there is a wide variety of algorithms that use the eigenvectors in slightly ..."
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Cited by 1102 (14 self)
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Despite many empirical successes of spectral clustering methods  algorithms that cluster points using eigenvectors of matrices derived from the distances between the points  there are several unresolved issues. First, there is a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well. We also show surprisingly good experimental results on a number of challenging clustering problems.
On Clusterings: Good, Bad and Spectral
, 2000
"... We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic has polylogarithmic worstcase guarantees under the new measure. The main result of the paper is the analysis of a popular spe ..."
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Cited by 254 (12 self)
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We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic has polylogarithmic worstcase guarantees under the new measure. The main result of the paper is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worstcase guarantees
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.
Finding a large hidden clique in a random graph
, 1998
"... ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomia ..."
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Cited by 83 (5 self)
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ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k�cn0.5 ˇ, for
A Comparison of Spectral Clustering Algorithms
, 2003
"... Spectral Clustering has become quite popular over the last few years and several new algorithms have been published. In this paper, we compare several of the bestknown algorithms from the point of view of clustering quality over arti cial and real datasets. We implement many variations of the ex ..."
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Cited by 59 (2 self)
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Spectral Clustering has become quite popular over the last few years and several new algorithms have been published. In this paper, we compare several of the bestknown algorithms from the point of view of clustering quality over arti cial and real datasets. We implement many variations of the existing spectral algorithms and compare their performance to see which features are more important. We also demonstrate that spectral methods show competitive performance on real dataset with respect to existing methods.
Approximation Algorithms for Channel Assignment in Radio Networks
 Wireless Networks
, 1998
"... We consider the channel assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with certain geometric structure. The channel assignment problem can be cast as a variant of the vertex coloring problem (called the distance2 coloring problem) ..."
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Cited by 24 (1 self)
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We consider the channel assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with certain geometric structure. The channel assignment problem can be cast as a variant of the vertex coloring problem (called the distance2 coloring problem) on the graph that models a given packet radio network. We present efficient approximation algorithms for the distance2 coloring problem for several classes of graphs including a class of geometric graphs that naturally model a large class of packet radio networks. The classes of graphs considered include (r, s)civilized graphs, planar graphs, graphs with bounded genus, etc. Many of the approximation results presented here are the first such results in the literature.
New Spectral Bounds on kPartitioning of Graphs
"... When executing processes on parallel computer systems they encounter as a major bottleneck interprocessor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the kpartitioning problem of the c ..."
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When executing processes on parallel computer systems they encounter as a major bottleneck interprocessor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the kpartitioning problem of the corresponding process graph, where k is the number of processors. The classical spectral lower bound of jV j 2k P k i=1 i for the ksection width of a graph is wellknown. We show new relations between the structure and the eigenvalues of a graph and present a new method to get tighter lower bounds on the ksection width. This method makes use of the level structure defined by the ksection. We define some global expansion property and prove that for graphs with the same ksection width the spectral lower bound increases with this global expansion. We also present examples of graphs for which our new bounds are tight up to a constant factor.