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Breaking the Small Cluster Barrier of Graph Clustering Supplementary Material
"... In this supplementary material, we present proof details. 1 Notation and Conventions We use the following notation and conventions throughout the supplement. For a real n × n matrix M, we use the unadorned norm ‖M ‖ to denote its spectral norm. The notation ‖M‖F refers to the Frobenius norm, ‖M‖1 is ..."
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Cited by 7 (1 self)
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In this supplementary material, we present proof details. 1 Notation and Conventions We use the following notation and conventions throughout the supplement. For a real n × n matrix M, we use the unadorned norm ‖M ‖ to denote its spectral norm. The notation ‖M‖F refers to the Frobenius norm, ‖M‖1 is ∑ i,j M(i, j)  and ‖M‖ ∞ is maxij M(i, j). We will also study operators on the space of matrices. To distinguish them from the matrices studied in this work, we will simply call these objects “operators”, and will denote them using a calligraphic font, e.g. P. The norm ‖P ‖ of an operator is defined as where the supremum is over matrices M. ‖P ‖ = sup ‖PM‖F, M:‖M‖F =1 For a fixed, real n × n matrix M, we define the matrix linear subspace T (M) as follows: T (M): = {Y M + MX: X, Y ∈ R n×n}. In words, this subspace is the set of matrices spanned by matrices each row of which is in the row space of M, and matrices each column of which is in the column space of M. For any given subspace of matrices S ⊆ Rn×n, we let PS denote the orthogonal projection onto S with respect to the the inner product 〈X, Y 〉 = ∑n i,j=1 X(i, j)Y (i, j) = tr XtY. This means that for any matrix M, PSM = argminX∈S ‖M − X‖F. For a matrix M, we let Γ(M) denote the set of matrices supported on a subset of the support of M. Note that for any matrix X,
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices
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 1697 (13 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
Adaptive clustering for mobile wireless networks
 IEEE Journal on Selected Areas in Communications
, 1997
"... This paper describes a selforganizing, multihop, mobile radio network, which relies on a code division access scheme for multimedia support. In the proposed network architecture, nodes are organized into nonoverlapping clusters. The clusters are independently controlled and are dynamically reconfig ..."
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Cited by 556 (11 self)
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This paper describes a selforganizing, multihop, mobile radio network, which relies on a code division access scheme for multimedia support. In the proposed network architecture, nodes are organized into nonoverlapping clusters. The clusters are independently controlled and are dynamically
Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering
 Advances in Neural Information Processing Systems 14
, 2001
"... Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a higher ..."
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Cited by 664 (8 self)
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Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a
GPFS: A SharedDisk File System for Large Computing Clusters
 In Proceedings of the 2002 Conference on File and Storage Technologies (FAST
, 2002
"... GPFS is IBM's parallel, shareddisk file system for cluster computers, available on the RS/6000 SP parallel supercomputer and on Linux clusters. GPFS is used on many of the largest supercomputers in the world. GPFS was built on many of the ideas that were developed in the academic community ove ..."
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Cited by 518 (3 self)
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GPFS is IBM's parallel, shareddisk file system for cluster computers, available on the RS/6000 SP parallel supercomputer and on Linux clusters. GPFS is used on many of the largest supercomputers in the world. GPFS was built on many of the ideas that were developed in the academic community
ModelBased Clustering, Discriminant Analysis, and Density Estimation
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2000
"... Cluster analysis is the automated search for groups of related observations in a data set. Most clustering done in practice is based largely on heuristic but intuitively reasonable procedures and most clustering methods available in commercial software are also of this type. However, there is little ..."
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Cited by 557 (28 self)
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Cluster analysis is the automated search for groups of related observations in a data set. Most clustering done in practice is based largely on heuristic but intuitively reasonable procedures and most clustering methods available in commercial software are also of this type. However
Scatter/Gather: A Clusterbased Approach to Browsing Large Document Collections
, 1992
"... Document clustering has not been well received as an information retrieval tool. Objections to its use fall into two main categories: first, that clustering is too slow for large corpora (with running time often quadratic in the number of documents); and second, that clustering does not appreciably ..."
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Cited by 772 (12 self)
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Document clustering has not been well received as an information retrieval tool. Objections to its use fall into two main categories: first, that clustering is too slow for large corpora (with running time often quadratic in the number of documents); and second, that clustering does not appreciably
Factor Graphs and the SumProduct Algorithm
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... A factor graph is a bipartite graph that expresses how a "global" function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple c ..."
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Cited by 1787 (72 self)
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A factor graph is a bipartite graph that expresses how a "global" function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple
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