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296
METIS  Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 2.0
, 1995
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On the equivalence of nonnegative matrix factorization and spectral clustering
 in SIAM International Conference on Data Mining
, 2005
"... Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X ..."
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Cited by 159 (20 self)
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Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X = FGT is equivalent to simultaneous clustering of rows and columns of a bipartite graph. Algorithms are given for computing these symmetric NMFs. 1
Some Applications of Laplace Eigenvalues of Graphs
 GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 129 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the maxcut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Empirical and theoretical comparisons of selected criterion functions for document clustering
 Machine Learning
"... Abstract. This paper evaluates the performance of different criterion functions in the context of partitional clustering algorithms for document datasets. Our study involves a total of seven different criterion functions, three of which are introduced in this paper and four that have been proposed i ..."
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Cited by 116 (6 self)
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Abstract. This paper evaluates the performance of different criterion functions in the context of partitional clustering algorithms for document datasets. Our study involves a total of seven different criterion functions, three of which are introduced in this paper and four that have been proposed in the past. We present a comprehensive experimental evaluation involving 15 different datasets, as well as an analysis of the characteristics of the various criterion functions and their effect on the clusters they produce. Our experimental results show that there are a set of criterion functions that consistently outperform the rest, and that some of the newly proposed criterion functions lead to the best overall results. Our theoretical analysis shows that the relative performance of the criterion functions depends on (i) the degree to which they can correctly operate when the clusters are of different tightness, and (ii) the degree to which they can lead to reasonably balanced clusters. Keywords:
SPECTRAL CLUSTERING AND THE HIGHDIMENSIONAL STOCHASTIC BLOCKMODEL
 SUBMITTED TO THE ANNALS OF STATISTICS
"... Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the stru ..."
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Cited by 105 (7 self)
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Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasible method to discover these communities. The Stochastic Blockmodel (Holland, Laskey and Leinhardt, 1983) is a social network model with well defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, we bound the number of nodes “misclustered” by spectral clustering. The asymptotic results in this paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name highdimensional. In order to study spectral clustering under the Stochastic Blockmodel, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a “population” normalized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original.
Solving Cluster Ensemble Problems by Bipartite Graph Partitioning
 IN PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON MACHINE LEARNING
, 2004
"... A critical problem in cluster ensemble research is how to combine multiple clusterings to yield a final superior clustering result. Leveraging advanced graph partitioning techniques, we solve this problem by reducing it to a graph partitioning problem. We introduce a new reduction method that constr ..."
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Cited by 103 (3 self)
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A critical problem in cluster ensemble research is how to combine multiple clusterings to yield a final superior clustering result. Leveraging advanced graph partitioning techniques, we solve this problem by reducing it to a graph partitioning problem. We introduce a new reduction method that constructs a bipartite graph from a given cluster ensemble. The resulting graph models both instances and clusters of the ensemble simultaneously as vertices in the graph. Our approach retains all of the information provided by a given ensemble, allowing the similarity among instances and the similarity among clusters to be considered collectively in forming the final clustering. Further, the resulting graph partitioning problem can be solved efficiently. We empirically evaluate the proposed approach against two commonly used graph formulations and show that it is more robust and achieves comparable or better performance in comparison to its competitors.
Multilevel Circuit Partitioning
 IN PROC. OF THE 34TH ACM/IEEE DESIGN AUTOMATION CONFERENCE
, 1998
"... Many previous works in partitioning have used some underlying clustering algorithm to improve performance. As problem sizes reach new levels of complexity, a single application of a clustering algorithm is insufficient to produce excellent solutions. Recent work has illustrated the promise of multi ..."
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Cited by 89 (8 self)
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Many previous works in partitioning have used some underlying clustering algorithm to improve performance. As problem sizes reach new levels of complexity, a single application of a clustering algorithm is insufficient to produce excellent solutions. Recent work has illustrated the promise of multilevel approaches. A multilevel partitioning algorithm recursively clusters the instance until its size is smaller than a given threshold, then unclusters the instance while applying a partitioning refinement algorithm. In this paper, we propose a new multilevel partitioning algorithm that exploits some of the latest innovations of classical iterative partitioning approaches. Our method also uses a new technique to control the number of levels in our matchingbased clustering algorithm. Experimental results show that our heuristic outperforms numerous existing bipartitioning heuristics with improvements ranging from 6.9 to 27.9 % for 100 runs and 3.0 to 20.6 % for just ten runs (while also using less CPU time). Further, our algorithm generates solutions better than the best known mincut bipartitionings for seven of the ACM/SIGDA benchmark circuits, including golem3 (which has over 100 000 cells). We also present quadrisection results which compare favorably to the partitionings obtained by the GORDIAN cell placement tool. Our work in multilevel quadrisection has been used as the basis for an effective cell placement package.
Spectral Partitioning: The More Eigenvectors, the Better
 PROC. ACM/IEEE DESIGN AUTOMATION CONF
, 1995
"... The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) w ..."
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Cited by 75 (3 self)
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The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widelyused spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a kway partitioning). Our result motivates a simple ordering heuristic that is a multipleeigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multiway VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.
Learning with hypergraphs: Clustering, classification, and embedding
 Advances in Neural Information Processing Systems (NIPS) 19
, 2006
"... We usually endow the investigated objects with pairwise relationships, which can be illustrated as graphs. In many realworld problems, however, relationships among the objects of our interest are more complex than pairwise. Naively squeezing the complex relationships into pairwise ones will inevita ..."
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Cited by 74 (2 self)
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We usually endow the investigated objects with pairwise relationships, which can be illustrated as graphs. In many realworld problems, however, relationships among the objects of our interest are more complex than pairwise. Naively squeezing the complex relationships into pairwise ones will inevitably lead to loss of information which can be expected valuable for our learning tasks however. Therefore we consider using hypergraphs instead to completely represent complex relationships among the objects of our interest, and thus the problem of learning with hypergraphs arises. Our main contribution in this paper is to generalize the powerful methodology of spectral clustering which originally operates on undirected graphs to hypergraphs, and further develop algorithms for hypergraph embedding and transductive classification on the basis of the spectral hypergraph clustering approach. Our experiments on a number of benchmarks showed the advantages of hypergraphs over usual graphs. 1
Isoperimetric graph partitioning for image segmentation
 IEEE TRANS. ON PAT. ANAL. AND MACH. INT
, 2006
"... Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations o ..."
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Cited by 74 (12 self)
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Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations of spectral methods, but with improved speed and stability.