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65
Consistency of spectral clustering
, 2004
"... Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spe ..."
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Cited by 289 (15 self)
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Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacianbased methods in a statistical setting.
A Polylogarithmic Approximation of the Minimum Bisection
, 2001
"... A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets. ..."
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Cited by 74 (8 self)
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A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets.
Permuting Sparse Rectangular Matrices into BlockDiagonal Form
 SIAM Journal on Scientific Computing
, 2002
"... We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. W ..."
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Cited by 57 (19 self)
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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using stateoftheart graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
Graph Partitioning Algorithms With Applications To Scientific Computing
 Parallel Numerical Algorithms
, 1997
"... Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of su ..."
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Cited by 40 (0 self)
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Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...
Greedy, Prohibition, and Reactive Heuristics for Graph Partitioning
 IEEE Transactions on Computers
, 1998
"... New heuristic algorithms are proposed for the Graph Partitioning problem. A greedy construction scheme with an appropriate tiebreaking rule (MINMAXGREEDY) produces initial assignments in a very fast time. For some classes of graphs, independent repetitions of MINMAXGREEDY are sufficient to rep ..."
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Cited by 30 (6 self)
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New heuristic algorithms are proposed for the Graph Partitioning problem. A greedy construction scheme with an appropriate tiebreaking rule (MINMAXGREEDY) produces initial assignments in a very fast time. For some classes of graphs, independent repetitions of MINMAXGREEDY are sufficient to reproduce solutions found by more complex techniques. When the method is not competitive, the initial assignments are used as starting points for a prohibitionbased scheme, where the prohibition is chosen in a randomized and reactive way, with a bias towards more successful choices in the previous part of the run. The relationship between prohibitionbased diversification (Tabu Search) and the variabledepth KernighanLin algorithm is discussed. Detailed experimental results are presented on benchmark suites used in the previous literature, consisting of graphs derived from parametric models (random graphs, geometric graphs, etc.) and of "realworld " graphs of large size. On the first series ...
A scalable graphcut algorithm for nd grids
 In Proceedings of CVPR
, 2008
"... Global optimisation via st graph cuts is widely used in computer vision and graphics. To obtain highresolution output, graph cut methods must construct massive ND gridgraphs containing billions of vertices. We show that when these graphs do not fit into physical memory, current maxflow/mincut ..."
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Cited by 24 (0 self)
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Global optimisation via st graph cuts is widely used in computer vision and graphics. To obtain highresolution output, graph cut methods must construct massive ND gridgraphs containing billions of vertices. We show that when these graphs do not fit into physical memory, current maxflow/mincut algorithms—the workhorse of graph cut methods—are totally impractical. Others have resorted to banded or hierarchical approximation methods that get trapped in local minima, which loses the main benefit of global optimisation. We enhance the pushrelabel algorithm for maximum flow [14] with two practical contributions. First, true global minima can now be computed on immense gridlike graphs too large for physical memory. These graphs are ubiquitous in computer vision, medical imaging and graphics. Second, for commodity multicore platforms our algorithm attains nearlinear speedup with respect to number of processors. To achieve these goals, we generalised the standard relabeling operations associated with pushrelabel. 1.
Spectral Techniques in Graph Algorithms
 Lecture Notes in Computer Science 1380
, 1998
"... The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs. ..."
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Cited by 23 (2 self)
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The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs. 1
Parameterized graph separation problems
 In Proc. 1st IWPEC, volume 3162 of LNCS
, 2004
"... We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminal ..."
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Cited by 22 (2 self)
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We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are