Results 1 - 10
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341
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
, 2003
"... One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondenc ..."
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Cited by 1230 (15 self)
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One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient ap-proach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
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 572 (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 Laplacian-based methods in a statistical setting.
A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems, in
- Proc. 6th SIAM Conf. Parallel Processing for Scientific Computing,
, 1993
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Spectral partitioning works: planar graphs and finite element meshes, in:
- Proceedings of the 37th Annual Symposium on Foundations of Computer Science,
, 1996
"... Abstract Spectral partitioning methods use the Fiedler vector-the eigenvector of the second-smallest eigenvalue of the Laplacian matrix-to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to wo ..."
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Cited by 201 (10 self)
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Abstract Spectral partitioning methods use the Fiedler vector-the eigenvector of the second-smallest eigenvalue of the Laplacian matrix-to 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 bounded-degree 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( √ n) for bounded-degree planar graphs and two-dimensional meshes and O(n 1/d ) for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs: we prove a bound of O(1/n) for bounded-degree planar graphs and O(1/n 2/d ) for well-shaped d-dimensional meshes.
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 max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Geometric Mesh Partitioning: Implementation and Experiments
"... We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain ..."
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Cited by 112 (20 self)
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We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of “well-shaped” finite element meshes have good separators. The geometric method is quite simple to implement: we describe a Matlab code for it in some detail. The method is also quite efficient and effective: we compare it with some other methods, including spectral bisection.
How Good is Recursive Bisection?
- SIAM J. Sci. Comput
, 1995
"... . The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Becaus ..."
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Cited by 100 (5 self)
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. The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NP-complete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a p-way partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as well-shaped finite element and finite difference...
Aspects of Unstructured Grids and FiniteVolume Solvers for Euler and Navier-Stokes Equations,
- [VKI/NASA/AGARD Special Courses on Unstructured Grid Methods for Advection Dominated Flows AGARD Publication R-787],
, 1995
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A spectral algorithm for envelope reduction of sparse matrices
- ACM/IEEE CONFERENCE ON SUPERCOMPUTING
, 1993
"... The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelope-reducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the ..."
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Cited by 85 (5 self)
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The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelope-reducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. This Laplacian eigenvector solves a continuous relaxation of a discrete problem related to envelope minimization called the minimum 2-sum problem. The permutation vector computed by the spectral algorithm is a closest permutation vector to the specified Laplacian eigenvector. Numerical results show that the new reordering algorithm usually computes smaller envelope sizes than those obtained from the current standards such as the Gibbs-Poole-Stockmeyer (GPS) algorithm or the reverse Cuthill-McKee (RCM) algorithm in SPARSPAK, in some cases reducing the envelope by more than a factor of two.
Adaptive Local Refinement with Octree Load-Balancing for the Parallel Solution of Three-Dimensional Conservation Laws
- J. Parallel Distrib. Comput
, 1997
"... Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive space-time mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to method ..."
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Cited by 68 (17 self)
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Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive space-time mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalaces, introduced at adaptive enrichment steps, are corrected by using traversals of an octtee representing a spatial decomposition of the domain. To accommodate the variable time steps, octtee partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octtee ad inertial partitioning procedures ae presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.
