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23
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
 Neural Computation
, 2003
"... Abstract 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 corr ..."
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Cited by 725 (15 self)
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Abstract 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 approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. Consider, for example, gray scale images of an object taken under fixed lighting conditions with a moving camera. Each such image would typically be represented by a brightness value at each pixel. If there were n 2
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) which ..."
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Cited by 69 (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.
Partitioning very large circuits using analytical placement techniques
 in Proceedings 31st ACM/IEEE Design Automation Conference
, 1994
"... A new partitioning approach for very large circuits is described. We demonstrate that applying a recently developed analytical placement algorithm, that pro ts from a linear objective function, signi cantly improves the partitioning quality compared to the wellknown eigenvector approach, which mini ..."
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Cited by 35 (1 self)
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A new partitioning approach for very large circuits is described. We demonstrate that applying a recently developed analytical placement algorithm, that pro ts from a linear objective function, signi cantly improves the partitioning quality compared to the wellknown eigenvector approach, which minimizes a quadratic objective function. For the rst time, results of benchmark circuits with up to 100,000 cells are presented. The cutsize and the minimum ratio cut is improved up to 90%. The average improvement is about 50%. 1
A Computational Study of Graph Partitioning
, 1993
"... Let G = (N; E) be an edgeweighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical s ..."
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Cited by 35 (10 self)
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Let G = (N; E) be an edgeweighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvaluebased techniques to find upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.
Geometric Embeddings for Faster and Better MultiWay Netlist Partitioning
 Proc. ACM/IEEE Design Automation Conf
, 1993
"... We give new, effective algorithms for kway circuit partitioning in the two regimes of k ø n and k = \Theta(n), where n is the number of modules in the circuit. We show that partitioning an appropriately designed geometric embedding of the netlist, rather than a traditional graph representation, yi ..."
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Cited by 30 (15 self)
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We give new, effective algorithms for kway circuit partitioning in the two regimes of k ø n and k = \Theta(n), where n is the number of modules in the circuit. We show that partitioning an appropriately designed geometric embedding of the netlist, rather than a traditional graph representation, yields improved results as well as large speedups. We derive d dimensional geometric embeddings of the netlist via (i) a new "partitioningspecific" net model for constructing the Laplacian of the netlist, and (ii) computation of d eigenvectors of the netlist Laplacian; we then apply (iii) fast topdown and bottomup geometric clustering methods. 1 Preliminaries In topdown layout synthesis of complex VLSI systems, the goal of partitioning/clustering is to reveal the natural circuit structure, via a decomposition into k subcircuits which minimizes connectivity between subcircuits. A generic problem statement is as follows: kWay Partitioning: Given a circuit netlist G = (V; E) with jV j...
A Hypergraph Framework For Optimal ModelBased Decomposition Of Design Problems
 Computational Optimization and Applications
, 1997
"... Decomposition of large engineering system models is desirable since increased model size reduces reliability and speed of numerical solution algorithms. The article presents a methodology for optimal modelbased decomposition (OMBD) of design problems, whether or not initially cast as optimization p ..."
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Cited by 30 (20 self)
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Decomposition of large engineering system models is desirable since increased model size reduces reliability and speed of numerical solution algorithms. The article presents a methodology for optimal modelbased decomposition (OMBD) of design problems, whether or not initially cast as optimization problems. The overall model is represented by a hypergraph and is optimally partitioned into weakly connected subgraphs that satisfy decomposition constraints. Spectral graphpartitioning methods together with iterative improvement techniques are proposed for hypergraph partitioning. A known spectral Kpartitioning formulation, which accounts for partition sizes and edge weights, is extended to graphs with also vertex weights. The OMBD formulation is robust enough to account for computational demands and resources and strength of interdependencies between the computational modules contained in the model. KEYWORDS: Model decomposition, multidisciplinary design, hypergraph partitioning, larges...
SEMIDEFINITE PROGRAMMING RELAXATIONS FOR THE GRAPH PARTITIONING PROBLEM
, 1999
"... A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 co ..."
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Cited by 26 (7 self)
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A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 constraints in GP. The special structure of the relaxation is exploited in order to project onto the minimal face of the cone of positive semidefinite matrices which contains the feasible set. This guarantees that the Slater constraint qualification holds, which allows for a numerically stable primaldual interiorpoint solution technique. A gangster operator is the key to providing an efficient representation of the constraints in the relaxation. An incomplete preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. Only dual feasibility is enforced, which results in the desired lower bounds, but avoids the expensive primal feasibility calculations.
Optimal ModelBased Decomposition of Powertrain System Design
, 1995
"... Optimal design of large engineering systems modeled as nonlinear programming problems remains a challenge because increased size reduces reliability and speed of numerical optimization algorithms. Decomposition of the original model into smaller coordinated submodels is desirable or even necessary. ..."
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Cited by 22 (12 self)
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Optimal design of large engineering systems modeled as nonlinear programming problems remains a challenge because increased size reduces reliability and speed of numerical optimization algorithms. Decomposition of the original model into smaller coordinated submodels is desirable or even necessary. The article presents a methodology for optimal modelbased decomposition of design problems, whether or not initially cast as optimization models. The overall model is represented by a hypergraph that is optimally partitioned into weaklyconnected subgraphs satisfying partitioning constraints. The formulation is robust enough to account for computational demands and resources, and the strength of interdependencies between the design relations contained in the model. This decomposition methodology is applied to a vehicle powertrain system design model consisting of engine, torque converter, transmission, and wheeltire assemblies, with 87 design relations and 119 design and state/behavior variables.
MultiWay Partitioning Via Geometric Embeddings, Orderings, and Dynamic Programming
 Orderings, and Dynamic Programming’, in IEEE Trans. on CAD
, 1995
"... This paper presents effective algorithms for multiway partitioning. Confirming ideas originally due to Hall [27], we demonstrate that geometric embeddings of the circuit netlist can lead to highquality kway partitionings. The netlist embeddings are derived via the computation of d eigenvectors o ..."
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Cited by 15 (1 self)
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This paper presents effective algorithms for multiway partitioning. Confirming ideas originally due to Hall [27], we demonstrate that geometric embeddings of the circuit netlist can lead to highquality kway partitionings. The netlist embeddings are derived via the computation of d eigenvectors of the Laplacian for a graph representation of the netlist. As [27] did not specify how to partition such geometric embeddings, we explore various geometric partitioning objectives and algorithms, and find that they are limited because they do not integrate topological information from the netlist. Thus, we also present a new partitioning algorithm that exploits both the geometric embedding and netlist information, as well as a Restricted Partitioning formulation that requires each cluster of the kway partitioning to be contiguous in a given linear ordering. We begin with a ddimensional spectral embedding and construct a heuristic 1dimensional ordering of the modules (combining spacefillin...