C. Alpert and S. Yao. Spectral partitioning: The more eigenvectors, the better. In Proc. ACM/IEEE Design Automation Conference, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [9, 6] cuts a graph in two by thresholding the second eigenvector of the graph s normalized Laplacian matrix, and numerous clustering algorithms use selected eigenvectors of dot product or kernel matrices to re represent the data for clustering by simpler heuristics such as thresholding or Kmeans [25, 2, 5, 7, 27, 21, 29, 13, 3, 18, 19, 4, 20, 22]. While the statistical basis and optimality of PCA is well understood, virtually all other spectral inethods are motivated by imperfect analogies between data derived graphs and physical problems (e.g. harmonic analysis 2 and random walks3) or as approximations to other problems (e.g. vector ....

.... statistical basis and optimality of PCA is well understood, virtually all other spectral inethods are motivated by imperfect analogies between data derived graphs and physical problems (e.g. harmonic analysis 2 and random walks3) or as approximations to other problems (e.g. vector quantization [2], min cut [27] or max flow [6] Underlying all this work is the notion that the truncated eigenvector basis somehow makes the problem simpler for the subsequent analysis. Our theoretical goal is to explain how and why this works. Embeddings and clusterings imply loss of information, but there ....

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C. Alpert and S. Yao. Spectral partitioning: The more eigenvectors, the better. In Proc. ACM/IEEE Design Automation ConJrence, 1995.

....One of them is model selection. Several of the current techniques partition a graph in two sets, and are recursively applied to nd K clusters [16, 7] However, it has been experimentally observed that using more eigenvectors and computing directly a K way partitioning provides better results [1]. In [18] Weiss performed a comparative analysis of four spectral methods employed in computer vision. His analysis led other authors to propose a new algorithm [10] that uses K eigenvectors simultaneously and combines the advantages of two other algorithms [15, 16] demonstrating theoretically ....

C. Alpert, A. Kahng, and S.Z Yao, Spectral partitioning : The more eigenvectors, the better, Discrete Applied Math, , no. 90, pp. 326, 1999.

....alternative that has recently emerged in a number of elds is to use spectral methods for clustering. Here, one uses the top eigenvectors of a matrix derived from the distance between points. Such algorithms have been successfully used in many applications including computer vision and VLSI design [5, 1]. But despite their empirical successes, di erent authors still disagree on exactly which eigenvectors to use and how to derive clusters from them (see [11] for a review) Also, the analysis of these algorithms, which we brie y review below, has tended to focus on simpli ed algorithms that only ....

.... graph partitioning appear to deal with partitioning the graph into exactly two parts, these methods are then typically applied recursively to nd k clusters (e.g. 9] Experimentally it has been observed that using more eigenvectors and directly computing a k way partitioning is better (e.g. [5, 1]) Here, we build upon the recent work of Weiss [11] and Meila and Shi [6] who analyzed algorithms that use k eigenvectors simultaneously in simple settings. We propose a particular manner to use the k eigenvectors simultaneously, and give conditions under which the algorithm can be expected to ....

C. Alpert, A. Kahng, and S. Yao. Spectral partitioning: The more eigenvectors, the better. Discrete Applied Math, 90:3-26, 1999.

....alternative that has recently emerged in a number of elds is to use spectral methods for clustering. Here, one uses the top eigenvectors of a matrix derived from the distance between points. Such algorithms have been successfully used in many applications including computer vision and VLSI design [3, 1]. But despite their empirical successes, di erent authors still disagree on exactly which eigenvectors to use and how to derive clusters from them (see [9] for a review) Also, the analysis of these algorithms, which we brie y review below, has tended to focus on simpli ed algorithms that only use ....

.... graph partitioning appear to deal with partitioning the graph into exactly two parts, these methods are then typically applied recursively to nd k clusters (e.g. 7] Experimentally it has been observed that using more eigenvectors and directly computing a k way partitioning is better (e.g. [3, 1]) Here, we build upon the recent work of Weiss (1999) and Meila and Shi (2001) who analyzed algorithms that use k eigenvectors simultaneously in simple settings. We propose a particular manner to use the k eigenvectors simultaneously, and give conditions under which the algorithm can be ....

Charles Alpert, Andrew Kahng, and So-Zen Yao. Spectral partitioning: The more eigenvectors, the better. Discrete Applied Math, 90:3-26, 1999.

....system must be partitioned into subsystems such that elements in the same subsystem are strongly interconnected, whereas elements in different subsystems are weakly interconnected. Such applications include computer logic and page partitioning (Donath 1988) VLSI layout and packaging of circuits (Alpert and Yao, 1995; Dunlop and Kernighan, 1985) machine layout in manufacturing systems, assignment of computations to multiple processors (Hendrickson and Leland, 1992) and domain decomposition of finite element or finite volume meshes for parallel computation (Barnard and Simon, 1993; Farhat and Lesoinne, ....

....applied to hypergraphs, so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research, but only recently spectrum based methods has been successfully applied to graph partitioning (Alpert and Yao, 1995; Hendrickson and Leland, 1992; Pothen et al. 1990) Michelena and Papalambros (1995a) have extended formulations given by Rendl and Wolkowicz (1990) and Falkner et al. 1994) to account for weighted vertices. In the powertrain application below, the software package Chaco (Hendrickson and ....

Alpert, C.J. and Yao, S.-Z., 1995, Spectral Partitioning: The More Eigenvectors, The Better, Proceedings, 32nd ACM/IEEE Design Automation Conference.

....in ordering the nonzeros of a given matrix into approximate block diagonal form via permutations. This problem corresponds directly to the partitioning problem in graph theory, and so is often referred to as matrix partitioning. The partitioning problem has been well studied in the symmetric case [1, 2, 4, 11, 12, 13, 14, 15, 16, 21, 22, 23]. The rectangular partitioning problem, however, has received very little attention; the primary reference in this area is Berry, Hendrickson, and Raghavan [3] on envelope reduction for hypertext matrices. Let A denote a sparse rectangular m Theta n matrix. We will assume throughout that we are ....

....12, 13, 15] We develop a multilevel method specific for bipartite graphs with various refinement strategies including the alternating partitioning method and a version of Kernighan Lin for bipartite graphs. Eventually, we would also like to examine handling four or eight diagonal blocks directly [1, 11]. Acknowledgments The author is indebted to Bruce Hendrickson and Dianne O Leary for many helpful discussions. The author also thanks Eduardo D Azevedo, Chuck Romine, and the anonymous referees for their reviews, and Mike Berry for providing data. ....

Charles J. Alpert and So-Zen Yao. Spectral partitioning: The more eigenvectors, the better. In 32nd ACM/IEEE Design Automation Conference, pages 195--200, 1995.

....feasible solution by exchanging modules between partitions [1] 2] Some of these techniques utilize clustering [3] or cell replication [4] 5] 6] to improve the results. Other techniques also exist which effectively solve the multi way partitioning problem, such as spectral methods [7] [8], Boolean programming [9] geometric embeddings [10] placement techniques [11] and others [12] 13] 14] Approaches that focus on FPGA partitioning, reporting benchmark results, have been proposed in [15] 16] Contributions that address performance oriented partitioning of FPGA are only ....

C. J. Alpert and So-Zen Yao. Spectral Partitioning: The More Eigenvectors, The Better. In 32th DAC, ACM/IEEE, pages 195--200, June 1995.

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C. Alpert and S. Yao. Spectral partitioning: The more eigenvectors, the better. In Proc. ACM/IEEE Design Automation Conference, 1995.

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C. Alpert and S. Yao. Spectral partitioning: The more eigenvectors, the better, 1994.

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C. Alpert and S. Yao. Spectral partitioning: The more eigenvectors, the better. In Proc. ACM/IEEE Design Automation Conference, 1995.

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C. J. Alpert and S. Z. Yao. Spectral partitioning: the more eigenvectors, the better. In Proc. ACM/IEEE Design Automation Conf. (to appear), 1995.

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