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Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto 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 extr ..."
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Cited by 201 (10 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto 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 boundeddegree 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( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Regularization and semisupervised learning on large graphs
 In COLT
, 2004
"... Abstract. We consider the problem of labeling a partially labeled graph. This setting may arise in a number of situations from survey sampling to information retrieval to pattern recognition in manifold settings. It is also of potential practical importance, when the data is abundant, but labeling i ..."
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Cited by 147 (1 self)
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Abstract. We consider the problem of labeling a partially labeled graph. This setting may arise in a number of situations from survey sampling to information retrieval to pattern recognition in manifold settings. It is also of potential practical importance, when the data is abundant, but labeling is expensive or requires human assistance. Our approach develops a framework for regularization on such graphs. The algorithms are very simple and involve solving a single, usually sparse, system of linear equations. Using the notion of algorithmic stability, we derive bounds on the generalization error and relate it to structural invariants of the graph. Some experimental results testing the performance of the regularization algorithm and the usefulness of the generalization bound are presented. 1
Combinatorial and Spectral Aspects of Nearest Neighbor Graphs in Doubling Dimensional and NearlyEuclidean Spaces
"... Abstract. Miller, Teng, Thurston, and Vavasis proved that every knearest neighbor graph (kNNG) in R d has a balanced vertex separator of size O(n 1−1/d k 1/d). Later, Spielman and Teng proved that the Fiedler value — the second smallest eigenvalue of the graph — of the Laplacian matrix of a kNNG i ..."
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Abstract. Miller, Teng, Thurston, and Vavasis proved that every knearest neighbor graph (kNNG) in R d has a balanced vertex separator of size O(n 1−1/d k 1/d). Later, Spielman and Teng proved that the Fiedler value — the second smallest eigenvalue of the graph — of the Laplacian matrix of a kNNG in R d is at O ( 1 n2/d). In this paper, we extend these two results to nearest neighbor graphs in a metric space with doubling dimension γ and in nearlyEuclidean spaces. We prove that for every l> 0, each kNNG in a metric space with doubling dimension γ has a vertex separator of size O(k 2 l(32l + 8) 2γ 2 L n log log n +), where L and S l S are respectively the maximum and minimum distances between any two points in P. We show how to use the singular value decomposition method to approximate a kNNG in a nearly Euclidean space by an Euclidean kNNG. This approximation enables us to obtain an upper bound on the Fiedler value of the kNNG in a nearly Euclidean space.