On clusterings-good, bad and spectral

Kannan, Ravindran; Vempala, Santosh; Veta, A.

doi:10.1109/sfcs.2000.892125

Cited by 485 publications

(618 citation statements)

References 11 publications

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“…One is the hierarchical method, which re-partitions the set until a stopping condition is met, or an aggregation presses which begins by considering each point as a cluster and then merging close clusters until a stopping condition is met [11,15,18,23,24]. The second is the k-means heuristic, where a mean of a cluster is the average of the clusters points.…”

Section: Clusteringmentioning

confidence: 99%

A Maximum Profit Coverage Algorithm with Application to Small Molecules Cluster Identification

Hassin

2006

Experimental Algorithms

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Section: Clusteringmentioning

confidence: 99%

A Maximum Profit Coverage Algorithm with Application to Small Molecules Cluster Identification

Hassin

2006

Experimental Algorithms

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“…The m-cut problem asks for the minimum weight of edges whose deletion leaves m disjoint parts. Closer to ours is the (α, )-clustering problem from [KVV04] that asks for a partition where each part has conductance at least α and the total weight of edges removed is minimized.…”

Section: Generalizations Of Sparsest Cutmentioning

confidence: 99%

Algorithmic Extensions of Cheeger’s Inequality to Higher Eigenvalues and Partitions

Louis

Raghavendra

Tetali

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. We consider two generalizations of the problem of finding a sparsest cut in a graph. The first is to find a partition of the vertex set into m parts so as to minimize the sparsity of the partition (defined as the ratio of the weight of edges between parts to the total weight of edges incident to the smallest m − 1 parts). The second, that has appeared in the context of understanding the unique games conjecture, is to find a subset of minimum sparsity that contains at most a 1/m fraction of the vertices. Our main results are extensions of Cheeger's classical inequality to these problems via higher eigenvalues of the graph Laplacian. In particular, for the sparsest m-partition, we prove that the sparsity is at most 8 √ 1 − λm log m where λm is the m th largest eigenvalue of the normalized adjacency matrix. For sparsest small-set, we bound the sparsity by O( (1 − λ m 2 ) log m). Our results are algorithmic, with the first using a recursive spectral decomposition and the second using a convex relaxation.

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“…Intuitively the quality of a clustering is assessed by how much similar points are grouped in the same cluster. In [68] the question is posed: how good is the clustering which is produced by a clustering algorithm? As already discussed the k-median clustering may produce a very bad clustering in case the "hidden" clusters are far from spherical.…”

Section: Quality Of Clusteringmentioning

confidence: 99%