Introduction to Clustering Large and

There is a growing need for a more automated system of partitioning data sets into groups, or clusters. For example, digital libraries and the World Wide Web continue to grow exponentially, the ability to find useful information increasingly depends on the indexing infrastructure or search engine. Clustering techniques can be used to discover natural groups in data sets and to identify abstract structures that might reside there, without having any background knowledge of the characteristics of the data. Clustering has been used in a variety of areas, including computer vision, VLSI design, data mining, bio-informatics (gene expression analysis), and information retrieval, to name just a few. This book focuses on a few of the most important clustering algorithms, providing a detailed account of these major models in an information retrieval context. The beginning chapters introduce the classic algorithms in detail, while the later chapters describe clustering through divergences and show recent research for more advanced audiences.

Jacob Kogan is an Associate Professor in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County

Foreword by Michael W. Berry

xi

Preface

xiii

1 Introduction and motivation

1

(8)

1.1 A way to embed ASCII documents into a finite dimensional Euclidean space

3

(2)

1.2 Clustering and this book

5

(1)

1.3 Bibliographic notes

6

(3)

2 Quadratic k-means algorithm

9

(32)

2.1 Classical batch k-means algorithm

10

(11)

2.1.1 Quadratic distance and centroids

12

(1)

2.1.2 Batch k-means clustering algorithm

13

(1)

2.1.3 Batch k-means: advantages and deficiencies

14

(7)

2.2 Incremental algorithm

21

(8)

2.2.1 Quadratic functions

21

(4)

2.2.2 Incremental k-means algorithm

25

(4)

2.3 Quadratic k-means: summary

29

(8)

2.3.1 Numerical experiments with quadratic k-means

29

(2)

2.3.2 Stable partitions

31

(4)

2.3.3 Quadratic k-means

35

(2)

2.4 Spectral relaxation

37

(1)

2.5 Bibliographic notes

38

(3)

3 BIRCH

41

(10)

3.1 Balanced iterative reducing and clustering algorithm

41

(3)

3.2 BIRCH-like k-means

44

(5)

3.3 Bibliographic notes

49

(2)

4 Spherical k-means algorithm

51

(22)

4.1 Spherical batch k-means algorithm

51

(6)

4.1.1 Spherical batch k-means: advantages and deficiencies

53

(2)

4.1.2 Computational considerations

55

(2)

4.2 Spherical two-cluster partition of one-dimensional data

57

(7)

4.2.1 One-dimensional line vs. the unit circle

57

(3)

4.2.2 Optimal two cluster partition on the unit circle

60

(4)

4.3 Spherical batch and incremental clustering algorithms

64

(8)

4.3.1 First variation for spherical k-means

65

(3)

4.3.2 Spherical incremental iterations–computations complexity

68

(1)

4.3.3 The "ping-pong" algorithm

69

(2)

4.3.4 Quadratic and spherical k-means

71

(1)

4.4 Bibliographic notes

72

(1)

5 Linear algebra techniques

73

(18)

5.1 Two approximation problems

73

(1)

5.2 Nearest line

74

(3)

5.3 Principal directions divisive partitioning

77

(10)

5.3.1 Principal direction divisive partitioning (PDDP)

77

(3)

5.3.2 Spherical principal directions divisive partitioning (sPDDP)

80

(2)

5.3.3 Clustering with PDDP and sPDDP

82

(5)

5.4 Largest eigenvector

87

(2)

5.4.1 Power method

88

(1)

5.4.2 An application: hubs and authorities

88

(1)

5.5 Bibliographic notes

89

(2)

6 Information theoretic clustering

91

(10)

6.1 Kullback–Leibler divergence

91

(3)

6.2 k-means with Kullback–Leibler divergence

94

(2)

6.3 Numerical experiments

96

(2)

6.4 Distance between partitions

98

(1)

6.5 Bibliographic notes

99

(2)

7 Clustering with optimization techniques

101

(24)

7.1 Optimization framework

102

(1)

7.2 Smoothing k-means algorithm

103

(6)

7.3 Convergence

109

(5)

7.4 Numerical experiments

114

(8)

7.5 Bibliographic notes

122

(3)

8 k-means clustering with divergences

125

(30)

8.1 Bregman distance

125

(3)

8.2 φ-divergences

128

(4)

8.3 Clustering with entropy-like distances

132

(3)

8.4 BIRCH-type clustering with entropy-like distances

135

(5)

8.5 Numerical experiments with (upsilon, μ) k-means

140

(4)

8.6 Smoothing with entropy-like distances

144

(2)

8.7 Numerical experiments with (upsilon, μ) smoka

146

(6)

8.8 Bibliographic notes

152

(3)

9 Assessment of clustering results

155

(6)

9.1 Internal criteria

155

(1)

9.2 External criteria

156

(4)

9.3 Bibliographic notes

160

(1)

10 Appendix: Optimization and linear algebra background

161

(18)

10.1 Eigenvalues of a symmetric matrix

161

(2)

10.2 Lagrange multipliers

163

(1)

10.3 Elements of convex analysis

164

(14)

10.3.1 Conjugate functions

166

(3)

10.3.2 Asymptotic cones

169

(4)

10.3.3 Asymptotic functions

173

(3)

10.3.4 Smoothing

176

(2)

10.4 Bibliographic notes

178

(1)

11 Solutions to selected problems

179

(10)

Bibliography

189

(14)

Index

203

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Introduction to Clustering Large and High-Dimensional Data

Summary

Author Biography

Table of Contents

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