The Clustering Problem

Yongsub LimApplied Algorithm Laboratory

KAIST

04/19/2023 The Clustering Problem 2

Contents

• The Clustering Problem• Basic Algorithms

K-Means K-Clustering of Max. Spacing

• Two-Phase Algorithms• Other Algorithms

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The Clustering Problem

• Given data, it is to discover “mean-ingful” groups

• Data in same group are similar, and• Data between different groups are

not similar

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Example of clustering

1x

2x

04/19/2023 The Clustering Problem 5

Example of clustering

1x

2x

04/19/2023 The Clustering Problem 6

Example of clustering

1x

2x

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Applications of Clustering

• The image segmentation problem can be considered as a clustering of pixels of an image

• In unsupervised learning, before making a decision rule, we classify unlabeled training data through clus-tering

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Applications of Clustering

• In a network or a graph, we can do grouping vertices which are highly connected within one group

• Clustering is also useful in biology to classify genes

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Basic Algorithms

• Two algorithms will be introduced

• K-Means computes iteratively centers of K clusters

• K-Clustering of Max. Spacing uses a minimum spanning tree

• Objective functions of theses are dif-ferent

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K-Means

• Determine means of K clusters ran-domly

• At each iteration, Every data belongs to a cluster whose

mean is the nearest one among K means

Re-compute means of all clusters

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K-Means

• Objective is to minimize the sum of distance of centers of clusters and their members

• It is clustering for high density in one cluster

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K-Means Algorithm

• Worst caseInitial two cen-ters randomly chosen

This may be not what we want!!!

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K-Clustering of Max. Spacing

• Given data, find K clusters which maximize the minimum distances be-tween all pairs of clusters

• spacing: min. distance between any pair of data in different clusters

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K-Clustering of Max. Spacing

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K-Clustering of Max. Spacing

• Consider given data to a complete graph with Euclidean distance

• Compute a MST

• Delete the K-1 most expensive edges of a MST

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K-Clustering of Max. Spacing

Calg

Copt

≤ spacing of Calg

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K-Clustering of Max. Spacing

• It is no randomness

• Objective seems to be better or more reasonable than K-means

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K-Means vs. Max. Spacing

• Good clustering is High density in one cluster (K-Means) Long dist. between clusters (Max. Spac-

ing)

>

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K-Means vs. Max. Spacing

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Two-Phase Algorithms

• Two algorithms will be introduced

• In the first phase, both do clustering without restriction on K

• In second phase, if # of clusters are larger than K, merge using Max. Spacing

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Hierarchical EMST

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

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Hierarchical EMST

• HEMST removes all edges with weights greater than the threshold (mean+std. of edges)

• If # of clusters is less than a given K, same with Max. Spacing

• If not, it runs Max. Spacing on data set each of which is nearest to the center of its cluster

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

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Hierarchical EMST

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

04/19/2023 The Clustering Problem 24

Hierarchical EMST

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

04/19/2023 The Clustering Problem 25

Hierarchical EMST

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

04/19/2023 The Clustering Problem 26

Hierarchical EMST

Oleksandr Grygorash, Yan zhou, Zach Jorgensen, Minimum spanning Tree Based Clustering Algorithms

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Modified K-Means Process

• MKF, in the first phase, is similar to K-Means

• The difference is that if data is far enough from all clusters, it becomes the center of the new cluster

• While running, if # of clusters is larger than a threshold, the two nearest clus-ters are merged

• In the second phase, apply Max. Spaing

M.F. Jiang, S.S. Tseng, C.M. Su, Two-phase clustering process for outliers detection

04/19/2023 The Clustering Problem 28

Modified K-Means Process

• This scheme can identify outliers by using Max. Spacing

M.F. Jiang, S.S. Tseng, C.M. Su, Two-phase clustering process for outliers detection

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Modified K-Means Process

M.F. Jiang, S.S. Tseng, C.M. Su, Two-phase clustering process for outliers detection

04/19/2023 The Clustering Problem 30

Modified K-Means Process

M.F. Jiang, S.S. Tseng, C.M. Su, Two-phase clustering process for outliers detection

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Two-Phase Algorithms

• Both give more weights to members in small sets in the first phase

• A small set will be the most likely clustered data, so it is reasonable to decrease distances between them

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Other Algorithms

• HCS uses min-cut of a graph

• It recursively separate data to dis-joint two subsets (min-cut) until all clusters are highly connected

• A graph is highly connected if the min. # of edges whose removal disconnects the graph is greater than |V|/2

Erez Hartuv, Ron Shamir, a clustering algorithm based on graph connectivity

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Other Algorithms

• Voting

• Apply K-Means N times

• If any pair of data belonged to same cluster greater than threshold t times, they are grouped

Ana L.N. Fred, Anil K. Jain, Data Clustering Using Evidence ac-cumulation

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Thanks