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Linked Based Clustering and Its Theoretical Foundations Paper written by Margareta Ackerman and Shai Ben-David Yan T. Yang Presented by Yan T. Yang

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Outline Hierarchical clustering Linkage based clustering Theoretical foundation of clustering Characterization of linkage based clustering

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Outline Hierarchical clustering Linkage based clustering Theoretical foundation of clustering Characterization of linkage based clustering

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Hierarchical clustering Agglomerative Divisive

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Hierarchical clustering A D E B C ABCDE Raw Data Dendrogram F F

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Agglomerative clustering Raw Data Dendrogram ABCDEF A D E B C F

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Agglomerative clustering Raw Data Dendrogram ABCDEF A D E B C F

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Agglomerative clustering A D E B C Raw Data Dendrogram F ABCDEF

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Agglomerative clustering A D E B C Raw Data Dendrogram F ABCDEF

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Agglomerative clustering Raw Data Dendrogram ABCDEF A D E B C F

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Agglomerative clustering Raw Data Dendrogram ABCDEF A D E B C F General Complexity: O(n 3 )

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Agglomerative clustering Raw Data Dendrogram General Complexity: O(n 3 ) Linkage Based: O(n 2 )

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Linkage Based Clustering General Complexity: O(n 3 ) Linkage Based: O(n 2 ) Whenever a pair of points share the same cluster, they are connected via a tight chain of points. A D E B C F

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Linkage Based Clustering General Complexity: O(n 3 ) Linkage Based: O(n 2 ) Whenever a pair of points share the same cluster, they are connected via a tight chain of points. A D E B C F 3. Repeat until only k clusters remain. Steps: 1. Start: singletons 2. At each step, merge the closest pair of clusters

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Linkage-based clustering Linkage Based: O(n 2 ) Dissimilarity MeasureLinkage Criteria A measure of distance between pair of observations. A measure of distance between sets of observations as function of dissimilarity measure.

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Linkage-based clustering Linkage Based: O(n 2 ) Euclidean distance Minkowski distance (generalized Euclidean) Mahalahobis distance (scaled Euclidean) City block distance ( |*| version of Euclidean) Dissimilarity MeasureLinkage Criteria

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Linkage-based clustering Linkage Based: O(n 2 ) Dissimilarity MeasureLinkage Criteria Between two sets A and B

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Linkage-based clustering Linkage Based: O(n 2 ) Linkage Criteria Between two sets A and B Minimum Spanning Tree Successive Cliques UPGMA tree

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Linkage-based clustering Linkage Based: O(n 2 ) Linkage Criteria Between two sets A and B SLINK [Defays] CLINK [Sibson] UPGMA [Murtagh]

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster as long as there is a link

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Linkage-based clustering Add to cluster when a clique is formed

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Theoretical Foundation Many different clustering techniques: which one to use ? Identify properties that separate input-output behavior of different clustering paradigms: [Ackerman, Ben-David, Loker] 1) Be intuitive and meaningful 2) Differentiate algorithms The properties should

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Theoretical Foundation [Kleinberg, 2002]: abstract properties OR axioms. [Zadeh, Ben-David 2009]: properties to characterize single linkage. [Ackerman, Ben-David, Loker] : properties to characterize all linkage.

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical Clustering function Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition A clustering C is a refinement of clustering C’ if every cluster in C’ is a union of some clusters in C. C C’ Refinement Hierarchical Clustering function Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical A clustering function F(X,d,k): Clustering function Hierarchical 1. Take input X, all data points 2. Distance measure, d 3. Number of clusters desired, k. 4. Output the clusters

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical A clustering function is hierarchical if for all X and all d and every 1<= k <= k’ <= |X| F(X,d,k’) is a refinement of F(X,d,k) Clustering function ABCDEF 6 4 3 2 Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical Clustering function C Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical Clustering function dd’ Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical Clustering function Locality and outer consistency Not axioms Spectral clustering with Ratio region Normalized cut does not satisfy these two. Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Definition Refinement Hierarchical Clustering function Hierarchical

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Theoretical Foundation Properties Extended richness Outer consistency Locality Theorem: The clustering function is linkage based If and only if All of the properties are satisfied. Hierarchical

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Theoretical Foundation Theorem: The clustering function is linkage based If and only if All of the properties are satisfied. properties are satisfied. Properties Extended richness Outer consistency Locality Hierarchical linkage based function

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Theoretical Foundation Theorem: The clustering function is linkage based If and only if All of the properties are satisfied. linkage based functionproperties are satisfied. Straightforward Properties Extended richness Outer consistency Locality Hierarchical

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Theoretical Foundation Theorem: The clustering function is linkage based If and only if All of the properties are satisfied. properties are satisfied. Properties Extended richness Outer consistency Locality Hierarchical linkage based function

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Theoretical Foundation Properties Extended richness Outer consistency Locality Hierarchical linkage based function A linkage function is a function l : {(X 1,X 2,d) : d is a dissimilarity function over X 1 U X 2 } → R + that satisfies the following: - Representation independent - Monotonic - Any pair of clusters can be made arbitrarily distant

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Theoretical Foundation Properties Extended richness Outer consistency Locality Hierarchical linkage based function Goal Given a clustering function F that satisfies the properties, define a linkage function l so that the linkage-based clustering based on l coincides with F (for every X, d and k). F Linkage-based

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Theoretical Foundation Given a clustering function F that satisfies the properties, define a linkage function l so that the linkage-based clustering based on l coincides with F (for every X, d and k). F Linkage-based Define an operator < F : (A,B,d 1 ) < F (C,D,d 2 ) if there exists d that extends d 1 and d 2 such that when we run F on (A U B U C U D), A and B are merged before C and D. A B C D F(AUBUCUD,d,4) A B C D F(AUBUCUD,d,3)

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Theoretical Foundation Given a clustering function F that satisfies the properties, define a linkage function l so that the linkage-based clustering based on l coincides with F (for every X, d and k). F Linkage-based Define an operator < F : (A,B,d 1 ) < F (C,D,d 2 ) if there exists d that extends d 1 and d 2 such that when we run F on (A U B U C U D), A and B are merged before C and D. A B C D F(AUBUCUD,d,4) A B C D F(AUBUCUD,d,3)

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Theoretical Foundation - Prove that < F can be extended to a partial ordering - Use the ordering to define l Lemma (< F is cycle-free) Given a function F that is hierarchical, local, outer-consistent and satisfies extended richness, there are no (A 1,B 1,d 1 ), (A 2,B 2,d 2 ),… (A n,B n,d n ) so that (A 1,B 1,d 1 ) < F (A 2,B 2,d 2 ) … < F (A n,B n,d n ) and (A 1,B 1,d 1 ) = (A n,B n,d n ) By the above Lemma, the transitive closure of < F is a partial ordering. There exists an order preserving function l that maps pairs of data sets to R (since < F is defined over a countable set) It can be shown that l satisfies the properties of a linkage function.

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