Hierarchical Clustering Lecture 15 David Sontag New York - PowerPoint PPT Presentation

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Hierarchical Clustering Lecture 15 David Sontag New York University Agglomerative Clustering Agglomerative clustering: First merge very similar instances Incrementally build larger clusters out

Hierarchical ¡Clustering ¡ Lecture ¡15 ¡ David ¡Sontag ¡ New ¡York ¡University ¡
Agglomerative Clustering • Agglomerative clustering: – First merge very similar instances – Incrementally build larger clusters out of smaller clusters • Algorithm: – Maintain a set of clusters – Initially, each instance in its own cluster – Repeat: • Pick the two closest clusters • Merge them into a new cluster • Stop when there’s only one cluster left • Produces not one clustering, but a family of clusterings represented by a dendrogram
Agglomerative Clustering • How should we define “ closest ” for clusters with multiple elements?
Agglomerative Clustering • How should we define “ closest ” for clusters with multiple elements? • Many options: – Closest pair (single-link clustering) – Farthest pair (complete-link clustering) – Average of all pairs • Different choices create different clustering behaviors
Agglomerative Clustering • How should we define “ closest ” for clusters with multiple elements? Closest pair Farthest pair (single-link clustering) (complete-link clustering) 1 5 6 2 1 5 2 6 3 4 7 8 3 4 7 8 [Pictures from Thorsten Joachims]
Clustering ¡Behavior ¡ Average Farthest Nearest Mouse tumor data from [Hastie et al. ]
Agglomera<ve ¡Clustering ¡ When ¡can ¡this ¡be ¡expected ¡to ¡work? ¡ Strong separation property: Closest pair All points are more similar to points in (single-link clustering) their own cluster than to any points in any other cluster Then, the true clustering corresponds to some pruning of the tree obtained by 1 5 6 2 single-link clustering! Slightly weaker (stability) conditions are solved by average-link clustering 3 4 7 8 (Balcan et al., 2008)

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