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### Clustering

Rong Jin

What is Clustering?

- Identify the underlying structure for given data points
- Doc. clustering: groups documents of same topics into the same cluster

$$$

age

Improve IR by Document Clustering

- Cluster-based retrieval
- Cluster docs in collection a priori
- Only compute the relevance scores for docs in the cluster closest to the query
- Improve retrieval efficiency by only search a small portion of the document collection

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters

x2

x1

x4

x3

x5

x6

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters

x2

x1

x4

x3

x5

x6

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters

x2

x1

x4

x3

C1

C2

x5

x6

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters
- Membership indicators:
mi,j =1 if xi is assigned to Cj, and zero otherwise.

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters
- Membership indicators:
mi,j =1 if xi is assigned to Cj, and zero otherwise.

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters
- Find good clusters by minimizing the cluster compactness
- Cluster centers C1 and C2
- Membership mi,j

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Measure the compactness by the sum of distance square within clusters
- Find good clusters by minimizing the cluster compactness
- Cluster centers C1 and C2
- Membership mi,j

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Find good Clusters?

- Find good clusters by minimizing the cluster compactness
- Cluster centers C1 and C2
- Membership mi,j

x2

x1

C1

x4

x3

x5

x6

C2

x7

How to Efficiently Cluster Data?

Update mi,j: assign xi to the closest Cj

How to Efficiently Cluster Data?

Update mi,j: assign xi to the closest Cj

Update Cj as the average of xi assigned to Cj

How to Efficiently Cluster Data?

Update mi,j: assign xi to the closest Cj

K-means algorithm

Update Cj as the average of xi assigned to Cj

Example of k-means

- Identify the points that are closer to C2 than C1, and points that are closer to C1 than to C2

x2

x1

x4

x3

C1

x5

x6

C2

x7

Example of k-means

- Identify the points that are closer to C2 than C1, and points that are closer to C1 than to C2
- Update C1 and C2

x2

x1

C1

x4

x3

x5

x6

C2

x7

K-means for Clustering

- K-means
- Start with a random guess of cluster centers
- Determine the membership of each data points
- Adjust the cluster centers

K-means for Clustering

- K-means
- Start with a random guess of cluster centers
- Determine the membership of each data points
- Adjust the cluster centers

K-means for Clustering

- K-means
- Start with a random guess of cluster centers
- Determine the membership of each data points
- Adjust the cluster centers

K-means

- Ask user how many clusters they’d like. (e.g. k=5)

K-means

- Ask user how many clusters they’d like. (e.g. k=5)
- Randomly guess k cluster Center locations

K-means

- Ask user how many clusters they’d like. (e.g. k=5)
- Randomly guess k cluster Center locations
- Each datapoint finds out which Center it’s closest to. (Thus each Center “owns” a set of datapoints)

K-means

- Ask user how many clusters they’d like. (e.g. k=5)
- Randomly guess k cluster Center locations
- Each datapoint finds out which Center it’s closest to.
- Each Center finds the centroid of the points it owns

K-means

- Ask user how many clusters they’d like. (e.g. k=5)
- Randomly guess k cluster Center locations
- Each datapoint finds out which Center it’s closest to.
- Each Center finds the centroid of the points it owns

K-means

Any Computational Problem ?

K-means

Need to go through each data point at each iteration of k-means

Improve K-means

- Group nearby data points by region
- KD tree
- SR tree

- Try to update the membership for all the data points in the same region

Improved K-means

- Find the closest center for each rectangle
- Assign all the points within a rectangle to one cluster

A Mixture Model for Document Clustering

- Assume that data are generated from a mixture of multinomial distributions
- Estimate the mixture distribution from the observed documents

Gaussian Mixture Example: Start

Measure the probability for every data point to be associated with each cluster

Hierarchical Doc Clustering

- Goal is to create a hierarchy of topics
- Challenge: create this hierarchy automatically
- Approaches: top-down or bottom-up

Hierarchical Agglomerative Clustering (HAC)

- Given a similarity measure for determining the similarity between two clusters
- Start with each document in a separate cluster
- repeatedly merge the two most similar clusters
- Until there is only one cluster
- The history of merging forms a binary tree
- The standard way of depicting this history is a dendrogram.

An Example of Dendrogram

similarity

With an appropriately chosen similarity cut, we can convert the dendrogram into a flat clustering.

Similarity of Clusters

- Single-link: Maximum similarity
- Maximum over all document pairs

Similarity of Clusters

- Single-link: Maximum similarity
- Maximum over all document pairs

- Complete-link: Minimum similarity
- Minimum over all document pairs

Similarity of Clusters

- Single-link: Maximum similarity
- Maximum over all document pairs

- Complete-link: Minimum similarity
- Minimum over all document pairs

- Centroid: Average “intersimilarity”
- Average over all document pairs

Single Link vs. Complete Link

Single Link

Complete Link

- Complete link usually produces balanced clusters

Divisive Hierarchical Clustering

- Top-down (instead of bottom-up as in HAC)
- Start with all docs in one big cluster
- Then recursively split clusters
- Eventually each node forms a cluster on its own.
- Example: Bisecting K-means

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