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### Semi-Supervised Clustering

Jieping Ye

Department of Computer Science and Engineering

Arizona State University

http://www.public.asu.edu/~jye02

Outline

- Overview of clustering and classification
- What is semi-supervised learning?
- Semi-supervised clustering
- Semi-supervised classification
- Semi-supervised clustering
- What is semi-supervised clustering?
- Why semi-supervised clustering?
- Semi-supervised clustering algorithms

Supervised classification versus unsupervised clustering

- Unsupervised clustering Group similar objects together to find clusters
- Minimize intra-class distance
- Maximize inter-class distance
- Supervised classification Class label for each training sample is given
- Build a model from the training data
- Predict class label on unseen future data points

Inter-cluster distances are maximized

Intra-cluster distances are minimized

What is clustering?- Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups

Clustering algorithms

- K-Means
- Hierarchical clustering
- Graph based clustering (Spectral clustering)
- Bi-clustering

Classification algorithms

- K-Nearest-Neighbor classifiers
- Naïve Bayes classifier
- Linear Discriminant Analysis (LDA)
- Support Vector Machines (SVM)
- Logistic Regression
- Neural Networks

learning

Supervised

classification

Unsupervised

clustering

Semi-Supervised Learning- Combines labeled and unlabeled data during training to improve performance:
- Semi-supervised classification: Training on labeled data exploits additional unlabeled data, frequently resulting in a more accurate classifier.
- Semi-supervised clustering: Uses small amount of labeled data to aid and bias the clustering of unlabeled data.

Semi-Supervised Classification

- Algorithms:
- Semisupervised EM [Ghahramani:NIPS94,Nigam:ML00].
- Co-training [Blum:COLT98].
- Transductive SVM’s [Vapnik:98,Joachims:ICML99].
- Graph based algorithms
- Assumptions:
- Known, fixed set of categories given in the labeled data.
- Goal is to improve classification of examples into these known categories.

Semi-supervised clustering: problem definition

- Input:
- A set of unlabeled objects, each described by a set of attributes (numeric and/or categorical)
- A small amount of domain knowledge
- Output:
- A partitioning of the objects into k clusters (possibly with some discarded as outliers)
- Objective:
- Maximum intra-cluster similarity
- Minimum inter-cluster similarity
- High consistency between the partitioning and the domain knowledge

Why semi-supervised clustering?

- Why not clustering?
- The clusters produced may not be the ones required.
- Sometimes there are multiple possible groupings.
- Why not classification?
- Sometimes there are insufficient labeled data.
- Potential applications
- Bioinformatics (gene and protein clustering)
- Document hierarchy construction
- News/email categorization
- Image categorization

Semi-Supervised Clustering

- Domain knowledge
- Partial label information is given
- Apply some constraints (must-links and cannot-links)
- Approaches
- Search-based Semi-Supervised Clustering
- Alter the clustering algorithm using the constraints
- Similarity-based Semi-Supervised Clustering
- Alter the similarity measure based on the constraints
- Combination of both

Search-Based Semi-Supervised Clustering

- Alter the clustering algorithm that searches for a good partitioning by:
- Modifying the objective function to give a reward for obeying labels on the supervised data [Demeriz:ANNIE99].
- Enforcing constraints (must-link, cannot-link) on the labeled data during clustering [Wagstaff:ICML00, Wagstaff:ICML01].
- Use the labeled data to initialize clusters in an iterative refinement algorithm (kMeans,) [Basu:ICML02].

Overview of K-Means Clustering

- K-Means is a partitional clustering algorithm based on iterative relocation that partitions a dataset into K clusters.

Algorithm:

Initialize K cluster centers randomly. Repeat until convergence:

- Cluster Assignment Step: Assign each data point x to the cluster Xl, such that L2 distance of x from (center of Xl) is minimum
- Center Re-estimation Step: Re-estimate each cluster center as the mean of the points in that cluster

K-Means Objective Function

- Locally minimizes sum of squared distance between the data points and their correspondingcluster centers:
- Initialization of K cluster centers:
- Totally random
- Random perturbation from global mean
- Heuristic to ensure well-separated centers

Semi-Supervised K-Means

- Partial label information is given
- Seeded K-Means
- Constrained K-Means
- Constraints (Must-link, Cannot-link)
- COP K-Means

Semi-Supervised K-Means for partially labeled data

- Seeded K-Means:
- Labeled data provided by user are used for initialization: initial center for cluster i is the mean of the seed points having label i.
- Seed points are only used for initialization, and not in subsequent steps.
- Constrained K-Means:
- Labeled data provided by user are used to initialize K-Means algorithm.
- Cluster labels of seed data are kept unchanged in the cluster assignment steps, and only the labels of the non-seed data are re-estimated.

Seeded K-Means

Use labeled data to find

the initial centroids and

then run K-Means.

The labels for seeded

points may change.

Constrained K-Means

Use labeled data to find

the initial centroids and

then run K-Means.

The labels for seeded

points will not change.

COP K-Means

- COP K-Means [Wagstaff et al.: ICML01] is K-Means with must-link (must be in same cluster) and cannot-link (cannot be in same cluster) constraints on data points.
- Initialization: Cluster centers are chosen randomly, but as each one is chosen any must-link constraints that it participates in are enforced (so that they cannot later be chosen as the center of another cluster).
- Algorithm: During cluster assignment step in COP-K-Means, a point is assigned to its nearest cluster without violating any of its constraints. If no such assignment exists, abort.

Similarity-based semi-supervised clustering

- Alter the similarity measure based on the constraints
- Paper: From Instance-level Constraints to Space-Level Constraints: Making the Most of Prior Knowledge in Data Clustering. D. Klein et al.

Two types of constraints: Must-links and Cannot-links

Clustering algorithm: Hierarchical clustering

Overview of Hierarchical Clustering Algorithm

- Agglomerative versusDivisive
- Basic algorithm of Agglomerative clustering
- Compute the distance matrix
- Let each data point be a cluster
- Repeat
- Merge the two closest clusters
- Update the distance matrix
- Until only a single cluster remains
- Key operation is the update of the distance between two clusters

p2

p3

p4

p5

. . .

p1

p2

p3

p4

p5

.

.

.

How to Define Inter-Cluster DistanceDistance?

- MIN
- MAX
- Group Average
- Distance Between Centroids

distance matrix

Must-link constraints

- Distance between must-links pair to zero.
- Derive a new metric by running an all-pairs-shortest distances algorithm.
- It is still a metric
- Faithful to the original metric
- Computational complexity: O(N2C)
- C: number of points involved in must-link constraints
- N: total number of points

p2

p3

p4

p5

. . .

p1

p2

p3

p4

p5

.

.

.

New distance matrix based on must-link constraintsHierarchical clustering can be carried out based on the new distance matrix.

What is missing?

New distance matrix

Cannot-link constraint

- Run hierarchical clustering with complete link (MAX)
- The distance between two clusters is determined by the largest distance.
- Set the distance between cannot-link pair to be
- The new distance matrix does not define a metric.
- Work very well in practice

Constrained complete-link clustering algorithm

Derive a new distance

Matrix based on both

Types of constraints

Hierarchical clustering

1 and 2 form a cluster, and 3 and 4 form another cluster

1,2

3,4

5

1,2

3,4

1 2 3 4 5

5

Summary

- Seeded and Constrained K-Means: partially labeled data
- COP K-Means: constraints (Must-link and Cannot-link)
- Constrained K-Means and COP K-Means require all the constraints to be satisfied.
- May not be effective if the seeds contain noise.
- Seeded K-Means use the seeds only in the first step to determine the initial centroids.
- Less sensitive to the noise in the seeds.
- Semi-supervised hierarchical clustering

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