Integrating Constraints and Metric Learning in Semi-Supervised Clustering

Mikhail Bilenko, Sugato Basu, Raymond J. Mooney

ICML 2004

Presented by Xin Li

Semi-Supervised Clustering

K=4

Semi-Supervised Clustering

Semi-Supervised Clustering

How to exploit supervision in clustering

Incorporate supervision as constraints Learn a distance metric using

supervision Integration of these two approaches

K-means Clustering

X = {x1,x2,…}

L = {l1,l2,…,lk}

Euclidean Distance:

Minimizing:

Clustering with constraints

Pairwise constraints: M – Must-link pairs

(xi, xj) should be in the same cluster

C -- Cannot-link pairs (xi, xj) should be in different

clusters

Learning a pairwise distance metricBinary Classification: (xi, xj) 0/1 M positive examples

(xi, xj) are the same cluster

C negative examples (xi, xj) are in different clusters

Apply the learned distance metric in clustering Metric learning and clustering are disjointed

Unsupervised Clustering with Metric Learning

Maximizing the complete data log-likelihood under generalized K-means

Learn a distance metric that optimize a quality function

Integrating Constraints and Metric Learning

Combining the previous two equations leads to the following objective function that minimizes cluster dispersion under that learned metrics while reducing constraint violations.

Penalty for violating constraints

Penalty for violating a must-link constraints between distant points should be higher than that between nearby points.

Penalty for violating a cannot-link constraints between nearby points should be lower than that between nearby points.

MPCK-MEANS Algorithm

Constraints are utilized during cluster initialization and when assigning points to clusters.

The distance metric is adapted by re-estimating the weights in matrices Ah.

Initialization An initial guess of the clusters. Assign each point x to one of K clusters in a way that satisfies the

constraints. Compute the centroid of each cluster.

E-step

Every point x is assigned to the cluster that minimizes the sum of the distance of x to the cluster centroid according to the local metric and the cost of any constraint violations incurred by the cluster assignment.

M-Step

= 0

Update Metrics:

Experimental Setting

Single Metric, Diagonal Matrix A

Single Metric, Diagonal Matrix A

Multiple Metrics, Full Matrix A

Multiple Metrics, Full Matrix A

Conclusion and Discussion

This paper has presented MPCK-MEANS, a new approach to semi-supervised clustering.

Supervision and metric learning are helpful in clustering and multiple distance metrics are not necessary in most cases.

Question 1: If we have supervision in clustering, why not utilize supervision in the same way as in a typical classification task ?

Question 2: If there are infinite number of classes, can we gain from supervision on part of them ?