1. X-means: Extending K-means with Efficient Estimation of the Number of Clusters
Dan Phelleg, Andrew Moore
Carnegie Mellon University
Published: ICML 2000
Presentation by:
Payam Refaeilzadeh

2. Problems with K-means Need to know K
Searching for K is expensive
Even K-means with fixed-K scales poorly
Need to calculate the distance from each point to each centroid to find new cluster assignments

3. Remedies
Forward search for the appropriate value of k in a given range
Recursively split each cluster and use BIC score to decide if we should keep each split
Use kd-trees to accelerate individual rounds of K-means

4. Splitting Use local BIC score to decide on keeping a split
Use global BIC score to decide which K to output at the end

5. BIC (Bayesian Information Criterion)
Adjusted Log-likelihood of the model.
The likelihood that the data is “explained by” the clusters according to the spherical-Gaussian assumption of k-means

6. Kd-trees
Points to be clustered are put into a binary hierarchical structure
Each node represents a subset of points and stores
The minimal hyper-rectangle enclosing all points in the subset
The vector-sum of all the points in the subset
The number of points in the subset

7. Using kd-trees
For each centroid store a counter containing the vector sum of all the points belonging to it and the number of points
Update the above by scanning the kd-tree only once
Start with the root node and all centroids
As you walk down the tree centroids start to get black-listed (when the points in that node could not possibly belong to a centroid)
When only one centroid remains, the counter for that centroid can be updated using the statistics stored in the node
At the end of the scan we have enough info to recalculate the centroid coordinates

8. Results