This document describes a MapReduce algorithm for determining the optimal k value in k-means clustering. It presents the G-means algorithm, which uses recursive k-means clustering and normality testing to split clusters until all points are normally distributed around cluster centers. The document outlines challenges in implementing G-means in MapReduce, and describes solutions to reduce I/O, jobs, maximize parallelism and limit memory usage. It compares the proposed MapReduce G-means approach to existing multi-k-means methods, finding it has better quality and comparable speed on synthetic datasets.

1. Determining the k in k-means
with MapReduce
Thibault Debatty, Pietro Michiardi,
Wim Mees & Olivier Thonnard
Algorithms for MapReduce and Beyond 2014

2. Determining the k in k-means with MapReduce 2
Clustering & k-means
● Clustering
● K-means
[Stuart P. Lloyd. Least squares quantization in pcm. IEEE
Transactions on Information Theory, 28:129–137, 1982.]
– 1982 (a great year!)
– But still largely used
– Drawbacks (amongst others):
● Local minimum
● K is a parameter!

3. Determining the k in k-means with MapReduce 3
Clustering & k-means
● Determine k:
– VERY difficult
[Anil K Jain. Data Clustering : 50 Years Beyond K-Means.
Pattern Recognition Letters, 2009]
– Using cluster evaluation metrics:
Dunn's index, Elbow, Silhouette, “jump method” (based on
information theory), “Gap statistic”,...
O(k²)

4. Determining the k in k-means with MapReduce 4
G-means
● G-means
[Greg Hamerly and Charles Elkan. Learning the k in k-
means. In Neural Information Processing Systems. MIT
Press, 2003]
● K-means : points in each cluster are
spherically distributed around the center
Source:scikit-learn

5. Determining the k in k-means with MapReduce 5
G-means
● G-means
[Greg Hamerly and Charles Elkan. Learning the k in k-
means. In Neural Information Processing Systems. MIT
Press, 2003]
● K-means : points in each cluster are
spherically distributed around the center
normality test &
recursion

6. Determining the k in k-means with MapReduce 6
G-means
Dataset

7. Determining the k in k-means with MapReduce 7
G-means
1. Pick 2 centers

8. Determining the k in k-means with MapReduce 8
G-means
2. k-means

9. Determining the k in k-means with MapReduce 9
G-means
3. Project

10. Determining the k in k-means with MapReduce 10
G-means
3. Project

11. Determining the k in k-means with MapReduce 11
G-means
Normal?
No
=> recursion
4. Normality test

12. Determining the k in k-means with MapReduce 12
G-means
5. Recursion

13. Determining the k in k-means with MapReduce 13
MapReduce G-means
● Challenges:
1. Reduce I/O operations
2. Reduce number of jobs
3. Maximize parallelism
4. Limit memory usage

14. Determining the k in k-means with MapReduce 14
MapReduce G-means
● Challenges:
1. Reduce I/O operations
2. Reduce number of jobs
3. Maximize parallelism
4. Limit memory usage

15. Determining the k in k-means with MapReduce 15
MapReduce G-means
2. Reduce number of jobs
PickInitialCenters
while Not ClusteringCompleted do
KMeans
KMeansAndFindNewCenters
TestClusters
end while

16. Determining the k in k-means with MapReduce 16
MapReduce G-means
TestClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Emit(cluster, projection)
end procedure
Reduce(cluster, projections)
Build a vector
ADtest(vector)
if normal then
Mark cluster
end if
end procedure
3. Maximize
parallelism
4. Limit memory
usage

17. Determining the k in k-means with MapReduce 17
MapReduce G-means
TestClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Emit(cluster, projection)
end procedure
Reduce(cluster, projections)
Build a vector
ADtest(vector)
if normal then
Mark cluster
end if
end procedure
Bottleneck
3. Maximize
parallelism
4. Limit memory
usage (risk of crash)

18. Determining the k in k-means with MapReduce 18
MapReduce G-means
TestClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Emit(cluster, projection)
end procedure
Reduce(cluster, projections)
Build a vector
ADtest(vector)
if normal then
Mark cluster
end if
end procedure
TestFewClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Add projection to list
end procedure
Close()
For each list do
Build a vector
A2 = ADtest(vector)
Emit(cluster, A2)
End for each
end procedure
In memory combiner

19. Determining the k in k-means with MapReduce 19
MapReduce G-means
TestClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Emit(cluster, projection)
end procedure
Reduce(cluster, projections)
Build a vector
ADtest(vector)
if normal then
Mark cluster
end if
end procedure
TestFewClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Add projection to list
end procedure
Close()
For each list do
Build a vector
A2 = ADtest(vector)
Emit(cluster, A2)
End for each
end procedure
#clusters > #reducers
&
Estimated required memory < Java heap

20. Determining the k in k-means with MapReduce 20
MapReduce G-means
TestClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Emit(cluster, projection)
end procedure
Reduce(cluster, projections)
Build a vector
ADtest(vector)
if normal then
Mark cluster
end if
end procedure
TestFewClusters
Map(key, point)
Find cluster
Find vector
Project point on vector
Add projection to list
end procedure
Close()
For each list do
Build a vector
A2 = ADtest(vector)
Emit(cluster, A2)
End for each
end procedure
#clusters > #reducers
&
Estimated required memory < Java heap
Experimentally:
64 Bytes / point

21. Determining the k in k-means with MapReduce 21
Comparison
MR multi-k-means MR G-means
Speed
Quality
all possible values of k
in a single job

22. Determining the k in k-means with MapReduce 22
Comparison
MR multi-k-means MR G-means
Speed O(nk²) computations O(nk) computations
But:
● more iterations
● more dataset reads
● log2
(k)
Quality New centers added if and
where needed
But:
tends to overestimate k!

23. Determining the k in k-means with MapReduce 23
Experimental results : Speed
● Hadoop
● Synthetic dataset
● 10M points in R10
● Euclidean distance
● 8 machines

24. Determining the k in k-means with MapReduce 24
Experimental results : Quality
● Hadoop
● Synthetic dataset
● 10M points in R10
● Euclidean distance
● 8 machines
k 100 200 400
kfound
150 279 639
Within Cluster Sum of Square
(less is better)
MR G-means 3.34 3.33 3.23
multi-k-means 3.71 3.6 3.39
(with same k)
x ~1.5

25. Determining the k in k-means with MapReduce 25
Conclusions & future work...
● MapReduce algorithm to determine k
● Running time proportional to k
● Future:
– Overestimation of k
– Test on real data
– Test scalability
– Reduce I/O (using Spark)
– Consider skewed data
– Consider impact of machine failure

26. Determining the k in k-means with MapReduce 26
Thank you!