Question

What's the math behind divisive clustering?

Answer 1

Divisive Clustering :

Divisive Clustering is a top-down approach.There is no requirement for this algorithm to pre-specify the number of clusters.A method is required for splitting the cluster that contains the whole data, and this process of splitting clusters occurs recursively until each document is in its own singleton cluster.

-run k-means algorithm on the original data x₁...x_n

-for each of the resulting clusters c_i : i = 1...k

• recursively run k-means on points in c_i

Algorithm :
Given a dataset  (x₁...x_n) of size n
all data is stored in the cluster at the top
this cluster is split using a flat clustering method such as k-means

repeat :
choosing the best cluster among all the clusters
splitting that cluster using flat clustering algorithm untill each individual data has been splitted into a singleton cluster

end

Efficiency :Recursive calls operate on a slice (O(knd log_kn) where k is the branching factor, but divisive clustering can be made more efficient( with fixed number of top levels using flat algorithms like k-means linear time complexity can be achieved) if we do not generate a whole hierarchy all the way down to the individual data leaves.