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Figure 22.1: K-Means Clustering "hard" boundary locations, with feature vector centroids
marked as white crosses
The goal of K-Means Clustering is to minimise the Within-Cluster Variation (WCV), also
known as the Within-Cluster Sum of Squares (WCSS). This concept represents the sum across
clusters of the sum of distances to each point in the cluster to its mean. That is, it measures how
much observations within a cluster differ from each other. This translates into an optimisation
problem, the goal of which is to minimise the following expression:
arg min
S
K∑
k=1
∑
x i∈S k
‖x i − µ k ‖ (22.1)
Where µ k represents the mean feature vector of cluster k and x i is the ith feature vector in
cluster k.
Unfortunately this particular minimisation is difficult to solve globally. That is, finding a
global minimum to this problem is NP-hard, in the complexity sense.
Fortunately however there exists useful heuristic algorithms for finding acceptable local optima,
one of which will be outlined below.
22.1.1 The Algorithm
The heuristic algorithm to solve K-Means Clustering is, understandably, known as the K-Means
Algorithm. It is relatively straightforward to conceptualise. It consists of two steps, the second
of which is iterated until completion:
1. Assign each observation x i to a random cluster k.