Selected Applications of Convex Optimization by Li Li

Author:Li Li
Language: eng
Format: epub, pdf
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

3.6 K-Means Clustering

Finally, we would like to introduce K-means clustering algorithm that is widely used for clustering [15].

Suppose we have a data set consisting of n observations of Euclidean variable . Our goal is to partition the data set into K clusters. That is, K-means clustering can be viewed as a special GMM model: in which we first assign the data points to clusters and then perform a maximum likelihood estimation of the mean of spherical Gaussians clusters (all of which have the same covariance matrix equal to a scalar multiple of the identity).

Intuitively, we assume a cluster is comprised by a group of data points whose interpoint distances are smaller than the distances to points outside of this cluster.

We can formalize this notion by first introducing a set of d-dimensional Euclidean vectors , where , in which is the prototype associated with the kth cluster. Our goal is then to find a partition of data points, as well as a set of vectors , such that the sum of the squares of the distances of each data point to its closest vector μ k is a minimum. This leads to the following optimization problem

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