Lecture 10 : k - means clustering

The sets Sj are the sets of points to which μj is the closest center. In each step of the algorithm the potential function is reduced. Let’s examine that. First, if the set of centers μj are fixed, the best assignment is clearly the one which assigns each data point to its closest center. Also, assume that μ is the center of a set of points S. Then, if we move μ to 1 |S| ∑ i∈S xi then we only reduce the potential. This is because 1 |S| ∑ i∈S xi is the best possible value for μ (can easily be seen by derivation of the cost function). The algorithm therefore terminates in a local minimum. The question of course is whether we can guaranty that the solution is close to optimal and under what computational cost.