K-Means Clustering Algorithm

This post is dedicated to K-Means Clustering Algorithm, used for unsupervised learning. Here is a short introduction into the unsupervised learning subject. This video shows the algorithm at work.

So, how does it work?

1. First of all, we have the number K (the number of clusters) and the data set X₁,...,X_m as inputs, K < m and X_j ∈ ℜⁿ for ∀j=1..m. Considering that clustering is used as a task in data mining (e.g., including text), it is a nice exercise indeed to formalise a problem in a way where data is represented by vectors :)
2. At this step, we randomly initialise K cluster centroids μ₁,...,μ_K, μ_j ∈ ℜⁿ for ∀j=1..K. The recommended way is to randomly initialise the centroids from the data set X₁,...,X_m and make sure that μ_i≠μ_j for ∀ i≠j.
3. At this step, we execute the loop:

   Repeat {
     //cluster assignment step
     For i=1..m {
       Find the closest centroid for X<sub>i</sub>, i.e.
       min{||X<sub>i</sub>-μ<sub>j</sub>||}, ∀j=1..K, e.g. it is μ<sub>t</sub>;
       Assign c<sub>i</sub>=t;
       //in this case c<sub>i</sub> is the index of 
       //the closest centroid for X<sub>i</sub>
     }
   
     //move centroids step, if a particular cluster 
     //centroid has no points assigned to it, 
     //then eliminate it
     For j=1..K {
       old_μ<sub>j</sub> = μ<sub>j</sub>;
       μ<sub>j</sub> = average (mean) of all X<sub>i</sub> where c<sub>i</sub>=j, ∀i=1..m;
     }
   }

This loop ends when all the centroids stop "moving", i.e., ||old_μ_j - μ_j|| < ε, ∀j=1..K, where ε is an error we are happy with (e.g. 0.01 or 0.001).

This is pretty much the algorithm. However, in this form, there is a risk to get stuck in a local minima. By local minima, I mean the local minima of the cost function:
J(X₁,...,X_m,c₁,...c_m,μ₁,...,μ_K) = (1 ⁄ m)⋅∑||X_i – μ_ci||², sum is for i=1..m

In order to address this, the algorithm (steps 2, 3) are executed several times (e.g. 100). Each time the cost function (J(...)) is computed and the clustering with the lowest J(...) is picked up. E.g.

For p=1..100 {
  Perform step 2;
  Perform step 3;
  Calculate cost function J(...);
}
Pick up clustering with the lowest cost J(...);

In order to make it more interactive, I and my son spent couple of hours implementing a JavaScript version of this algorithm using <canvas> tag from HTML5 (my son studies HTML at school, so it was an interesting exercise). Here is the link to the page (excuse me and my son for our non OOP approach to JavaScript :)). If you want to have a look at the sources, feel free to download that page (we didn't provide comments, but hopefully this article explains everything). We tested it with Google Chrome and Internet Explorer 9 (if you have problems with IE9, please consult the following link).

Note: While I was writing this article, a friend of mine suggested this nice JavaScript library for plotting www.jqplot.com.