If you have a set of $n$ observations or data represented with $\{x_1,x_2,\dots ,x_n\}$, and each data includes $d$ features and no labels, k-mean clustering provides a simple method to group the whole data set into k clusters $\{c_1,c_2,\dots ,c_k\}$, where data in the same cluster are more similar to each other.

Each data ($x_{\mathrm{i}}$) is labeled with the cluster $\mathrm{j}(1\dots k)$, where the distance of the data point $i$ to the centroid of cluster $j$ is minimum compare to all other clusters.

The objective is $\min \sum _{i=1}^k\sum _{x\in C}{\Vert x-{\mu }_i\Vert }^2$, where ${\mu }_i$ is the mean of the data points in cluster $i$.

Different methods for initializing this unsupervised clustering algorithm exist, and this initialization is important for reducing the complexity of the algorithm (the number of iterations required) and the results of clustering (local optimum).

Typically, if the initial centroids are closer to the final centroid, you can achieve better clustering results and lower complexity. This application note does not cover initialization methods.