On the quality of k-means clustering based on grouped data

Let us have a probability distribution P (possibly empirical) on the real line R . Consider the problem of finding the k-mean of P, i.e. a set A of at most k points that minimizes given loss-function. It is known that the k-mean can be found using an iterative algorithm by Lloyd [1982. Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 129–136]. However, depending on the complexity of the distribution P, the application of this algorithm can be quite resource-consuming. One possibility to overcome the problem is to group the original data and calculate the k-mean on the basis of the grouped data. As a result, the new k-mean will be biased, and our aim is to measure the loss of the quality of approximation caused by such approach.