I am working on a quantization problem which could be express in these terms :
Given a set of positive reals $\{x_1, x_2,\dots,x_M\}$, I need to find another set $\{y_1,y_2,\dots,y_N\}$ of size $N < M$ such that :
$ \left\{ \begin{array}{c} \forall k\leq M, y_{f(k)}^{2}-x_{k}^{2}\geq0\\ \underset{y,f}{min}\underset{k}{\sum}y_{f(k)}^{2}-x_{k}^{2} \end{array}\right. $
What would be a good/fast algorithmic approach to solve this kind of problem ?