Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 7 / c

Combining the first two parts gives the centroid sum-of-squares identityThe added term is nonnegative and, since $n_k>0$, vanishes exactly at $z={\bar{x}}_k$. Thus the unique minimizing centroid in this step of K-means clustering isAn empty cluster needs a separate algorithmic convention because its sample mean is undefined and its objective is identically zero.