Given a cluster mean \( \boldsymbol{m_k} \) we define the total cluster variance
$$ \begin{equation}\tag{5} \min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2 \end{equation} $$Now we have all the pieces necessary to formally revisit the \( k \)-means algorithm.