them. We describe the whole procedure of Ward’s method in Algorithm \[alg:wards\].

Let $u_k$ and $v_k$, respectively, be the $k$-th hidden layer unit ($k=1, \cdots , k_0$) and its corresponding feature vector, and let $\{C^{(t)}_m\}$ be the unit set assigned to the $m$-th cluster in the $t$-th iteration ($m=1, \cdots , k_0 -t+1$). Initially, we set $t\gets 1$ and $C^{(1)}_m \gets \{u_m\}$. $(C^{(t-1)}_{m_1}, C^{(t-1)}_{m_2})\gets {\mathop{\rm arg~min}\limits}_{(C^{(t-1)}_{i}, C^{(t-1)}_{j})} \Delta ESS(C^{(t-1)}_{i}, C^{(t-1)}_{j})$, where $$\begin{aligned}
  \Delta ESS(C, C') \equiv \frac{|C||C'|}{|C|+|C'|}\Bigl\|\frac{1}{|C|}\sum_{k: u_k\in C} v_k-\frac{1}{|C'|}\sum_{k: u_k\in C'} v_k\Bigr\|^2.
  \end{aligned}$$ Here, we assume $m_1<m_2$. Update the clusters as follows: $$\begin{aligned}
  C^{(t)}_{m}\gets \left\{ \begin{array}{ll}
  C^{(t-1)}_{m_1}\cup C^{(t-1)}_{m_2} & (m=m_1) \\
  C^{(t-1)}_{m} & (1\leq m\leq m_2-1, m\neq m_1) \\
  C^{(t-1)}_{m+1} & (m_2\leq m \leq k_0 -t+1) \\
  \end{array} \right. .
  \end{aligned}$$

This procedure to combine a pair of clusters is repeated until all the hidden layer units are assigned to one cluster, and from the clustering result $\{C^{(t)}_m\}$ in each iteration $t=1, \cdots , k_0-1$, we can obtain a hierarchical modular representation of an LNN, which connects the two extreme resolutions given by “all units are in a single cluster” and “all clusters consist of a single unit.” The role of each extracted cluster can be determined from the 