assumptions A1p and A5u in (Choromanska, Henaff, Mathieu, Arous & LeCun, 2015), respectively. Assumption A5u-m is used for the proof of Corollary 3.2 presented by Kawaguchi (2016). Strictly speaking, the following assumption A5u-m-1 suffices for the proof instead of the assumption A5u-m described above

[**A5u-m-1**]{} [ *For any $i$ and any $(j,p)$, $[{\mbox{\boldmath $Z$}}_i]_{(j,p)} \in \{0, 1\}$ is independent of the $i$-th input training pattern ${\mbox{\boldmath $X$}}_i \in {\mbox{\boldmath $R$}}^{d_x}$ and the sequence of the weights on the $p$-th path $\{ w_{(j,p)}^{(k)} \in {\mbox{\boldmath $R$}}\}_{k=1}^{H+1}$ where $w_{(j,p)}^{(k)}$ is the weight between the layer $k-1$ and the layer $k$ on the $p$-th path ($k=1, \cdots, H+1$).* ]{}

Actually, according to assumption A5u-m-1, $$\begin{aligned}
\label{eqn2-3}
& & E_{{\mbox{\boldmath $Z$}}} \left[\hat{{\mbox{\boldmath $Y$}}}_i({\mbox{\boldmath $\Theta$}}, {\mbox{\boldmath $X$}}_i)_j \right] \nonumber\\
&=& E_{{\mbox{\boldmath $Z$}}} \left[q\sum_{p=1}^\Psi [{\mbox{\boldmath $X$}}_i]_{(j,p)} [{\mbox{\boldmath $Z$}}_i]_{(j,p)} 
\prod_{k=1}^{H+1} w_{(j,p)}^{(k)} \right] \hspace*{1cm} \mbox{(from Eq.} (\ref{eqn2-1})) \nonumber\\
&=& q\sum_{p=1}^\Psi [{\mbox{\boldmath $X$}}_i]_{(j,p)} E_{{\mbox{\boldmath $Z$}}} \left[ [{\mbox{\boldmath $Z$}}_i]_{(j,p)} \right] 
\prod_{k=1}^{H+1} w_{(j,p)}^{(k)}.  \hspace*{1cm} \mbox{(from the assumption A5u-m-1) }
$$

Even Initialization {#even_weight}
-------------------

This subsection proposes a new weight initialization method for the Kawaguchi model described in the subsection \[kawa-model\]. We assume that the width of the deep nonlinear neural network is sufficiently large, that is, $d_0 (=d_x), d_1, \cdots, 