$w_{2l}$ obeys ($k=1, \cdots, M, l=1, \cdots, N$) (Fig. \[fig3\]).

Analysis
--------

This subsection presents an analysis of the initial loss landscape of the Kawaguchi model initialized by the even initialization method described in Section \[even\_weight\].

We prove in the following that the two assumptions A1p-m and A5u-m (A5u-m-1) are satisfied in the Kawaguchi model initialized by the even initialization.

\[thm1\] For any training pattern $i$, any output neuron $j$, and any path $p$ from an input neuron to the output neuron $j$ in the Kawaguchi model initialized by the even initialization method, $$\label{eqn4-1}
P\left([Z_i]_{(j,p)} = 1 \right) = \frac{1}{2^H}$$ where $H$ is the number of hidden layers ($H \geq 1$). [$\Box$]{}

[*Proof*]{}.   Denote by $j_1, \cdots, j_H$ the hidden neurons on path $p$ where $j_k$ is the hidden neuron in the $k$-th hidden layer $(k=1, \cdots, H$) (Fig. \[fig4\]). Then, $$\begin{aligned}
\label{eqn4-2}
P\Big( [Z_i]_{(j,p)} = 1 \Big) 
&=& P\Big( \mbox{Net input to the hidden neuron}\ j_k > 0 \ \ 
(k=1, \cdots, H) \Big) \nonumber\\
&=& P \left( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0, \left[ \varphi( U_1^{(1)} ) \cdots \varphi(U_{d_1}^{(1)} ) \right] 
{\mbox{\boldmath $w$}}_{j_2} > 0, \right. \nonumber\\ 
& & \left. 
\cdots, \left[ \varphi( U_1^{(H-1)} ) \cdots \varphi(U_{d_{H-1}}^{(H-1)}) \right] 
