H+1$. Let ${\mbox{\boldmath $\Theta$}}$ denote the one-dimensional vector which consists of all the weight parameters of the deep nonlinear neural network.

Kawaguchi specifically examined a path from an input neuron to an output neuron of the deep nonlinear neural network (Fig. \[fig2\]), and expressed the actual output of output neuron $j$ of the output layer of the deep nonlinear neural network for the $i$-th input training pattern ${\mbox{\boldmath $X$}}_i \in {\mbox{\boldmath $R$}}^{d_x}$ as $$\begin{aligned}
\label{eqn2-1}
\hat{{\mbox{\boldmath $Y$}}}_i({\mbox{\boldmath $\Theta$}}, {\mbox{\boldmath $X$}}_i)_j = q\sum_{p=1}^\Psi [{\mbox{\boldmath $X$}}_i]_{(j,p)} [{\mbox{\boldmath $Z$}}_i]_{(j,p)} 
\prod_{k=1}^{H+1} w_{(j,p)}^{(k)} \in {\mbox{\boldmath $R$}}\end{aligned}$$ where $\Psi$ represents the total number of paths from the input layer to output neuron $j$, $[{\mbox{\boldmath $X$}}_i]_{(j,p)} \in {\mbox{\boldmath $R$}}$ denotes the component of the $i$-th input training pattern ${\mbox{\boldmath $X$}}_i \in {\mbox{\boldmath $R$}}^{d_x}$ that is used in the $p$-th path to the $j$-th output neuron. Also, $[{\mbox{\boldmath $Z$}}_i]_{(j,p)} \in \{0, 1\}$ represents whether the $p$-th path to the output neuron $j$ is active or not for each training pattern $i$ as a result of ReLU activation. $[{\mbox{\boldmath $Z$}}_i]_{(j,p)} = 1$ means that the path is active, and $[{\mbox{\boldmath $Z$}}_i]_{(j,p)} = 0$ means that the path is inactive. $w_{(j,p)}^{(k)} \in {\mbox{\boldmath $R$}}$ is the component of the weight 