]^T \in {\mbox{\boldmath $R$}}^{d_1}$ such that $U_{j_1}^{(1)} = {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0$ and $w_{j_1 j_0} = \lambda_1$ is active. Also, for any $1 \leq s \leq d_1$, by denoting ${\mbox{\boldmath $X$}}_i = (x_1 \cdots x_{d_0})^T$ and ${\mbox{\boldmath $w$}}_s = (w_1 \cdots w_{d_0})$ for the sake of simplicity, $\vert \varphi(U_s^{(1)}) \vert \leq \vert U_s^{(1)} \vert 
=\vert {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_s \vert 
\leq \sum_{t=1}^{d_0} \vert x_t \vert \vert w_t \vert 
\leq \alpha \sum_{t=1}^{d_0} (1/d_0) = \alpha$. Hence, according to the assumption of mathematical induction, the first term of the right-hand-side of Eq. (\[eqn4-8\]) is nearly equal to $(1/2)^{H-1}$. In addition, the second term of the right-hand-side of Eq. (\[eqn4-8\]) is nearly equal to 1/2 from Eq.(\[eqn4-7\]). So, $$\begin{aligned}
\label{eqn4-9}
&P& \hspace*{-0.3cm} \left( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0, \left[ \varphi( U_1^{(1)} ) 
\cdots \varphi(U_{d_1}^{(1)} ) \right] {\mbox{\boldmath $w$}}_{j_2} > 0, 
\cdots, \left[ \varphi( U_1^{(H-1)} ) \cdots \varphi(U_{d_{H-1}}^{(H-1)}) \right] 
{\mbox{\boldmath $w$}}_{j_H} > 0 \right. \nonumber\\
& & \hspace*{7.0cm} \left. \Big\vert \ w_{j_1 j_0} = \lambda_1, 
\cdots, w_{j_H j_{H-1} } = \lambda_H \right) \nonumber\\
&\fallingdotseq& \left(\frac{1}{2}\right)^{H-1} \cdot \frac{1}{2} \nonumber\\
&=& \frac{1}{2^H}, \end{aligned}$$ which means that the case of $H$ indeed holds. Thus, by mathematical induction, Eq. (\[eqn4-6\]) holds for any $H \geq 