\begin{equation}
    \label{eq:gen_process_intermediate_noise}
    Z^{(\ell+1)}= X^{(\ell)} W^{(\ell)T}+ \epsilon^{(\ell+1)}_Z, \quad X^{(\ell+1)}=\sigma(Z^{(\ell+1)})+ \epsilon^{(\ell+1)}_X, \quad \ell\in[L].
\end{equation}
