End-to-end, input-to-output transformation in a neural network:
\[\begin{equation*}
\hat{Y}
= \hat{f}(X)
= A^{[\ell]} \in \mathbb{R}^{p^{\scriptstyle [\ell]} \times p^{\scriptstyle [\ell - 1]}}
\end{equation*}
\]
with
\[\begin{equation*}
A^{[i]} =
\begin{cases}
X \in \mathbb{R}^{n \times m}
& i = 0
\\
\boxed{g^{[i]}({W^{[i]}} A^{[i - 1]} + \vec{b}^{[i]})}
\in \mathbb{R}^{p^{\scriptstyle [i]} \times p^{\scriptstyle [i - 1]}}
& i > 0
\end{cases}
\end{equation*}
\]
and
\[\begin{equation*}
p^{[1]} = p^{[0]}
\end{equation*}
\]
where