bpp.tex.tex

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\begin{document}\definecolor{fgC}{rgb}{0,0,0}\color{fgC}\definecolor{bgC}{rgb}{1,1,1}\pagecolor{bgC} \begin{align*}\frac{\partial E}{\partial \left[ \bold{W}_\ell \right]_{ij}} =&
\frac{\partial E}{\partial \left[ \boldsymbol{\omega}_\ell \right]_i } 
\frac{\partial \left[ \boldsymbol{\omega}_\ell \right]_i }{ \partial \left[ \textbf{\textit{n}}_\ell \right]_i }
\frac{ \partial \left[ \textbf{\textit{n}}_\ell \right]_i }{ \partial \left[ \bold{W}_\ell \right]_{ij} } \\ 
\\
\frac{ \partial \left[ \textbf{\textit{n}}_\ell \right]_i }{ \partial \left[ \bold{W}_\ell \right]_{ij} } =& \left[ \boldsymbol{\omega}_{\ell - 1} \right]_j \\
\frac{\partial \left[ \boldsymbol{\omega}_\ell \right]_i }{ \partial \left[ \textbf{\textit{n}}_\ell \right]_i } =& 
\varphi'(\left[ \boldsymbol{n}_\ell \right]_i) = \left[ \boldsymbol{\omega}_\ell \right]_i(1 - \left[ \boldsymbol{\omega}_\ell \right]_i) \\
\frac{\partial E}{\partial \left[ \boldsymbol{\omega}_L \right]_i } =& \left[ \boldsymbol{\omega}_L \right]_i - \left[ \bold{t} \right]_i \\ 
\\
\frac{\partial E}{\partial \left[ \bold{W}_\ell \right]_{ij}} =& 
\frac{\partial E}{\partial \left[ \boldsymbol{\omega}_\ell \right]_i } 
\left[ \boldsymbol{\omega}_\ell \right]_i(1 - \left[ \boldsymbol{\omega}_\ell \right]_i) 
\left[ \boldsymbol{\omega}_{\ell - 1} \right]_j \\
\\
\frac{\partial E}{\partial \left[ \boldsymbol{\omega}_{\ell - 1} \right]_j} =& 
\frac{\partial E}{\partial \left[ \boldsymbol{\omega}_{\ell} \right]_j}
\frac{\partial \left[ \boldsymbol{\omega}_{\ell} \right]_j}{\partial \textbf{\textit{n}}_\ell}
\frac{\partial \textbf{\textit{n}}_\ell}{\partial \left[ \boldsymbol{\omega}_{\ell - 1} \right]_j} \\
\left[ \boldsymbol{\delta}_{\ell} \right]_j =& \frac{\partial E}{\partial \left[ \boldsymbol{\omega}_{\ell} \right]_j} \\
\left[ \boldsymbol{\delta}_{\ell - 1} \right]_j =& \left[ \boldsymbol{\delta}_{\ell} \right]_j
\frac{\partial \left[ \boldsymbol{\omega}_{\ell} \right]_j}{\partial \textbf{\textit{n}}_\ell}
\frac{\partial \textbf{\textit{n}}_\ell}{\partial \left[ \boldsymbol{\omega}_{\ell - 1} \right]_j} \\
\\
\frac{\partial E}{\partial \left[ \bold{W}_\ell \right]_{ij}} =& 
\left[ \boldsymbol{\delta}_{\ell} \right]_i
\left[ \boldsymbol{\omega}_\ell \right]_i(1 - \left[ \boldsymbol{\omega}_\ell \right]_i) 
\left[ \boldsymbol{\omega}_{\ell - 1} \right]_j \\
\boldsymbol{\Delta}_\ell =& \left[ \frac{\partial E}{\partial \left[ \bold{W}_\ell \right]_{ij}} \right] \\
=& \begin{bmatrix}
a_{11} & \cdots & a_{1n}\\ 
\vdots &  \ddots & \vdots\\ 
a_{m1} & \cdots & a_{mn}
\end{bmatrix}
\\
\\
\boldsymbol{\delta}_{\ell} =& \frac{\partial E}{\partial \boldsymbol{\omega}_{\ell}} \\
\boldsymbol{\delta}_{\ell - 1} =& \boldsymbol{\delta}_{\ell} 
\frac{\partial \boldsymbol{\omega}_{\ell}}{\partial \textbf{\textit{n}}_\ell}
\frac{\partial \textbf{\textit{n}}_\ell}{\partial \boldsymbol{\omega}_{\ell - 1}} \\\end{align*}\end{document}