From 7710efc305682f35cbc8d69d9b1e5739dbb89f0d Mon Sep 17 00:00:00 2001
From: jvech <jmvalenciae@unal.edu.co>
Date: Mon, 24 Jul 2023 20:06:05 -0500
Subject: doc: backpropagation notes added

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+%acmart
+%IEEEtran
+%rbt-mathnotes-formula-sheet
+\documentclass{rbt-mathnotes-formula-sheet}
+\usepackage[utf8]{inputenc}
+\usepackage[pdf]{graphviz}
+\usepackage{derivative}
+\title{Deep learning notes}
+
+\begin{document}
+
+\section{Observations}
+\begin{eqnarray}
+    i,j,k,l,L,m,M,n,N,o \in & \mathcal{N} \\
+    X \in & \mathcal{R}^{n \times o} \\
+    Y \in & \mathcal{R}^{n \times m}
+\end{eqnarray}
+
+\section{Neural Network}
+
+\includegraphics[width=0.3\textwidth]{net.pdf}
+
+\begin{eqnarray}
+    a^0 = & x_{1 \times p}(n) \\
+    a^L = & d_{1 \times m}(n) \\
+    a^l = & \varphi (z^l) \\
+    z^l = &  a^{l - 1} W^l
+\end{eqnarray}
+
+\section{Gradient Descent}
+
+\begin{eqnarray}
+    e(n) = & y(n) - d(n) \\
+    \xi(n) = & \frac{1}{2} e e^{\top}\\
+    \xi(n) = & \frac{1}{2} \sum_{j=1}^{M} (e_j(n))^2 \\
+    W_{(k + 1)} = & W_{(k)} - \nabla_{W} \xi(d,y) \\
+    \xi_{avg}(n) = & \frac{1}{2n} \sum_{n=1}^N \sum_{j=1}^{M} (e_j(n))^2 \\
+\end{eqnarray}
+
+\section{Backpropagation}
+
+\begin{eqnarray}
+    \pdv{\xi}{\omega^l_{ij}} = & \delta_j^l \pdv{z_j^l}{\omega_{ij}} \\
+    \delta_j^l = & \pdv{\xi}{z_j^l} \\ 
+    \pdv{z_j^l}{\omega_{ij}} = & a_i^{l-1} \\ 
+\end{eqnarray}
+
+Output Layer
+
+\begin{eqnarray}
+    \delta_j^L =& \pdv{\xi}{z_j^L} = \pdv{\xi}{a_j^L} \pdv{a_j^L}{z_j^L}\\
+    \delta_j^L =& \pdv{\xi}{a_j^L} \dot{\varphi}(z_j^L)\\
+               =& - e_j \dot{\varphi}(z_j^L)
+\end{eqnarray}
+
+Hidden Layer
+
+\begin{eqnarray}
+    \delta_j^l = & \pdv{\xi}{z_j^l} = \sum_k \pdv{\xi}{z_k^{l+1}} \pdv{z_k^{l+1}}{z_j^l}\\ 
+    \delta_j^l = & \sum_k \delta_k^{l+1} \pdv{z_k^{l+1}}{z_j^l}\\
+    \pdv{z_k^{l+1}}{z_j^l} = &
+    \frac{\partial}{\partial z_j^l} \left( \sum_j \omega_{jk}^{l+1} \varphi(z_j^l) \right)\\
+    \pdv{z_k^{l+1}}{z_j^l} = & \omega_{jk} \dot{\varphi}(z_j^l)\\
+    \delta_j^l = & \sum_k \delta_k^{l+1} \omega_{jk}^{l+1} \dot{\varphi}(z_j^l)\\
+\end{eqnarray}
+
+\end{document}
-- 
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