From 26c9fe7c966b2ef04d22c50182f270d58895d402 Mon Sep 17 00:00:00 2001 From: julienChemillier Date: Tue, 7 Feb 2023 21:44:59 +0100 Subject: [PATCH] Add backpropagation explaination (conv not here) --- doc/cnn/2d_pooling_layer.png | Bin 0 -> 88809 bytes doc/cnn/backpropagation_explaination.md | 58 ++++++++++++++++++++++++ doc/cnn/fully_connected.png | Bin 0 -> 104739 bytes doc/cnn/last_layer.png | Bin 0 -> 96181 bytes 4 files changed, 58 insertions(+) create mode 100644 doc/cnn/2d_pooling_layer.png create mode 100644 doc/cnn/backpropagation_explaination.md create mode 100644 doc/cnn/fully_connected.png create mode 100644 doc/cnn/last_layer.png diff --git a/doc/cnn/2d_pooling_layer.png b/doc/cnn/2d_pooling_layer.png new file mode 100644 index 0000000000000000000000000000000000000000..9d900952d77daa101a94192a5ebadff52c601580 GIT binary patch literal 88809 zcmeGE^;=a@+cu1Xq%;cBElNpAr${MCEV>&h=~B8o1SvsUq`O(36(`gX=}CRWy##&7Hm?Tn4B>`kp54iTD# 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\dfrac{e^{a_2}}{e^{a_1}+e^{a_2}+e^{a_3}}$ +$l_3 = \dfrac{e^{a_3}}{e^{a_1}+e^{a_2}+e^{a_3}}$ +$E = \dfrac{1}{2}((l_1-o_1)^2+(l_2-o_2)^2+(l_3-o_3)^2)$ +- $\dfrac{\partial E}{\partial l1} = o_1 - l_1$ +$\dfrac{\partial E}{\partial l2} = o_2 - l_2$ +$\dfrac{\partial E}{\partial l3} = o_3 - l_3$ +- $\dfrac{\partial l_1}{\partial a_1} = l_1(1-l_1)$ +$\dfrac{\partial E}{\partial a_1} = \dfrac{\partial E}{\partial l_1} \dfrac{\partial l_1}{\partial a_1} = (o_1-l_1)l_1(1-l_1)$ + +> Derivatives: +$\dfrac{\partial E}{\partial a_i} = \dfrac{\partial E}{\partial l_i} \dfrac{\partial l_i}{\partial a_i} = (o_i-l_i)l_1(1-l_i)$ +$\dfrac{\partial E}{\partial b_i} = \dfrac{\partial E}{\partial a_i}$ + +--- +--- + +## Backpropagation of a fully connected layer + + + +Soit f la fonction d'activation de la première couche et g la fonction d'activation de la deuxième couche. +- $d_1 =g(c_1)$ +$d_2 = g(c2)$ +$c_1 = w_{11}l_1 + w_{21}l_2 + w_{31}l_3 + b'_1$ +$c_2 = w_{12}l_1 + w_{22}l_2 + w_{32}l_3 + b'_2$ +$l_1 = f(a_1)$ +$l_2 = f(a_2)$ +$l_3 = f(a_3)$ +- $\dfrac{\partial E}{\partial a_1} = \dfrac{\partial E_{c_1}}{\partial c_1} \dfrac{\partial c_1}{\partial l_1} \dfrac{\partial l_1}{\partial a_1} + \dfrac{\partial E_{c_2}}{\partial c_2} \dfrac{\partial c_2}{\partial l_1} \dfrac{\partial l_1}{\partial a_1}$ +$\dfrac{\partial c_2}{\partial l_1} = w_{12}$ +$\dfrac{\partial c_1}{\partial l_1} = w_{11}$ +$\dfrac{\partial l_1}{\partial a_1} = f'(a_1)$ + +> Derivatives: +$\dfrac{\partial E}{\partial b_j} = \dfrac{\partial E}{\partial l_i} $ +$\dfrac{\partial E}{\partial w_{ij}} = \dfrac{\partial E}{\partial c_j}l_i$ +$\dfrac{\partial E}{\partial a_i} = \dfrac{\partial E_{c_1}}{\partial c_1} w_{i1} + \dfrac{\partial E_{c_2}}{\partial c_2} w_{i2}$ +--- +--- + +## Backpropagation of an average 2d pooling layer + + + +$\forall i,j: \space b_{i j} = \dfrac{a_{2i \space 2j} + a_{2i+1 \space 2j} + a_{2i \space 2j+1} + a_{2i+1 \space 2j+1}}{4}$ + +> Derivatives: +$\forall i,j: \space \dfrac{\partial E}{\partial a_{i \space j}} = \dfrac{1}{4} \dfrac{\partial E}{\partial b_{k \space l}} $ +where k = i//2 and l = j//2 + diff --git a/doc/cnn/fully_connected.png b/doc/cnn/fully_connected.png new file mode 100644 index 0000000000000000000000000000000000000000..9f795fd6553fab5066c9772753be311d18e709e2 GIT binary patch literal 104739 zcmce81y_}S6D<}BD4;YF(kQKTDvc5%-3`*+Wq>pYNT-A#-QC?C($d`}edqk&d%wbU zEtkrAe9kW>_RQ>k{N788Jwhc!MM6S)Bq1*R0SW1LDEyXtfDAu*k0xaYf84bdl#qV_ zzuX?^`@+8o>_n997tDnc=#F4db(y%r9R(V|mHV#>~z7O3=$p90}yl0uGrM zrr-mztk)U#x8P4N%%)RuU+n+>ra(>h0r3+-`4}(p{~nVyctY-tcw9I_;{Ve_tyC(x zivIroC5%FdwP=b9rXjsn7%d_t3GunSbmO*Ldo3@Iy}rI)F|%Vcj(8+RM^0Yi?s4SQ z_OImRPbn$nLqkKH9+#ZIfB!a{DhrB_Cn7?>PenzgoU3TY{?wtGII7#beX((I|F0D% z5;I}Y+OdlrH!^VpHI6~mCJVJAqol#y=sZ?k001Y#l_F*>C*}dFtbPf+ZTTi$6|{# z>K0-W5lOy&jU*Mz`qTMvr6-!@>z_Z$A1FKe%(>A@;&1B$$RFtAJBJ)w{pg zK2hz+WIC2J-w{edNkbDJ9&S5fY$zf-uz9RZkE>c{CJ}(kP@qhI_wHRxY;0j=Wqf!! z8LSZ2?;j8l!)A($)X#rGzdcpnzHVo;GtC7{>+J5P5D-XOSkT>a;4$ipC!?bB4hst_ 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