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    Image classification

    The codes are available in the different .py files. This readme.md file will only present the mathematical part of the Artificial Neural Network.

    Artificial Neural Network

    We have \sigma(x) = \frac{1}{1+e^{-x}}

    ie \sigma(x)' = \frac{e^{-x}}{(1+e^{-x})^2}

    ie \sigma(x)' = \frac{1}{1+e^{-x}} * \frac{1+e^{-x}-1}{1+e^{-x}}

    ie \sigma(x)' = \frac{1}{1+e^{-x}} * (1-\frac{1}{1+e^{-x}})

    ie \sigma(x)' = \sigma(x) (1-\sigma(x))

    Calculating the partial derivative we have :

    \frac{\partial C}{\partial A^{(2)}}=\frac{2}{N_{out}} * (A^{(2)}-Y)

    We have :

    \frac{\partial C}{\partial Z^{(2)}} = \frac{\partial C}{\partial A^{(2)}} * \frac{\partial A^{(2)}}{\partial Z^{(2)}}

    but also :

    A^{(2)} = \sigma(Z^{(2)})

    So this gives :

    \frac{\partial A^{(2)}}{\partial Z^{(2)}} = \sigma'(Z^{(2)}) = \sigma(Z^{(2)}) (1-\sigma(Z^{(2)})) = A^{(2)} (1-A^{(2)})

    We then have :

    \frac{\partial C}{\partial Z^{(2)}} = \frac{\partial C}{\partial A^{(2)}} * A^{(2)} (1-A^{(2)})

    1. With the same method we have :

    \frac{\partial C}{\partial W^{(2)}} = \frac{\partial C}{\partial Z^{(2)}} \frac{\partial Z^{(2)}}{\partial W^{(2)}}

    and also : Z^{(2)} = W^{(2)} A^{(1)} + B^{(2)}

    So this gives :

    \frac{\partial C}{\partial W^{(2)}} = A^{(1)}.T matmul (\frac{\partial C}{\partial Z^{(2)}})

    1. With the same reasoning,

    \frac{\partial C}{\partial B^{(2)}} = \frac{\partial C}{\partial Z^{(2)}} \frac{\partial Z^{(2)}}{\partial B^{(2)}}

    and also :

    \frac{\partial C}{\partial B^{(2)}} = \frac{\partial C}{\partial Z^{(2)}} I_{N}

    \frac{\partial C}{\partial B^{(2)}} = \frac{\partial C}{\partial Z^{(2)}}

    1. We have :

    \frac{\partial C}{\partial A^{(1)}} = \frac{\partial C}{\partial Z^{(2)}} \frac{\partial Z^{(2)}}{\partial A^{(1)}}

    ie :

    \frac{\partial C}{\partial A^{(1)}} = \frac{\partial C}{\partial Z^{(2)}} mmul (W^{(2)}.T)

    \frac{\partial C}{\partial Z^{(1)}} = \frac{\partial C}{\partial A^{(1)}} \frac{\partial A^{(1)}}{\partial Z^{(1)}}

    ie :

    \frac{\partial C}{\partial Z^{(1)}} = \frac{\partial C}{\partial A^{(1)}} A^{(1)} (1-A^{(1)}) \frac{\partial C}{\partial Z^{(1)}} = \frac{\partial C}{\partial Z^{(2)}} mmul(W^{(2)}.T A^{(1)} (1-A^{(1)}))

    \frac{\partial C}{\partial W^{(1)}} = \frac{\partial C}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial W^{(1)}}

    ie :

    \frac{\partial C}{\partial W^{(1)}} = A^{(0)} mmul (\frac{\partial C}{\partial Z^{(1)}})

    \frac{\partial C}{\partial B^{(1)}} = \frac{\partial C}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial B^{(1)}}

    ie :

    \frac{\partial C}{\partial B^{(1)}} = \frac{\partial C}{\partial Z^{(1)}}

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