# Neural Networks

### From Ufldl

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- | + | <div align=center> | |

- | [[Image:Sigmoid_Function.png|400px| | + | [[Image:Sigmoid_Function.png|400px|top|Sigmoid activation function.]] |

- | [[Image:Tanh_Function.png|400px| | + | [[Image:Tanh_Function.png|400px|top|Tanh activation function.]] |

+ | </div> | ||

The <math>\tanh(z)</math> function is a rescaled version of the sigmoid, and its output range is | The <math>\tanh(z)</math> function is a rescaled version of the sigmoid, and its output range is | ||

<math>[-1,1]</math> instead of <math>[0,1]</math>. | <math>[-1,1]</math> instead of <math>[0,1]</math>. | ||

- | Note that unlike | + | Note that unlike some other venues (including the OpenClassroom videos, and parts of CS229), we are not using the convention |

here of <math>x_0=1</math>. Instead, the intercept term is handled separately by the parameter <math>b</math>. | here of <math>x_0=1</math>. Instead, the intercept term is handled separately by the parameter <math>b</math>. | ||

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A neural network is put together by hooking together many of our simple | A neural network is put together by hooking together many of our simple | ||

- | + | "neurons," so that the output of a neuron can be the input of another. For | |

example, here is a small neural network: | example, here is a small neural network: | ||

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In this figure, we have used circles to also denote the inputs to the network. The circles | In this figure, we have used circles to also denote the inputs to the network. The circles | ||

- | labeled | + | labeled "+1" are called '''bias units''', and correspond to the intercept term. |

The leftmost layer of the network is called the '''input layer''', and the | The leftmost layer of the network is called the '''input layer''', and the | ||

rightmost layer the '''output layer''' (which, in this example, has only one | rightmost layer the '''output layer''' (which, in this example, has only one | ||

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In the sequel, we also let <math>z^{(l)}_i</math> denote the total weighted sum of inputs to unit <math>i</math> in layer <math>l</math>, | In the sequel, we also let <math>z^{(l)}_i</math> denote the total weighted sum of inputs to unit <math>i</math> in layer <math>l</math>, | ||

- | including the bias term (e.g., <math>z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i</math>), so that | + | including the bias term (e.g., <math>\textstyle z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i</math>), so that |

<math>a^{(l)}_i = f(z^{(l)}_i)</math>. | <math>a^{(l)}_i = f(z^{(l)}_i)</math>. | ||

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We have so far focused on one example neural network, but one can also build neural | We have so far focused on one example neural network, but one can also build neural | ||

- | networks with other | + | networks with other '''architectures''' (meaning patterns of connectivity between neurons), including ones with multiple hidden layers. |

- | architectures | + | |

The most common choice is a <math>\textstyle n_l</math>-layered network | The most common choice is a <math>\textstyle n_l</math>-layered network | ||

where layer <math>\textstyle 1</math> is the input layer, layer <math>\textstyle n_l</math> is the output layer, and each | where layer <math>\textstyle 1</math> is the input layer, layer <math>\textstyle n_l</math> is the output layer, and each | ||

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example of a '''feedforward''' neural network, since the connectivity graph | example of a '''feedforward''' neural network, since the connectivity graph | ||

does not have any directed loops or cycles. | does not have any directed loops or cycles. | ||

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patient, and the different outputs <math>y_i</math>'s might indicate presence or absence | patient, and the different outputs <math>y_i</math>'s might indicate presence or absence | ||

of different diseases.) | of different diseases.) | ||

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+ | {{Sparse_Autoencoder}} | ||

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+ | {{Languages|神经网络|中文}} |