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mhjensen 05f321f985 update on neural networks
2018-10-18 05:50:28 +02:00

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<!-- navigation toc: --> <li><a href="._NeuralNet-bs001.html#___sec0" style="font-size: 80%;"><b>Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs002.html#___sec1" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs003.html#___sec2" style="font-size: 80%;"><b>Neural network types</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs004.html#___sec3" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs005.html#___sec4" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs007.html#___sec6" style="font-size: 80%;"><b>Other types of networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs008.html#___sec7" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs009.html#___sec8" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs010.html#___sec9" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs011.html#___sec10" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs012.html#___sec11" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs013.html#___sec12" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs014.html#___sec13" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs015.html#___sec14" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs016.html#___sec15" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation and activation</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs017.html#___sec16" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs018.html#___sec17" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs019.html#___sec18" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs020.html#___sec19" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs021.html#___sec20" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs022.html#___sec21" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs023.html#___sec22" style="font-size: 80%;"><b>Definitions</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs024.html#___sec23" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs025.html#___sec24" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs026.html#___sec25" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs027.html#___sec26" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs028.html#___sec27" style="font-size: 80%;"><b>Bringing it together</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs029.html#___sec28" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs030.html#___sec29" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs031.html#___sec30" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs032.html#___sec31" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs033.html#___sec32" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs034.html#___sec33" style="font-size: 80%;"><b>The Softmax function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs035.html#___sec34" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs036.html#___sec35" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs037.html#___sec36" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs038.html#___sec37" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs039.html#___sec38" style="font-size: 80%;"><b>Layers</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs040.html#___sec39" style="font-size: 80%;"><b>Weights and biases</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs041.html#___sec40" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs042.html#___sec41" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs043.html#___sec42" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs044.html#___sec43" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs045.html#___sec44" style="font-size: 80%;"><b>Regularization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs046.html#___sec45" style="font-size: 80%;"><b>Matrix multiplication</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs047.html#___sec46" style="font-size: 80%;"><b>Improving performance</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs048.html#___sec47" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs049.html#___sec48" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs050.html#___sec49" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs051.html#___sec50" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs052.html#___sec51" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs053.html#___sec52" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs054.html#___sec53" style="font-size: 80%;"><b>Building neural networks in Tensorflow and Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs055.html#___sec54" style="font-size: 80%;"><b>Tensorflow</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs056.html#___sec55" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs057.html#___sec56" style="font-size: 80%;"><b>Using TensorFlow backend</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs058.html#___sec57" style="font-size: 80%;"><b>Optimizing and using gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs059.html#___sec58" style="font-size: 80%;"><b>Using Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs060.html#___sec59" style="font-size: 80%;"><b>Which activation function should I use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs061.html#___sec60" style="font-size: 80%;"><b>Is the Logistic activation function (Sigmoid) our choice?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs062.html#___sec61" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs063.html#___sec62" style="font-size: 80%;"><b>The RELU function family</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs064.html#___sec63" style="font-size: 80%;"><b>Which activation function should we use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs065.html#___sec64" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs066.html#___sec65" style="font-size: 80%;"><b>Limitations of supervised learning with deep networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs067.html#___sec66" style="font-size: 80%;"><b>Convolutional Neural Networks (recognizing images)</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec67" style="font-size: 80%;"><b>Regular NNs don’t scale well to full images</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs069.html#___sec68" style="font-size: 80%;"><b>3D volumes of neurons</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs070.html#___sec69" style="font-size: 80%;"><b>Layers used to build CNNs</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs071.html#___sec70" style="font-size: 80%;"><b>CNNs in brief</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs072.html#___sec71" style="font-size: 80%;"><b>CNNs in more detail</b></a></li>
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<h2 id="___sec67" class="anchor">Regular NNs don&#8217;t scale well to full images </h2>
<p>
As an example, consider
an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a
single fully-connected neuron in a first hidden layer of a regular
Neural Network would have \( 32*32*3 = 3072 \) weights. This amount still
seems manageable, but clearly this fully-connected structure does not
scale to larger images. For example, an image of more respectable
size, say \( 200\times 200\times 3 \), would lead to neurons that have
\( 200*200*3 =120,000 \) weights. Moreover, we would almost certainly want to have
several such neurons, so the parameters would add up quickly! Clearly,
this full connectivity is wasteful and the huge number of parameters
would quickly lead to overfitting.
<p>
<center> <!-- FIGURE -->
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<center><p class="caption">Figure 1: A regular 3-layer Neural Network. </p></center>
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