Morten Hjorth-Jensen
448 lines
25 KiB
HTML
448 lines
25 KiB
HTML
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'sections': [('Plan for week 41', 2, None, 'plan-for-week-41'),
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('Videos on Neural Networks',
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2,
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None,
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'videos-on-neural-networks'),
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('Setting up the Back propagation algorithm',
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2,
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None,
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('Setting up a Multi-layer perceptron model for classification',
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None,
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('Defining the cost function',
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('Example: binary classification problem',
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('The Softmax function', 2, None, 'the-softmax-function'),
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('Developing a code for doing neural networks with back '
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('Collect and pre-process data',
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('Define model and architecture',
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('Weights and biases', 2, None, 'weights-and-biases'),
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('Feed-forward pass', 2, None, 'feed-forward-pass'),
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('Matrix multiplications', 2, None, 'matrix-multiplications'),
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('Choose cost function and optimizer',
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2,
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None,
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'choose-cost-function-and-optimizer'),
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('Optimizing the cost function',
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None,
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'optimizing-the-cost-function'),
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('Regularization', 2, None, 'regularization'),
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('Matrix multiplication', 2, None, 'matrix-multiplication'),
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('Improving performance', 2, None, 'improving-performance'),
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('Full object-oriented implementation',
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('Evaluate model performance on test data',
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('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
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('Visualization', 2, None, 'visualization'),
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('scikit-learn implementation',
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'scikit-learn-implementation'),
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('Visualization', 2, None, 'visualization'),
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('Testing our code for the XOR, OR and AND gates',
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('The AND and XOR Gates', 2, None, 'the-and-and-xor-gates'),
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('Representing the Data Sets',
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2,
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'representing-the-data-sets'),
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('Setting up the Neural Network',
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('The Code using Scikit-Learn',
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('Building neural networks in Tensorflow and Keras',
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('Tensorflow', 2, None, 'tensorflow'),
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('Using Keras', 2, None, 'using-keras'),
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('Collect and pre-process data',
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2,
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None,
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'collect-and-pre-process-data'),
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('The Breast Cancer Data, now with Keras',
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2,
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None,
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'the-breast-cancer-data-now-with-keras'),
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('The Mathematics of Neural Networks',
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2,
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None,
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'the-mathematics-of-neural-networks'),
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('Fine-tuning neural network hyperparameters',
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'fine-tuning-neural-network-hyperparameters'),
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('Hidden layers', 2, None, 'hidden-layers'),
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('Which activation function should I use?',
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2,
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None,
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'which-activation-function-should-i-use'),
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('Is the Logistic activation function (Sigmoid) our choice?',
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None,
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'is-the-logistic-activation-function-sigmoid-our-choice'),
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('The derivative of the Logistic funtion',
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2,
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None,
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'the-derivative-of-the-logistic-funtion'),
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('The RELU function family', 2, None, 'the-relu-function-family'),
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('Which activation function should we use?',
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2,
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'which-activation-function-should-we-use'),
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('More on activation functions, output layers',
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('Batch Normalization', 2, None, 'batch-normalization'),
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('Dropout', 2, None, 'dropout'),
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('Gradient Clipping', 2, None, 'gradient-clipping'),
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('A very nice website on Neural Networks',
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('A top-down perspective on Neural networks',
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('Limitations of supervised learning with deep networks',
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('Overarching Views, a personal note',
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'overarching-views-a-personal-note'),
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('From a Spherical Cow to a real one',
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('Convolutional Neural Networks (recognizing images)',
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'regular-nns-don-t-scale-well-to-full-images'),
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('3D volumes of neurons', 2, None, '3d-volumes-of-neurons'),
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('Layers used to build CNNs',
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('Transforming images', 2, None, 'transforming-images'),
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('CNNs in brief', 2, None, 'cnns-in-brief'),
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('CNNs in more detail, building convolutional neural networks in '
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'cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras'),
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('Setting it up', 2, None, 'setting-it-up'),
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('The MNIST dataset again', 2, None, 'the-mnist-dataset-again'),
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('Strong correlations', 2, None, 'strong-correlations'),
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('Layers of a CNN', 2, None, 'layers-of-a-cnn'),
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('Systematic reduction', 2, None, 'systematic-reduction'),
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('Prerequisites: Collect and pre-process data',
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('Importing Keras and Tensorflow',
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('Running with Keras', 2, None, 'running-with-keras'),
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('Final part', 2, None, 'final-part'),
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('Final visualization', 2, None, 'final-visualization'),
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('Fun links', 2, None, 'fun-links')]}
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<a class="navbar-brand" href="week41-bs.html">Week 41 Constructing a Neural Network code, Tensor flow and start Convolutional Neural Networks</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week41-bs001.html#plan-for-week-41" style="font-size: 80%;">Plan for week 41</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs002.html#videos-on-neural-networks" style="font-size: 80%;">Videos on Neural Networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs003.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;">Setting up the Back propagation algorithm</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs004.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;">Setting up a Multi-layer perceptron model for classification</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs005.html#defining-the-cost-function" style="font-size: 80%;">Defining the cost function</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs006.html#example-binary-classification-problem" style="font-size: 80%;">Example: binary classification problem</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs007.html#the-softmax-function" style="font-size: 80%;">The Softmax function</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs008.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;">Developing a code for doing neural networks with back propagation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs035.html#collect-and-pre-process-data" style="font-size: 80%;">Collect and pre-process data</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs010.html#train-and-test-datasets" style="font-size: 80%;">Train and test datasets</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs011.html#define-model-and-architecture" style="font-size: 80%;">Define model and architecture</a></li>
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||
<!-- navigation toc: --> <li><a href="#layers" style="font-size: 80%;">Layers</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs013.html#weights-and-biases" style="font-size: 80%;">Weights and biases</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs014.html#feed-forward-pass" style="font-size: 80%;">Feed-forward pass</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs015.html#matrix-multiplications" style="font-size: 80%;">Matrix multiplications</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs016.html#choose-cost-function-and-optimizer" style="font-size: 80%;">Choose cost function and optimizer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs017.html#optimizing-the-cost-function" style="font-size: 80%;">Optimizing the cost function</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs018.html#regularization" style="font-size: 80%;">Regularization</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs019.html#matrix-multiplication" style="font-size: 80%;">Matrix multiplication</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs020.html#improving-performance" style="font-size: 80%;">Improving performance</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs021.html#full-object-oriented-implementation" style="font-size: 80%;">Full object-oriented implementation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs022.html#evaluate-model-performance-on-test-data" style="font-size: 80%;">Evaluate model performance on test data</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs023.html#adjust-hyperparameters" style="font-size: 80%;">Adjust hyperparameters</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs026.html#visualization" style="font-size: 80%;">Visualization</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs025.html#scikit-learn-implementation" style="font-size: 80%;">scikit-learn implementation</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs026.html#visualization" style="font-size: 80%;">Visualization</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs027.html#testing-our-code-for-the-xor-or-and-and-gates" style="font-size: 80%;">Testing our code for the XOR, OR and AND gates</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs028.html#the-and-and-xor-gates" style="font-size: 80%;">The AND and XOR Gates</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs029.html#representing-the-data-sets" style="font-size: 80%;">Representing the Data Sets</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs030.html#setting-up-the-neural-network" style="font-size: 80%;">Setting up the Neural Network</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs031.html#the-code-using-scikit-learn" style="font-size: 80%;">The Code using Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs032.html#building-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">Building neural networks in Tensorflow and Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs033.html#tensorflow" style="font-size: 80%;">Tensorflow</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs034.html#using-keras" style="font-size: 80%;">Using Keras</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs035.html#collect-and-pre-process-data" style="font-size: 80%;">Collect and pre-process data</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs036.html#the-breast-cancer-data-now-with-keras" style="font-size: 80%;">The Breast Cancer Data, now with Keras</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs037.html#the-mathematics-of-neural-networks" style="font-size: 80%;">The Mathematics of Neural Networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs038.html#fine-tuning-neural-network-hyperparameters" style="font-size: 80%;">Fine-tuning neural network hyperparameters</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs039.html#hidden-layers" style="font-size: 80%;">Hidden layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs040.html#which-activation-function-should-i-use" style="font-size: 80%;">Which activation function should I use?</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs041.html#is-the-logistic-activation-function-sigmoid-our-choice" style="font-size: 80%;">Is the Logistic activation function (Sigmoid) our choice?</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs042.html#the-derivative-of-the-logistic-funtion" style="font-size: 80%;">The derivative of the Logistic funtion</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs043.html#the-relu-function-family" style="font-size: 80%;">The RELU function family</a></li>
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||
<!-- navigation toc: --> <li><a href="._week41-bs044.html#which-activation-function-should-we-use" style="font-size: 80%;">Which activation function should we use?</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs045.html#more-on-activation-functions-output-layers" style="font-size: 80%;">More on activation functions, output layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs046.html#batch-normalization" style="font-size: 80%;">Batch Normalization</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs047.html#dropout" style="font-size: 80%;">Dropout</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs048.html#gradient-clipping" style="font-size: 80%;">Gradient Clipping</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs049.html#a-very-nice-website-on-neural-networks" style="font-size: 80%;">A very nice website on Neural Networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs050.html#a-top-down-perspective-on-neural-networks" style="font-size: 80%;">A top-down perspective on Neural networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs051.html#limitations-of-supervised-learning-with-deep-networks" style="font-size: 80%;">Limitations of supervised learning with deep networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs052.html#overarching-views-a-personal-note" style="font-size: 80%;">Overarching Views, a personal note</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs053.html#from-a-spherical-cow-to-a-real-one" style="font-size: 80%;">From a Spherical Cow to a real one</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs054.html#convolutional-neural-networks-recognizing-images" style="font-size: 80%;">Convolutional Neural Networks (recognizing images)</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs055.html#regular-nns-don-t-scale-well-to-full-images" style="font-size: 80%;">Regular NNs don’t scale well to full images</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs056.html#3d-volumes-of-neurons" style="font-size: 80%;">3D volumes of neurons</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs057.html#layers-used-to-build-cnns" style="font-size: 80%;">Layers used to build CNNs</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs058.html#transforming-images" style="font-size: 80%;">Transforming images</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs059.html#cnns-in-brief" style="font-size: 80%;">CNNs in brief</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs060.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs061.html#setting-it-up" style="font-size: 80%;">Setting it up</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs062.html#the-mnist-dataset-again" style="font-size: 80%;">The MNIST dataset again</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs063.html#strong-correlations" style="font-size: 80%;">Strong correlations</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week41-bs064.html#layers-of-a-cnn" style="font-size: 80%;">Layers of a CNN</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs065.html#systematic-reduction" style="font-size: 80%;">Systematic reduction</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs066.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;">Prerequisites: Collect and pre-process data</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs067.html#importing-keras-and-tensorflow" style="font-size: 80%;">Importing Keras and Tensorflow</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs068.html#running-with-keras" style="font-size: 80%;">Running with Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs069.html#final-part" style="font-size: 80%;">Final part</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs070.html#final-visualization" style="font-size: 80%;">Final visualization</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs071.html#fun-links" style="font-size: 80%;">Fun links</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0012"></a>
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<!-- !split -->
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<h2 id="layers" class="anchor">Layers </h2>
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<ul>
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<li> Input</li>
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</ul>
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Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.
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<ul>
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<li> Hidden layer</li>
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</ul>
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We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
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Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.
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<ul>
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<li> Output</li>
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||
</ul>
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If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
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which could output 0 or 1 according to the Heaviside function. This would be an example of a <em>hard</em> classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a <em>soft</em> classifier, which outputs the probability of being in class 0 or 1.
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<p>
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For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.
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||
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||
<p>
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||
Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the <em>softmax</em> function:
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$$ P(\text{class \( j \)} \mid \text{input \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
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{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$
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||
|
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<p>
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i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) the weights of neuron \( j \) to the inputs.
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The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.
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The exponent is just the weighted sum of inputs as before:
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||
|
||
$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
|
||
|
||
<p>
|
||
Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
|
||
weights to the output layer.
|
||
|
||
<p>
|
||
<p>
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