Morten Hjorth-Jensen
453 lines
27 KiB
HTML
453 lines
27 KiB
HTML
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('Material for Lecture Thursday November 2',
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('Further Dimensionality Remarks',
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('Layers of a CNN', 2, None, 'layers-of-a-cnn'),
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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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('The CIFAR01 data set', 2, None, 'the-cifar01-data-set'),
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('Verifying the data set', 2, None, 'verifying-the-data-set'),
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('Set up the model', 2, None, 'set-up-the-model'),
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<a class="navbar-brand" href="week44-bs.html">Week 44, Convolutional Neural Networks (CNN)</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="._week44-bs001.html#plan-for-week-44" style="font-size: 80%;"><b>Plan for week 44</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs002.html#material-for-lecture-thursday-november-2" style="font-size: 80%;"><b>Material for Lecture Thursday November 2</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs003.html#convolutional-neural-networks-recognizing-images" style="font-size: 80%;"><b>Convolutional Neural Networks (recognizing images)</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs004.html#what-is-the-difference" style="font-size: 80%;"><b>What is the Difference</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs005.html#neural-networks-vs-cnns" style="font-size: 80%;"><b>Neural Networks vs CNNs</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs006.html#why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc" style="font-size: 80%;"><b>Why CNNS for images, sound files, medical images from CT scans etc?</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs007.html#regular-nns-don-t-scale-well-to-full-images" style="font-size: 80%;"><b>Regular NNs don’t scale well to full images</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs008.html#3d-volumes-of-neurons" style="font-size: 80%;"><b>3D volumes of neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs009.html#layers-used-to-build-cnns" style="font-size: 80%;"><b>Layers used to build CNNs</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs010.html#transforming-images" style="font-size: 80%;"><b>Transforming images</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs011.html#cnns-in-brief" style="font-size: 80%;"><b>CNNs in brief</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs012.html#key-idea" style="font-size: 80%;"><b>Key Idea</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs013.html#mathematics-of-cnns" style="font-size: 80%;"><b>Mathematics of CNNs</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs014.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;"><b>Convolution Examples: Polynomial multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs015.html#efficient-polynomial-multiplication" style="font-size: 80%;"><b>Efficient Polynomial Multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs016.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;"><b>A more efficient way of coding the above Convolution</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs017.html#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;"><b>Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs018.html#simple-code-example" style="font-size: 80%;"><b>Simple Code Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs019.html#wrapping-up-fourier-transforms" style="font-size: 80%;"><b>Wrapping up Fourier transforms</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs020.html#finding-the-coefficients" style="font-size: 80%;"><b>Finding the Coefficients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs021.html#final-words-on-fourier-transforms" style="font-size: 80%;"><b>Final words on Fourier Transforms</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs021.html#fourier-transforms-and-convolution" style="font-size: 80%;"> Fourier transforms and convolution</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs021.html#two-dimensional-objects" style="font-size: 80%;"><b>Two-dimensional Objects</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs022.html#more-on-dimensionalities" style="font-size: 80%;"><b>More on Dimensionalities</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs023.html#further-dimensionality-remarks" style="font-size: 80%;"><b>Further Dimensionality Remarks</b></a></li>
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<!-- navigation toc: --> <li><a href="#cnns-in-more-detail" style="font-size: 80%;"><b>CNNs in more detail</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs025.html#pooling" style="font-size: 80%;"><b>Pooling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs026.html#no-zero-padding-unit-strides" style="font-size: 80%;"><b>No zero padding, unit strides</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs027.html#zero-padding-unit-strides" style="font-size: 80%;"><b>Zero padding, unit strides</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs028.html#half-same-padding" style="font-size: 80%;"><b>Half (same) padding</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs029.html#full-padding" style="font-size: 80%;"><b>Full padding</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs030.html#pooling-arithmetic" style="font-size: 80%;"><b>Pooling arithmetic</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs031.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;"><b>CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs032.html#setting-it-up" style="font-size: 80%;"><b>Setting it up</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs033.html#the-mnist-dataset-again" style="font-size: 80%;"><b>The MNIST dataset again</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs034.html#strong-correlations" style="font-size: 80%;"><b>Strong correlations</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs035.html#layers-of-a-cnn" style="font-size: 80%;"><b>Layers of a CNN</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs036.html#systematic-reduction" style="font-size: 80%;"><b>Systematic reduction</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs037.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;"><b>Prerequisites: Collect and pre-process data</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs038.html#importing-keras-and-tensorflow" style="font-size: 80%;"><b>Importing Keras and Tensorflow</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs039.html#running-with-keras" style="font-size: 80%;"><b>Running with Keras</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs040.html#final-part" style="font-size: 80%;"><b>Final part</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs041.html#final-visualization" style="font-size: 80%;"><b>Final visualization</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs042.html#the-cifar01-data-set" style="font-size: 80%;"><b>The CIFAR01 data set</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs043.html#verifying-the-data-set" style="font-size: 80%;"><b>Verifying the data set</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs044.html#set-up-the-model" style="font-size: 80%;"><b>Set up the model</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs045.html#add-dense-layers-on-top" style="font-size: 80%;"><b>Add Dense layers on top</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs046.html#compile-and-train-the-model" style="font-size: 80%;"><b>Compile and train the model</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs047.html#finally-evaluate-the-model" style="font-size: 80%;"><b>Finally, evaluate the model</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#building-our-own-cnn-code" style="font-size: 80%;"><b>Building our own CNN code</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#list-of-contents" style="font-size: 80%;"> List of contents:</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#schedulers" style="font-size: 80%;"> Schedulers</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-schedulers" style="font-size: 80%;"> Usage of schedulers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#cost-functions" style="font-size: 80%;"> Cost functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-cost-functions" style="font-size: 80%;"> Usage of cost functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-activation-functions" style="font-size: 80%;"> Usage of activation functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution" style="font-size: 80%;"> Convolution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#layers" style="font-size: 80%;"> Layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution2dlayer-convolution-in-a-hidden-layer" style="font-size: 80%;"> Convolution2DLayer: convolution in a hidden layer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#backpropagation-in-the-convolutional-layer" style="font-size: 80%;"> Backpropagation in the convolutional layer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#demonstration" style="font-size: 80%;"> Demonstration</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#pooling-layer" style="font-size: 80%;"> Pooling Layer</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#flattening-layer" style="font-size: 80%;"> Flattening Layer</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#fully-connected-layers" style="font-size: 80%;"> Fully Connected Layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#optimized-convolution2dlayer" style="font-size: 80%;"> Optimized Convolution2DLayer</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#the-convolutional-neural-network-cnn" style="font-size: 80%;"> The Convolutional Neural Network (CNN)</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-cnn-code" style="font-size: 80%;"> Usage of CNN code</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#additional-remarks" style="font-size: 80%;"> Additional Remarks</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#remarks-on-the-speed" style="font-size: 80%;"> Remarks on the speed</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution-using-separable-kernels" style="font-size: 80%;"> Convolution using separable kernels</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution-in-the-fourier-domain" style="font-size: 80%;"> Convolution in the Fourier domain</a></li>
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</ul>
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</li>
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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="part0024"></a>
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<!-- !split -->
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<h2 id="cnns-in-more-detail" class="anchor">CNNs in more detail </h2>
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<p>Let assume we have an input matrix \( I \) of dimensionality \( 3\times 3 \)
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and a \( 2\times 2 \) filter \( W \) given by the following matrices
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</p>
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||
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$$
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\boldsymbol{I}=\begin{bmatrix}i_{00} & i_{01} & i_{02} \\
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i_{10} & i_{11} & i_{12} \\
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i_{20} & i_{21} & i_{22} \end{bmatrix},
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$$
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<p>and </p>
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$$
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\boldsymbol{W}=\begin{bmatrix}w_{00} & w_{01} \\
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w_{10} & w_{11}\end{bmatrix}.
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$$
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||
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<p>We introduce now the hyperparameter \( S \) <b>stride</b>. Stride represents how the filter \( W \) moves the convolution process on the matrix \( I \).
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We strongly recommend the repository on <a href="https://github.com/vdumoulin/conv_arithmetic" target="_self">Arithmetic of deep learning by Dumoulin and Visin</a>
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</p>
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||
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<p>Here we set the stride equal to \( S=1 \), which means that, starting with the element \( i_{00} \), the filter will act on \( 2\times 2 \) submatrices each time, starting with the upper corner and moving according to the stride value column by column. </p>
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||
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<p>Here we perform the operation</p>
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||
$$
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S_(i,j)=(I * W)(i,j) = \sum_m\sum_n I(i-m,j-n)W(m,n),
|
||
$$
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||
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<p>and obtain</p>
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||
$$
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\boldsymbol{S}=\begin{bmatrix}i_{00}w_{00}+i_{01}w_{01}+i_{10}w_{10}+i_{11}w_{11} & i_{01}w_{00}+i_{02}w_{01}+i_{11}w_{10}+i_{12}w_{11} \\
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i_{10}w_{00}+i_{11}w_{01}+i_{20}w_{10}+i_{21}w_{11} & i_{11}w_{00}+i_{12}w_{01}+i_{21}w_{10}+i_{22}w_{11}\end{bmatrix}.
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$$
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<p>We can rewrite this operation in terms of a matrix-vector multiplication by defining a new vector where we flatten out the inputs as a vector \( \boldsymbol{I}' \) of length \( 9 \) and
|
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a matrix \( \boldsymbol{W}' \) with dimension \( 4\times 9 \) as
|
||
</p>
|
||
|
||
$$
|
||
\boldsymbol{I}'=\begin{bmatrix}i_{00} \\ i_{01} \\ i_{02} \\ i_{10} \\ i_{11} \\ i_{12} \\ i_{20} \\ i_{21} \\ i_{22} \end{bmatrix},
|
||
$$
|
||
|
||
<p>and the new matrix</p>
|
||
$$
|
||
\boldsymbol{W}'=\begin{bmatrix} w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 & 0 & 0 & 0 \\
|
||
0 & w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 & 0 & 0 \\
|
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0 & 0 & 0 & w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 \\
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0 & 0 & 0 & 0 & w_{00} & w_{01} & 0 & w_{10} & w_{11}\end{bmatrix}.
|
||
$$
|
||
|
||
<p>We see easily that performing the matrix-vector multiplication \( \boldsymbol{W}'\boldsymbol{I}' \) is the same as the above convolution with stride \( S=1 \), that is</p>
|
||
|
||
$$
|
||
S=(\boldsymbol{W}*\boldsymbol{I}),
|
||
$$
|
||
|
||
<p>is now given by \( \boldsymbol{W}'\boldsymbol{I}' \) which is a vector of length \( 4 \) instead of the originally resulting \( 2\times 2 \) output matrix.</p>
|
||
|
||
<p>The collection of kernels/filters \( W \) defining a discrete convolution has a shape
|
||
corresponding to some permutation of \( (n, m, k_1, \ldots, k_N) \), where
|
||
</p>
|
||
|
||
$$
|
||
\begin{equation*}
|
||
\begin{split}
|
||
n &\equiv \text{number of output feature maps},\\
|
||
m &\equiv \text{number of input feature maps},\\
|
||
k_j &\equiv \text{kernel size along axis $j$}.
|
||
\end{split}
|
||
\end{equation*}
|
||
$$
|
||
|
||
<p>The following properties affect the output size \( o_j \) of a convolutional layer
|
||
along axis \( j \):
|
||
</p>
|
||
|
||
<ol>
|
||
<li> \( i_j \): input size along axis \( j \),</li>
|
||
<li> \( k_j \): kernel/filter size along axis \( j \),</li>
|
||
<li> stride (distance between two consecutive positions of the kernel/filter) along axis \( j \),</li>
|
||
<li> zero padding (number of zeros concatenated at the beginning and at the end of an axis) along axis \( j \).</li>
|
||
</ol>
|
||
<p>For instance, the above examples shows a \( 2\times 2 \) kernel/filter \( \boldsymbol{W} \) applied to a \( 3 \times 3 \) input padded with a \( 0 \times 0 \)
|
||
border of zeros using \( 1 \times 1 \) strides.
|
||
</p>
|
||
|
||
<p>Note that strides constitute a form of <b>subsampling</b>. As an alternative to
|
||
being interpreted as a measure of how much the kernel/filter is translated, strides
|
||
can also be viewed as how much of the output is retained. For instance, moving
|
||
the kernel by hops of two is equivalent to moving the kernel by hops of one but
|
||
retaining only odd output elements.
|
||
</p>
|
||
|
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<p>
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