486 lines
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HTML
486 lines
31 KiB
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('Defining the problem', 2, None, 'defining-the-problem'),
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('Setting up the network using Autograd; The full program',
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('Efficient Polynomial Multiplication',
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('Convolution Examples: Principle of Superposition and Periodic '
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'Forces (Fourier Transforms)',
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('Principle of Superposition',
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('Simple Code Example', 2, None, 'simple-code-example'),
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('Wrapping up Fourier transforms',
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('Finding the Coefficients', 2, None, 'finding-the-coefficients'),
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('Final words on Fourier Transforms',
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('Cross-Correlation', 2, None, 'cross-correlation'),
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('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
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('Further Dimensionality Remarks',
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'further-dimensionality-remarks'),
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('CNNs in more detail, Lecture from IN5400',
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2,
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None,
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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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2,
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None,
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'prerequisites-collect-and-pre-process-data'),
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('Importing Keras and Tensorflow',
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2,
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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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<ul class="nav navbar-nav navbar-right">
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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="._week42-bs001.html#plan-for-week-42" style="font-size: 80%;">Plan for week 42</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs002.html#using-automatic-differentiation" style="font-size: 80%;">Using Automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs003.html#solving-odes-with-deep-learning" style="font-size: 80%;">Solving ODEs with Deep Learning</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs004.html#ordinary-differential-equations" style="font-size: 80%;">Ordinary Differential Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs047.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs006.html#minimization-process" style="font-size: 80%;">Minimization process</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs007.html#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation" style="font-size: 80%;">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs008.html#example-exponential-decay" style="font-size: 80%;">Example: Exponential decay</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs009.html#the-function-to-solve-for" style="font-size: 80%;">The function to solve for</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs047.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs011.html#setup-of-network" style="font-size: 80%;">Setup of Network</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs012.html#reformulating-the-problem" style="font-size: 80%;">Reformulating the problem</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs013.html#more-technicalities" style="font-size: 80%;">More technicalities</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs038.html#more-details" style="font-size: 80%;">More details</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs015.html#a-possible-implementation-of-a-neural-network" style="font-size: 80%;">A possible implementation of a neural network</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs016.html#technicalities" style="font-size: 80%;">Technicalities</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs017.html#final-technicalities-i" style="font-size: 80%;">Final technicalities I</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs018.html#final-technicalities-ii" style="font-size: 80%;">Final technicalities II</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs019.html#final-technicalities-iii" style="font-size: 80%;">Final technicalities III</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs020.html#final-technicalities-iv" style="font-size: 80%;">Final technicalities IV</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs021.html#back-propagation" style="font-size: 80%;">Back propagation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs022.html#gradient-descent" style="font-size: 80%;">Gradient descent</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs023.html#the-code-for-solving-the-ode" style="font-size: 80%;">The code for solving the ODE</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs024.html#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer" style="font-size: 80%;">The network with one input layer, specified number of hidden layers, and one output layer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs025.html#example-population-growth" style="font-size: 80%;">Example: Population growth</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs026.html#setting-up-the-problem" style="font-size: 80%;">Setting up the problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs047.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs028.html#the-program-using-autograd" style="font-size: 80%;">The program using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs029.html#using-forward-euler-to-solve-the-ode" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs030.html#example-solving-the-one-dimensional-poisson-equation" style="font-size: 80%;">Example: Solving the one dimensional Poisson equation</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs031.html#the-specific-equation-to-solve-for" style="font-size: 80%;">The specific equation to solve for</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs032.html#solving-the-equation-using-autograd" style="font-size: 80%;">Solving the equation using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs033.html#comparing-with-a-numerical-scheme" style="font-size: 80%;">Comparing with a numerical scheme</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs034.html#setting-up-the-code" style="font-size: 80%;">Setting up the code</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs035.html#partial-differential-equations" style="font-size: 80%;">Partial Differential Equations</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs036.html#type-of-problem" style="font-size: 80%;">Type of problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs037.html#network-requirements" style="font-size: 80%;">Network requirements</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs038.html#more-details" style="font-size: 80%;">More details</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs039.html#example-the-diffusion-equation" style="font-size: 80%;">Example: The diffusion equation</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs040.html#defining-the-problem" style="font-size: 80%;">Defining the problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs041.html#setting-up-the-network-using-autograd" style="font-size: 80%;">Setting up the network using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs042.html#setting-up-the-network-using-autograd-the-trial-solution" style="font-size: 80%;">Setting up the network using Autograd; The trial solution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs043.html#why-the-jacobian" style="font-size: 80%;">Why the jacobian?</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs044.html#setting-up-the-network-using-autograd-the-full-program" style="font-size: 80%;">Setting up the network using Autograd; The full program</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs045.html#example-solving-the-wave-equation-with-neural-networks" style="font-size: 80%;">Example: Solving the wave equation with Neural Networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs046.html#the-problem-to-solve-for" style="font-size: 80%;">The problem to solve for</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs047.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs048.html#the-analytical-solution" style="font-size: 80%;">The analytical solution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs049.html#solving-the-wave-equation-the-full-program-using-autograd" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs050.html#resources-on-differential-equations-and-deep-learning" style="font-size: 80%;">Resources on differential equations and deep learning</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs051.html#convolutional-neural-networks-recognizing-images" style="font-size: 80%;">Convolutional Neural Networks (recognizing images)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs052.html#what-is-the-difference" style="font-size: 80%;">What is the Difference</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs053.html#neural-networks-vs-cnns" style="font-size: 80%;">Neural Networks vs CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs054.html#why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc" style="font-size: 80%;">Why CNNS for images, sound files, medical images from CT scans etc?</a></li>
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<!-- navigation toc: --> <li><a href="._week42-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>
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<!-- navigation toc: --> <li><a href="._week42-bs056.html#3d-volumes-of-neurons" style="font-size: 80%;">3D volumes of neurons</a></li>
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<!-- navigation toc: --> <li><a href="#layers-used-to-build-cnns" style="font-size: 80%;">Layers used to build CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs058.html#transforming-images" style="font-size: 80%;">Transforming images</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs059.html#cnns-in-brief" style="font-size: 80%;">CNNs in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs060.html#key-idea" style="font-size: 80%;">Key Idea</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs061.html#mathematics-of-cnns" style="font-size: 80%;">Mathematics of CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs062.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;">Convolution Examples: Polynomial multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs063.html#efficient-polynomial-multiplication" style="font-size: 80%;">Efficient Polynomial Multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs064.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;">A more efficient way of coding the above Convolution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs065.html#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;">Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs066.html#principle-of-superposition" style="font-size: 80%;">Principle of Superposition</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs067.html#simple-code-example" style="font-size: 80%;">Simple Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs068.html#wrapping-up-fourier-transforms" style="font-size: 80%;">Wrapping up Fourier transforms</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs069.html#finding-the-coefficients" style="font-size: 80%;">Finding the Coefficients</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs070.html#final-words-on-fourier-transforms" style="font-size: 80%;">Final words on Fourier Transforms</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs075.html#cnns-in-more-detail-lecture-from-in5400" style="font-size: 80%;">CNNs in more detail, Lecture from IN5400</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs076.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>
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<!-- navigation toc: --> <li><a href="._week42-bs077.html#setting-it-up" style="font-size: 80%;">Setting it up</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs078.html#the-mnist-dataset-again" style="font-size: 80%;">The MNIST dataset again</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs079.html#strong-correlations" style="font-size: 80%;">Strong correlations</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs080.html#layers-of-a-cnn" style="font-size: 80%;">Layers of a CNN</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs081.html#systematic-reduction" style="font-size: 80%;">Systematic reduction</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs082.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="._week42-bs083.html#importing-keras-and-tensorflow" style="font-size: 80%;">Importing Keras and Tensorflow</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs084.html#running-with-keras" style="font-size: 80%;">Running with Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs085.html#final-part" style="font-size: 80%;">Final part</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs086.html#final-visualization" style="font-size: 80%;">Final visualization</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs087.html#the-cifar01-data-set" style="font-size: 80%;">The CIFAR01 data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs088.html#verifying-the-data-set" style="font-size: 80%;">Verifying the data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs089.html#set-up-the-model" style="font-size: 80%;">Set up the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs090.html#add-dense-layers-on-top" style="font-size: 80%;">Add Dense layers on top</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs092.html#finally-evaluate-the-model" style="font-size: 80%;">Finally, evaluate the model</a></li>
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|
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<a name="part0057"></a>
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<!-- !split -->
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<h2 id="layers-used-to-build-cnns" class="anchor">Layers used to build CNNs </h2>
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<p>
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A simple CNN is a sequence of layers, and every layer of a CNN
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transforms one volume of activations to another through a
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differentiable function. We use three main types of layers to build
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CNN architectures: Convolutional Layer, Pooling Layer, and
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Fully-Connected Layer (exactly as seen in regular Neural Networks). We
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will stack these layers to form a full CNN architecture.
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<p>
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A simple CNN for image classification could have the architecture:
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||
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||
<ul>
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<li> <b>INPUT</b> (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.</li>
|
||
<li> <b>CONV</b> (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.</li>
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<li> <b>RELU</b> layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).</li>
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<li> <b>POOL</b> (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).</li>
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||
<li> <b>FC</b> (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.</li>
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