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[Convolutional Neural Networks] week1. Foundations of Convolutional Neural Networks
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<time datetime="2017-11-19T00:00:00+01:00"><i class="fa fa-calendar"></i> Sun, 19 Nov 2017</time>
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<span class="label label-default">Series</span>
Part 10 of «Andrew Ng Deep Learning MOOC»
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#computer-vision" title="Computer Vision">Computer Vision</a></li><li><a class="toc-href" href="#edge-detection-example" title="Edge Detection Example">Edge Detection Example</a></li><li><a class="toc-href" href="#more-edge-detection" title="More Edge Detection">More Edge Detection</a></li><li><a class="toc-href" href="#padding" title="Padding">Padding</a></li><li><a class="toc-href" href="#strided-convolutions" title="Strided Convolutions">Strided Convolutions</a></li><li><a class="toc-href" href="#convolutions-over-volume" title="Convolutions Over Volume">Convolutions Over Volume</a></li><li><a '="" class="toc-href" href="#filters-n_c_1" title="filters = n_c">filters = n_c'</a><ul><li><a class="toc-href" href="#one-layer-of-a-convolutional-network" title="One Layer of a Convolutional Network">One Layer of a Convolutional Network</a></li><li><a class="toc-href" href="#simple-convolutional-network-example" title="Simple Convolutional Network Example">Simple Convolutional Network Example</a></li><li><a class="toc-href" href="#pooling-layers" title="Pooling Layers">Pooling Layers</a></li><li><a class="toc-href" href="#cnn-example" title="CNN Example">CNN Example</a></li><li><a class="toc-href" href="#why-convolutions" title="Why Convolutions?">Why Convolutions?</a></li></ul></li></ul></div>
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</div>
<h3 id="computer-vision">Computer Vision</h3>
<p>CV with DL: rapid progress in past two years. </p>
<p>CV problems: </p>
<ul>
<li>image classification </li>
<li>object detection: bounding box of objects </li>
<li>neural style transfer </li>
</ul>
<p>input features could be very high dimension: e.g. 1000x1000 image → 3 million dim input ⇒ if 1st layer has 1000 hidden units → <em>3 billion params for first layer...</em> </p>
<p>foundamental operation: convolution. </p>
<h3 id="edge-detection-example">Edge Detection Example</h3>
<p>Motivating example for convolution operation: detecting vertical edges. </p>
<p>Convolve image with a <strong>filter(kernel) </strong>matrix:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image001.png"/><br/>
Each element in resulting matrix: sum(element-wise multiplication of filter and input image). </p>
<p>Why the filter can detect vertical edge?<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image003.png"/> </p>
<h3 id="more-edge-detection">More Edge Detection</h3>
<p><strong>Positive V.S. negative edges</strong>:<br/>
dark to light V.S. light to dark<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image004.png"/> </p>
<p>Instead of picking filter by hand, the actual parameters can be <em>learned</em> by ML.<br/>
Next: discuss some building blocks of CNN, padding/striding/pooling... </p>
<h3 id="padding">Padding</h3>
<p>Earlier: image <em>shrinks</em> after convolution.<br/>
Input <code>n*n</code> image, convolve with <code>f*f</code> filter ⇒ output shape = <code>(n-f+1) * (n-f+1)</code><br/>
downside: </p>
<ul>
<li>image shrinks on every step (if 100 layer → shrinks to very small images) </li>
<li>pixels <em>at corner</em> are less used in the output </li>
</ul>
<p>⇒ <em>pad the image</em> so that output shape is invariant. </p>
<p>if <code>p</code> = padding amount (width of padded border)<br/>
→ output shape = <code>(n+2p-f+1) * (n+2p-f+1)</code> </p>
<p>Terminology: <em>valid</em> and <em>same</em> convolutions: </p>
<ul>
<li><strong>valid</strong> convolution: no padding, output shape = (n-f+1) * (n-f+1) </li>
<li><strong>same</strong> convolution: output size equals input size. i.e. filter width <code>p = (f-1) / 2</code> (only works <em>when f is odd — </em>this is also a convention in CV, partially because this way there'll be a central filter) </li>
</ul>
<h3 id="strided-convolutions">Strided Convolutions</h3>
<p>Example <code>stride = 2</code> in convolution:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image005.png"/><br/>
(convention: stop moving if filter goes out of image border.) </p>
<p>if input image <code>n*n</code>, filter size <code>f*f</code>, padding = <code>f</code>, stride = <code>s</code><br/>
⇒ output shape = <code>(floor((n+2p-f)/s) + 1) * (floor((n+2p-f)/s) + 1)</code> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image006.png"/> </p>
<p><strong>Note on cross-correlation v.s. convolution</strong><br/>
In math books, "convolution" involves <em>flip filter in both direction</em> before doing "convolution" operation.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image007.png"/><br/>
The operation discribed before is called "cross-correlation". </p>
<p>(Why doing the flipping in math: to ensure assosative law for convolution — (A<em>B)</em>C=A<em>(B</em>C).) </p>
<h3 id="convolutions-over-volume">Convolutions Over Volume</h3>
<p>example: convolutions on RGB image<br/>
image size = 6<em>6</em>3 = height * width * #channels<br/>
filter size = 3<em>3</em>3, (convention: <em>filter's #channels matches the image</em>)<br/>
output size = 4<em>4 (</em>1) — <strong>output is 2D for each filter</strong>. </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image008.png"/> </p>
<p><strong>Multiple filters:</strong> </p>
<ul>
<li>take >1 filters </li>
<li>stack outputs together to form an <em>output volume</em>. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image009.png"/> </p>
<p><strong>Summary of dimensions</strong>:<br/>
input shape = <code>n*n*n_c</code><br/>
filter shape = <code>f*f*n_c</code> </p>
<h1 id="filters-n_c_1">filters = <code>n_c</code>'</h1>
<p>⇒ output shape = <code>(n-f+1) * (n-f+1) * n_c</code>' </p>
<h3 id="one-layer-of-a-convolutional-network">One Layer of a Convolutional Network</h3>
<p>For each filter's output: <em>add bias b, then apply nonlinear activation function.</em> </p>
<p>One layer of a CNN:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image011.png"/> </p>
<p>with analogy to normall NN: </p>
<ul>
<li>linear operation (matrix mul V.S. convolution) </li>
<li>bias </li>
<li>nonlinear activation </li>
<li>difference: Number of parameters <em>doesn't depend on input dimension</em>: even for very large images. </li>
</ul>
<p><strong>Notation summary:</strong><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image014.png"/><br/>
note: <em>ordering</em> of dimensions: example index, height, width, #channel. </p>
<h3 id="simple-convolutional-network-example">Simple Convolutional Network Example</h3>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image015.png"/> </p>
<p>general trend: as going to later layers, <em>image size shrinks, #channels increases</em>. </p>
<h3 id="pooling-layers">Pooling Layers</h3>
<p>Pooling layers makes CNN more robust. </p>
<p><strong>Max pooling</strong><br/>
divide input into regions, take max of each region. </p>
<ul>
<li>Hyperparams: </li>
</ul>
<p>(common choice) <em>filter size f=2 or 3, strid size s=2, padding p=0.</em> </p>
<ul>
<li>note: <em>no params to learn</em> for max pooling layer, pooling layer not counted in #layers (conv-pool as a single layer) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image017.png"/><br/>
Intuition: a large number indicats a detected feature in that region → preseved after pooling. </p>
<p>Formula of dimension <code>floor((n+2p-f+1)/s + 1)</code> holds for POOL layer as well. </p>
<p>Output of max pooling: the same #channels as input (i.e. do maxpooling on each channel). </p>
<p><strong>Average pooling</strong><br/>
Less often used than max pooling.<br/>
Typical usecase: collapse 7<em>7</em>1000 activation into 1<em>1</em>1000. </p>
<h3 id="cnn-example">CNN Example</h3>
<p><em>LeNet-5</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image018.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c4wk1/pasted_image020.png"/> </p>
<h3 id="why-convolutions">Why Convolutions?</h3>
<p>2 main advantages of CONV over FC: <em>param sharing; sparsity of connections.</em> </p>
<p><strong>Parameter sharing</strong>:<br/>
A feature detector useful in one part of img is probably useful in another part as well.<br/>
→ no need to learn separate feature detectors in different parts. </p>
<p><strong>Sparsity of connections</strong>:<br/>
For each output value depends only on a small number of inputs (the pixels near that position) </p>
<ul>
<li>Invarance to translation... </li>
</ul>
</div>
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<div id="toc"><ul><li><a class="toc-href" href="#computer-vision" title="Computer Vision">Computer Vision</a></li><li><a class="toc-href" href="#edge-detection-example" title="Edge Detection Example">Edge Detection Example</a></li><li><a class="toc-href" href="#more-edge-detection" title="More Edge Detection">More Edge Detection</a></li><li><a class="toc-href" href="#padding" title="Padding">Padding</a></li><li><a class="toc-href" href="#strided-convolutions" title="Strided Convolutions">Strided Convolutions</a></li><li><a class="toc-href" href="#convolutions-over-volume" title="Convolutions Over Volume">Convolutions Over Volume</a></li><li><a '="" class="toc-href" href="#filters-n_c_1" title="filters = n_c">filters = n_c'</a><ul><li><a class="toc-href" href="#one-layer-of-a-convolutional-network" title="One Layer of a Convolutional Network">One Layer of a Convolutional Network</a></li><li><a class="toc-href" href="#simple-convolutional-network-example" title="Simple Convolutional Network Example">Simple Convolutional Network Example</a></li><li><a class="toc-href" href="#pooling-layers" title="Pooling Layers">Pooling Layers</a></li><li><a class="toc-href" href="#cnn-example" title="CNN Example">CNN Example</a></li><li><a class="toc-href" href="#why-convolutions" title="Why Convolutions?">Why Convolutions?</a></li></ul></li></ul></div>
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