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2013-03-01-SVM.html
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2013-03-01-SVM.html
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<head>
<title>2013-03-01-SVM</title>
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<meta name="title" content="2013-03-01-SVM"/>
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</head>
<body>
<div id="preamble">
</div>
<div id="content">
<h1 class="title">2013-03-01-SVM</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#sec-1">1 Linear Regression</a></li>
<li><a href="#sec-2">2 Types of Models</a>
<ul>
<li><a href="#sec-2-1">2.1 Details</a></li>
</ul>
</li>
<li><a href="#sec-3">3 Case Study</a>
<ul>
<li><a href="#sec-3-1">3.1 Problem</a></li>
</ul>
</li>
<li><a href="#sec-4">4 Solution?</a>
<ul>
<li><a href="#sec-4-1">4.1 Prompts</a></li>
</ul>
</li>
<li><a href="#sec-5">5 Similarity</a>
<ul>
<li><a href="#sec-5-1">5.1 Challenge</a></li>
<li><a href="#sec-5-2">5.2 Log & Square</a></li>
</ul>
</li>
<li><a href="#sec-6">6 Point Distance</a>
<ul>
<li><a href="#sec-6-1">6.1 Error</a></li>
</ul>
</li>
<li><a href="#sec-7">7 Aggregate</a>
<ul>
<li><a href="#sec-7-1">7.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-8">8 Fitness Function</a>
<ul>
<li><a href="#sec-8-1">8.1 Fitness</a></li>
</ul>
</li>
<li><a href="#sec-9">9 Understanding Error</a>
<ul>
<li><a href="#sec-9-1">9.1 Error</a></li>
</ul>
</li>
<li><a href="#sec-10">10 Solution as Minimization</a></li>
<li><a href="#sec-11">11 Solution Approximation</a>
<ul>
<li><a href="#sec-11-1">11.1 Analytical</a></li>
</ul>
</li>
<li><a href="#sec-12">12 Gradient Descent</a>
<ul>
<li><a href="#sec-12-1">12.1 Steps</a></li>
</ul>
</li>
<li><a href="#sec-13">13 General Case</a>
<ul>
<li><a href="#sec-13-1">13.1 Approximate visualization</a></li>
</ul>
</li>
<li><a href="#sec-14">14 Support Vector Machines</a></li>
<li><a href="#sec-15">15 Decision Trees</a>
<ul>
<li><a href="#sec-15-1">15.1 Linearly Separable</a></li>
<li><a href="#sec-15-2">15.2 Details</a></li>
</ul>
</li>
<li><a href="#sec-16">16 Possibilities</a>
<ul>
<li><a href="#sec-16-1">16.1 Best?</a></li>
<li><a href="#sec-16-2">16.2 Best Separator</a></li>
<li><a href="#sec-16-3">16.3 Points, Vectors</a></li>
</ul>
</li>
<li><a href="#sec-17">17 Dimensions</a>
<ul>
<li><a href="#sec-17-1">17.1 Vocabulary</a></li>
</ul>
</li>
<li><a href="#sec-18">18 Expressing the Hyperplane</a>
<ul>
<li><a href="#sec-18-1">18.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-19">19 Challenge</a></li>
<li><a href="#sec-20">20 Maximizing Fitness Function</a></li>
<li><a href="#sec-21">21 Kernel Tricks</a>
<ul>
<li><a href="#sec-21-1">21.1 Polynomial Kernel</a>
<ul>
<li><a href="#sec-21-1-1">21.1.1 Square</a></li>
</ul>
</li>
<li><a href="#sec-21-2">21.2 Polynomial Kernel</a></li>
<li><a href="#sec-21-3">21.3 Details</a></li>
</ul>
</li>
<li><a href="#sec-22">22 <b>Break</b></a></li>
</ul>
</div>
</div>
<div id="outline-container-1" class="outline-2">
<h2 id="sec-1"><span class="section-number-2">1</span> Linear Regression <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-1">
</div>
</div>
<div id="outline-container-2" class="outline-2">
<h2 id="sec-2"><span class="section-number-2">2</span> Types of Models <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-2">
<ul>
<li>Classifiers
</li>
<li>Regressions
</li>
<li>Clustering
</li>
<li>Outlier
</li>
</ul>
</div>
<div id="outline-container-2-1" class="outline-3">
<h3 id="sec-2-1"><span class="section-number-3">2.1</span> Details <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-2-1">
<dl>
<dt>Classifiers</dt><dd>describes and distinguishes cases. Yelp may want to find a
category for a business based on the reviews and business description
</dd>
<dt>Regressions</dt><dd>Predict a continuous value. Eg. predict a home's selling
price given sq footage, # of bedrooms
</dd>
<dt>Clustering</dt><dd>find "natural" groups of data <b>without labels</b>
</dd>
<dt>Outlier</dt><dd>find anomalous transactions, eg. finding fraud for credit cards
</dd>
</dl>
</div>
</div>
</div>
<div id="outline-container-3" class="outline-2">
<h2 id="sec-3"><span class="section-number-2">3</span> Case Study <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-3">
<ul>
<li>Housing prices: square footage
</li>
</ul>
<p> <img src="img/housing-regression.gif" alt="img/housing-regression.gif" />
</p>
</div>
<div id="outline-container-3-1" class="outline-3">
<h3 id="sec-3-1"><span class="section-number-3">3.1</span> Problem <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-3-1">
<ul>
<li>We'd like to know how to price a house based on the square footage
</li>
<li>Let's pretend this is the data we have
</li>
<li>How would we guess that value for 2500 sq ft?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-4" class="outline-2">
<h2 id="sec-4"><span class="section-number-2">4</span> Solution? <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-4">
<ul>
<li>Find a line that represents the data
</li>
<li><code>y = m*x + b</code>
</li>
<li>A line that is not very far from the points
</li>
</ul>
</div>
<div id="outline-container-4-1" class="outline-3">
<h3 id="sec-4-1"><span class="section-number-3">4.1</span> Prompts <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-4-1">
<ul>
<li>In English, how would you solve this?
</li>
<li>How to mathematically represent the line?
</li>
<li>What is a good line?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-5" class="outline-2">
<h2 id="sec-5"><span class="section-number-2">5</span> Similarity <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-5">
<ul>
<li>Main challenges in data mining: defining a specific metric for an intuition
</li>
<li>Define distance for an individual point
</li>
<li>Define how to aggregate distances together
</li>
</ul>
</div>
<div id="outline-container-5-1" class="outline-3">
<h3 id="sec-5-1"><span class="section-number-3">5.1</span> Challenge <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-5-1">
<ul>
<li>This is big problem for engineering and math (stats) in general
</li>
<li>We'll cover some concepts, but if you're ever stuck, try looking in related
fields
</li>
<li>What are some of the ways we can measure distance between points?
Euclidian, Manhattan, Euclidian == L<sub>2</sub> norm
</li>
<li>What is a way to aggrgate numbers? sum, sum of squares, sum of logs
</li>
<li>Differences between the last two?
</li>
</ul>
</div>
</div>
<div id="outline-container-5-2" class="outline-3">
<h3 id="sec-5-2"><span class="section-number-3">5.2</span> Log & Square <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h3>
<div class="outline-text-3" id="text-5-2">
<dl>
<dt>Log</dt><dd>Useful for de-emphasizing large raw differences
</dd>
<dt>Square</dt><dd>Useful for taking the approximate absolute value
</dd>
</dl>
<p> <img src="img/logx.gif" alt="img/logx.gif" />
</p>
</div>
</div>
</div>
<div id="outline-container-6" class="outline-2">
<h2 id="sec-6"><span class="section-number-2">6</span> Point Distance <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-6">
<ul>
<li><code>y</code> distance from line
</li>
<li>Intuitively: error in estimate
</li>
<li><code>h(x) = m*x + b</code>
</li>
<li><code>err = h(x) - y</code>
</li>
</ul>
<p> <img src="img/error.gif" alt="img/error.gif" />
</p>
</div>
<div id="outline-container-6-1" class="outline-3">
<h3 id="sec-6-1"><span class="section-number-3">6.1</span> Error <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-6-1">
<ul>
<li>We want the difference from what we estimate to be the value to what the
value actually is
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-7" class="outline-2">
<h2 id="sec-7"><span class="section-number-2">7</span> Aggregate <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-7">
<ul>
<li><code>sum</code>
</li>
<li>What about negative error?
</li>
<li>Sum of squares
</li>
<li><code>err = sum( (h(x) - y)**2 for x,y in dataset) / len(dataset)</code>
</li>
</ul>
</div>
<div id="outline-container-7-1" class="outline-3">
<h3 id="sec-7-1"><span class="section-number-3">7.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-7-1">
<ul>
<li>Now we have info about all the errors from points, how to summarize?
</li>
<li>Some points have negative error, some positive? Do they cancel each other
out?
</li>
<li>Imagine data set of two points: one solutions covers lines, other divides
them. Which is better?
</li>
<li>Use our squaring trick to make sure we don't have any negative values
</li>
<li>Normalize by the number of points
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-8" class="outline-2">
<h2 id="sec-8"><span class="section-number-2">8</span> Fitness Function <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-8">
<ul>
<li>Measures the quality or cost of the solution
</li>
<li><b>Key</b> ingredient for data mining algorithms
</li>
<li>If you can measure it, you can find the best solution
</li>
</ul>
</div>
<div id="outline-container-8-1" class="outline-3">
<h3 id="sec-8-1"><span class="section-number-3">8.1</span> Fitness <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-8-1">
<ul>
<li>Function spits out a metric. Metric can be thought of as <b>fitness</b> or
<b>cost</b>
</li>
<li>Find the maximum or minimum of that metric
</li>
<li>Depending on your fitness function, this can be easy or difficult
</li>
<li>img: <a href="http://onlinestatbook.com">http://onlinestatbook.com</a>
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-9" class="outline-2">
<h2 id="sec-9"><span class="section-number-2">9</span> Understanding Error <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-9">
<p> <img src="img/Linear_regression.svg.png" alt="img/Linear_regression.svg.png" />
Several possible solutions
</p>
</div>
<div id="outline-container-9-1" class="outline-3">
<h3 id="sec-9-1"><span class="section-number-3">9.1</span> Error <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-9-1">
<ul>
<li>What happens to the error as we move line around?
</li>
<li>Decreases until best fit, then increases
</li>
<li>What happens if we plot this error? Say, slope (x) against error (y)?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-10" class="outline-2">
<h2 id="sec-10"><span class="section-number-2">10</span> Solution as Minimization <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-10">
<ul>
<li>Error is a parabola
</li>
<li>Several methods for finding the minimum
</li>
<li>Two categories: analytical, approximations
</li>
</ul>
<p><img src="img/parabola.png" alt="img/parabola.png" />
</p>
</div>
</div>
<div id="outline-container-11" class="outline-2">
<h2 id="sec-11"><span class="section-number-2">11</span> Solution Approximation <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-11">
<ul>
<li>Some fitness functions can be difficult to solve analytically
</li>
<li>Alternative: iteratively get closer to the solution
</li>
<li>Stop when answer is close enough
</li>
</ul>
</div>
<div id="outline-container-11-1" class="outline-3">
<h3 id="sec-11-1"><span class="section-number-3">11.1</span> Analytical <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-11-1">
<ul>
<li>How to find the minimum of functions in general?
</li>
<li>Take derivative, find 0
</li>
<li>Taking derivative can be complex or impossible (discontinuities) for some
functions, or solving for 0 is difficult
</li>
<li>Instead, well keep getting closer to the minimum using the function we
already have
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-12" class="outline-2">
<h2 id="sec-12"><span class="section-number-2">12</span> Gradient Descent <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-12">
<ol>
<li>Estimate current gradient (derivative)
</li>
<li>Take a step (<code>a * deriv</code>) in the direction of the gradient
</li>
<li>Step size is small, stop. Else repeat.
</li>
</ol>
<p> <img src="img/parabola.png" alt="img/parabola.png" />
</p>
</div>
<div id="outline-container-12-1" class="outline-3">
<h3 id="sec-12-1"><span class="section-number-3">12.1</span> Steps <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-12-1">
<ul>
<li>Take gradient by looking at the local derivative, or perturbating x
</li>
<li>Choose <code>a</code> as step size weight: big <code>a</code> is large step size
</li>
<li>If <code>deriv</code> is large, will also make you step size large.
</li>
<li>If <code>deriv</code> is large, probably means you are far away from minimum
</li>
<li>Keep repeating
</li>
<li>What happens if <code>a</code> is too small?
</li>
<li>What happens if <code>a</code> is too big?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-13" class="outline-2">
<h2 id="sec-13"><span class="section-number-2">13</span> General Case <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-13">
<ul>
<li>Formulate fitness function for your problem
</li>
<li>Use analytics or approximations to find min/max
</li>
<li>Approximations: Newton's Method, Gradient Descent
</li>
</ul>
<p> <img src="img/error-reduce.png" alt="img/error-reduce.png" />
</p>
</div>
<div id="outline-container-13-1" class="outline-3">
<h3 id="sec-13-1"><span class="section-number-3">13.1</span> Approximate visualization <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-13-1">
<ul>
<li>Desired output of the error as gradient descent runs
</li>
<li>maybe some local problems, as step size is too big, but slowly move down to
a small amount of error
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-14" class="outline-2">
<h2 id="sec-14"><span class="section-number-2">14</span> Support Vector Machines <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-14">
</div>
</div>
<div id="outline-container-15" class="outline-2">
<h2 id="sec-15"><span class="section-number-2">15</span> Decision Trees <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-15">
<ul>
<li>Great for separable attributes
</li>
<li>Rules operate on independent attributes
</li>
<li>Classes separable along an axis/attribute
</li>
</ul>
<p> <img src="img/tree.png" alt="img/tree.png" />
</p>
</div>
<div id="outline-container-15-1" class="outline-3">
<h3 id="sec-15-1"><span class="section-number-3">15.1</span> Linearly Separable <span class="tag"><span class="slide">slide</span></span></h3>
<div class="outline-text-3" id="text-15-1">
<ul>
<li>How to handle case where separator line is not along an axis?
</li>
</ul>
<p> <img src="img/dataset_linsep.png" alt="img/dataset_linsep.png" />
</p></div>
</div>
<div id="outline-container-15-2" class="outline-3">
<h3 id="sec-15-2"><span class="section-number-3">15.2</span> Details <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-15-2">
<ul>
<li>Could say if <code>x>2</code> and <code>y>2</code>, but not a great intuitive fit
</li>
<li>Draw a line that takes both into account
</li>
<li><code>y = m*x + b</code>
</li>
<li>img: <a href="http://www.eric-kim.net/eric-kim-net/posts/1/kernel_trick.html">http://www.eric-kim.net/eric-kim-net/posts/1/kernel_trick.html</a>
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-16" class="outline-2">
<h2 id="sec-16"><span class="section-number-2">16</span> Possibilities <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-16">
<ul>
<li>Many lines <b>could</b> separate these classes
</li>
</ul>
<p> <img src="img/dataset_linsep.png" alt="img/dataset_linsep.png" />
</p>
</div>
<div id="outline-container-16-1" class="outline-3">
<h3 id="sec-16-1"><span class="section-number-3">16.1</span> Best? <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-16-1">
<ul>
<li>Which is the best?
</li>
<li>Why?
</li>
</ul>
</div>
</div>
<div id="outline-container-16-2" class="outline-3">
<h3 id="sec-16-2"><span class="section-number-3">16.2</span> Best Separator <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h3>
<div class="outline-text-3" id="text-16-2">
<ul>
<li>Best line gives the most distance between the two classes
</li>
<li>Measure distance between closest points
</li>
<li>Closest points == support vectors
</li>
</ul>
<p> <img src="img/separable.jpg" alt="img/separable.jpg" />
</p></div>
</div>
<div id="outline-container-16-3" class="outline-3">
<h3 id="sec-16-3"><span class="section-number-3">16.3</span> Points, Vectors <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-16-3">
<ul>
<li>Points can be represented as vectors
</li>
<li>Vector math can be easier to express succinctly
</li>
<li>img: <a href="http://www.sciencedirect.com/science/article/pii/S1072751511001918">http://www.sciencedirect.com/science/article/pii/S1072751511001918</a>
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-17" class="outline-2">
<h2 id="sec-17"><span class="section-number-2">17</span> Dimensions <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-17">
<ul>
<li>When separating two dimensions, we need a line
</li>
<li>When separating 3 dimensions?
</li>
<li>4 dimensions?
</li>
</ul>
</div>
<div id="outline-container-17-1" class="outline-3">
<h3 id="sec-17-1"><span class="section-number-3">17.1</span> Vocabulary <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-17-1">
<ul>
<li>Plane
</li>
<li>Hyperplane
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-18" class="outline-2">
<h2 id="sec-18"><span class="section-number-2">18</span> Expressing the Hyperplane <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-18">
<ul>
<li><code>y = m*x + b</code>
</li>
<li><code>x_2 = m*x_1 + b</code>
</li>
<li><code>0 = m*x_1 + b - x_2</code>
</li>
<li><code>0 = [m -1] * [x_1, x_2] + b</code>
</li>
<li><code>0 = w * x + b</code>
</li>
</ul>
</div>
<div id="outline-container-18-1" class="outline-3">
<h3 id="sec-18-1"><span class="section-number-3">18.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-18-1">
<ul>
<li>How do you mathematically represent a line?
</li>
<li>Now, we're not going to think of a new letter for every dimension, we're
just going to say x<sub>1</sub> , x<sub>2</sub> , x<sub>3</sub> …
</li>
<li>Rewrite mathematically
</li>
<li>How to add more dimensions? x<sub>22</sub>? Express x as a vector of all attributes
</li>
<li>Again, don't want to come up with a bunch more letters after <code>m</code>, so use
<code>w</code> as the matrix representing all the <code>m</code> slopes
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-19" class="outline-2">
<h2 id="sec-19"><span class="section-number-2">19</span> Challenge <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-19">
<ul>
<li>Find <code>w</code>, <code>b</code> such that <code>w * x + b</code> maximizes the distance between the
support vectors
</li>
</ul>
<p> <img src="img/svm.png" alt="img/svm.png" />
</p>
</div>
</div>
<div id="outline-container-20" class="outline-2">
<h2 id="sec-20"><span class="section-number-2">20</span> Maximizing Fitness Function <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-20">
<ul>
<li>Now we have a fitness function and parameters we're trying to optimize
</li>
<li>Sound familiar?
</li>
</ul>
<p> <img src="img/svm.png" alt="img/svm.png" />
</p>
</div>
</div>
<div id="outline-container-21" class="outline-2">
<h2 id="sec-21"><span class="section-number-2">21</span> Kernel Tricks <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-21">
<ul>
<li>SVM good for linearly separable data
</li>
<li>How to handle other data?
</li>
</ul>
<p> <img src="img/svm-circular.jpg" alt="img/svm-circular.jpg" />
</p>
</div>
<div id="outline-container-21-1" class="outline-3">
<h3 id="sec-21-1"><span class="section-number-3">21.1</span> Polynomial Kernel <span class="tag"><span class="slide">slide</span></span></h3>
<div class="outline-text-3" id="text-21-1">
<ul>
<li>Transform it into linearly separable
</li>
<li>What function can we apply to these data points to make them separable?
</li>
</ul>
<p> <img src="img/svm-circular.jpg" alt="img/svm-circular.jpg" />
</p>
</div>
<div id="outline-container-21-1-1" class="outline-4">
<h4 id="sec-21-1-1"><span class="section-number-4">21.1.1</span> Square <span class="tag"><span class="notes">notes</span></span></h4>
<div class="outline-text-4" id="text-21-1-1">
<ul>
<li>Square all of them
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-21-2" class="outline-3">
<h3 id="sec-21-2"><span class="section-number-3">21.2</span> Polynomial Kernel <span class="tag"><span class="slide">slide</span></span></h3>
<div class="outline-text-3" id="text-21-2">
<p> <img src="img/kernel-trick.jpg" alt="img/kernel-trick.jpg" />
</p>
<p>
Now apply SVM
</p></div>
</div>
<div id="outline-container-21-3" class="outline-3">
<h3 id="sec-21-3"><span class="section-number-3">21.3</span> Details <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-21-3">
<ul>
<li>img: <a href="http://www.sciencedirect.com/science/article/pii/S1072751511001918">http://www.sciencedirect.com/science/article/pii/S1072751511001918</a>
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-22" class="outline-2">
<h2 id="sec-22"><span class="section-number-2">22</span> <b>Break</b> <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-22">
<script type="text/javascript" src="production/org-html-slideshow.js"></script>
</div>
</div>
</div>
<div id="postamble">
<p class="date">Date: 2013-03-15 10:19:25 PDT</p>
<p class="author">Author: Jim Blomo</p>
<p class="creator">Org version 7.8.02 with Emacs version 23</p>
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