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cs8850_07_curse.html
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<!doctype html>
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<title>Advanced Machine Learning</title>
<meta name="description" content="CS8850 GSU class">
<meta name="author" content="Sergey M Plis">
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<section>
<section>
<h2>Advanced Machine Learning</h2>
<h3>07: Curse of Dimensionality</h3>
</section>
<section>
<h3>Schedule</h3>
<row>
<col50>
<table style="font-size:14px">
<tr>
<th>#</th>
<th>date</th>
<th>topic</th>
<th>description</th>
</tr>
<tr><td>1</td>
<td> 22-Aug-2022 </td>
<td> Introduction </td>
<td></td>
</tr>
<tr>
<td> 2 </td>
<td> 24-Aug-2022 </td>
<td> Foundations of learning </td>
<td> </td>
</tr>
<tr><td> 3 </td><td> 29-Aug-2022 </td><td> PAC learnability </td><td> </td></tr>
<tr><td> 4 </td><td> 31-Aug-2022 </td><td> Linear algebra (recap) </td><td> hw1 released </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 05-Sep-2022 </td><td> <em>Holiday</em> </td><td> </td></tr>
<tr style='background-color: #E0E4CC;'><td> 5 </td><td> 07-Sep-2022 </td><td> Linear learning models </td><td> </td></tr>
<tr><td> 6 </td><td> 12-Sep-2022 </td><td> Principal Component Analysis </td><td> project ideas </td></tr>
<tr><td> 7 </td><td> 14-Sep-2022 </td><td> Curse of Dimensionality </td></td></td><td> <i class='fa fa-map-marker' style='color: #FA6900;'></i> hw1 due </td></tr>
<tr><td> 8 </td><td> 19-Sep-2022 </td><td> Bayesian Decision Theory </td><td>hw2 release</td></tr>
<tr><td> 9 </td><td> 21-Sep-2022 </td><td> Parameter estimation: MLE </td><td></td></tr>
<tr><td> 10 </td><td> 26-Sep-2022 </td><td> Parameter estimation: MAP & NB</td><td>finalize teams</td></tr>
<tr><td> 11 </td><td> 28-Sep-2022 </td><td> Logistic Regression </td><td> </td></tr>
<tr><td> 12 </td><td> 03-Oct-2022 </td><td> Kernel Density Estimation </td><td> </td></tr>
<tr><td> 13 </td><td> 05-Oct-2022 </td><td> Support Vector Machines </td><td> hw3, hw2 due </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 10-Oct-2022 </td><td> * Mid-point projects checkpoint </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 12-Oct-2022 </td><td> * Midterm: Semester Midpoint </td><td> exam </td></tr>
<tr><td> 14 </td><td> 17-Oct-2022 </td><td>Matrix Factorization</td><td> </td></tr>
<tr><td> 15 </td><td> 19-Oct-2022 </td><td>Stochastic Gradient Descent</td><td> </td></tr>
</table>
</col50>
<col50>
<table style="font-size:14px; vertical-align: top;">
<tr>
<th>#</th>
<th>date</th>
<th>topic</th>
<th>description</th>
</tr>
<tr><td> 16 </td><td> 24-Oct-2022 </td><td> k-means clustering </td><td> </td></tr>
<tr><td> 17 </td><td> 26-Oct-2022 </td><td> Expectation Maximization </td><td> hw4, hw3 due </td></tr>
<tr><td> 18 </td><td> 31-Oct-2022 </td><td> Automatic Differentiation </td><td> </td></tr>
<tr><td> 19 </td><td> 02-Nov-2022 </td><td> Nonlinear embedding approaches </td><td> </td></tr>
<tr><td> 20 </td><td> 07-Nov-2022 </td><td> Model comparison I </td><td> </td></tr>
<tr><td> 21 </td><td> 09-Nov-2022 </td><td> Model comparison II </td><td> hw5, hw4 due</td></tr>
<tr><td> 22 </td><td> 14-Nov-2022 </td><td> Model Calibration </td><td> </td></tr>
<tr><td> 23 </td><td> 16-Nov-2022 </td><td> Convolutional Neural Networks </td><td> </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 21-Nov-2022 </td><td> <em>Fall break</em> </td><td> </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 23-Nov-2022 </td><td> <em>Fall break</em> </td><td> </td></tr>
<tr><td> 24 </td><td> 28-Nov-2022 </td><td> Word Embedding </td><td> hw5 due </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 30-Nov-2022 </td><td> Presentation and exam prep day </td><td> </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 02-Dec-2022 </td><td> * Project Final Presentations </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 07-Dec-2022 </td><td> * Project Final Presentations </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 12-Dec-2022 </td><td> * Final Exam </td><td> * </td></tr>
<tr><td> </td><td> 15-Dec-2022 </td><td> Grades due </td><td> </td></tr>
</table>
</col50>
</row>
</section>
<section>
<h3>Outline of this lecture</h3>
<ul>
<li class="fragment roll-in" style="list-style-image: url('figures/watermelon_bullet.png');">Curse of dimensionality
</ul>
</section>
</section>
<!-- ------------------------------------------------------------------------- -->
<section>
<section>
<h2>Curse of dimensionality</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="500"
src="figures/watermelon.png" alt="watermelon cover">
<div class='slide-footer'>
in part based on: <a href="https://habr.com/ru/post/416551/" target="_blank">a blog post in Russian</a> and <a href="https://www.amazon.com/Art-Doing-Science-Engineering-Learning/dp/1732265178" target="_blank">The Art of Doing Science and Engineering: Learning to Learn</a>
</div>
</section>
<section>
<h3>How much rind?</h3>
<object id="watermelon" data="figures/watermelon.svg#1_0" ></object>
</section>
<section>
<h3>Generalized Pythagoras Theorem</h3>
<img width="700" src="figures/CD_GeneralizedPythagoras.png" alt="fraction">
$
D^2 = \sum_{i=1}^{n} x^2_{i}
$
<aside class="notes">
1. Derive d1
2. Derive d2 as a function of d1 and z
</aside>
</section>
<section>
<div id="header-right" style="right: -20%; top: -10%;">
<img style="margin-bottom: -5%" width="300" src="figures/CD_TrapezoidRule.png" alt="Trapezoid">
<br>
<small>Trapezoid rule</small>
</div>
<h3>Stirling approximation for $n!$</h3>
<ul style="list-style-type: none;">
<li class="fragment roll-in"> $\ln n! = \sum_{k=1}^n \ln k$
<li class="fragment roll-in"> $\int_1^n \ln x dx$
<li class="fragment roll-in"> $\int_1^n \ln x dx = n\ln n -n +1$
<li class="fragment roll-in"> $\int_1^n \ln x dx \sim {1 \over 2}\ln 1 + \ln 2 + \ln 3 + \dots + {1 \over 2}\ln n$
<li class="fragment roll-in"> $\sum_{k=1}^n \ln k \sim n\ln n - n + 1 + {1 \over 2}\ln n$
<li class="fragment roll-in"> $n! \sim Cn^ne^{-n}\sqrt{n}$
<li class="fragment roll-in"> $C = \sqrt{2\pi}$
<li class="fragment roll-in"> $n! \sim n^ne^{-n}\sqrt{2\pi n}$
</ul>
<aside class="notes">
Product is hard to handle and we will use logs<br>
Sum is related to an integral, lets consider one<br>
Show how to integrate by parts<br>
Add ${1 \over 2}\ln n$ to both terms<br>
C is trapezoid rule approximation error<br>
</aside>
</section>
<section>
<h3>Stirling approximation for $n!$</h3>
<table style="font-size: 32px;">
<thead>
<tr>
<th>$n$</th>
<th>Stirling</th>
<th>True</th>
<th>Stirling/True</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.92214</td>
<td>1</td>
<td>0.92214</td>
</tr>
<tr>
<td>2</td>
<td>1.91900</td>
<td>2</td>
<td>0.95950</td>
</tr>
<tr>
<td>3</td>
<td>5.83621</td>
<td>6</td>
<td>0.97270</td>
</tr>
<tr>
<td>4</td>
<td>23.50618</td>
<td>24</td>
<td>0.97942</td>
</tr>
<tr>
<td>5</td>
<td>118.01917</td>
<td>120</td>
<td>0.98349</td>
</tr>
<tr>
<td>6</td>
<td>710.07818</td>
<td>720</td>
<td>0.98622</td>
</tr>
<tr>
<td>7</td>
<td>4,980.3958</td>
<td>5,040</td>
<td>0.98817</td>
</tr>
<tr>
<td>8</td>
<td>39,902.3955</td>
<td>40,320</td>
<td>0.98964</td>
</tr>
<tr>
<td>9</td>
<td>359,536.87</td>
<td>362,880</td>
<td>0.99079</td>
</tr>
<tr>
<td>10</td>
<td>3,598,695.6</td>
<td>3,628,800</td>
<td>0.99170</td>
</tr>
</tbody>
</table>
<aside class="notes">
show that f(n) = n + sqrt(n) grows fast leading to ratio of 1 between itself and g(n) = n, while the difference is growing indefinitely as sqrt(n)
</aside>
</section>
<section>
<h3>$\Gamma$ (Gamma) function </h3>
<ul style="list-style-type: none;">
<li class="fragment roll-in"> $\Gamma(n) = \displaystyle\int_0^{\infty} x^{n-1}e^{-x}dx$
<li class="fragment roll-in"> Integration by parts: $\int udv = uv - \int vdu$
<li class="fragment roll-in"> $u=x^{n-1}$, $dv = e^{-x}dx$, $du = (n-1)x^{n-2}dx$, $v = -e^{-x}$
<li class="fragment roll-in"> $\Gamma(n) = -e^{-x}x^{n-1} \displaystyle|_0^{\infty} + (n-1)\int_0^{\infty} x^{n-2}e^{-x}dx$
<li class="fragment roll-in"> $\Gamma(n) = (n-1)\Gamma(n-1)$
<li class="fragment roll-in"> $\Gamma(1) = 1$
<li class="fragment roll-in"> $\Gamma(n) = (n - 1)!$
</ul>
</section>
<section data-vertical-align-top>
<h3>$\Gamma({1 \over 2})$</h3>
<ul style="list-style-type: none; font-size: 36px;">
<li class="fragment roll-in"> $\Gamma({1 \over 2}) = \displaystyle\int_0^{\infty} x^{-{1 \over 2}}e^{-x}dx$
<li class="fragment roll-in"> Set $x = t^2$, follows $dx = 2tdt$
<li class="fragment roll-in"> $\Gamma({1 \over 2}) = 2 \displaystyle\int_0^{\infty} e^{-t^2}dt = \displaystyle\int_{-\infty}^{\infty} e^{-t^2}dt$
<li class="fragment roll-in"> $\Gamma^2(\frac{1}{2}) = \displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} e^{-(x^2+y^2)dxdy}$
<li class="fragment roll-in"> $\Gamma^2(\frac{1}{2}) = \displaystyle\int_{0}^{2\pi}\int_{0}^{\infty} r e^{-r^2} dr d\theta$
<li class="fragment roll-in"> $\Gamma^2(\frac{1}{2}) = \pi $
<li class="fragment roll-in"> $\Gamma(\frac{1}{2}) = \sqrt{\pi} $
</ul>
<aside class="notes">
1. integration by parts <br>
2. symmetry of exp(-t^2) <br>
3. circle equation <br>
4. move to polar coordinates <br>
5. integral over theta is 2pi <br>
6. inner integral is also easy and it is -1/2 exp(-r^2). -2r/2 exp(-r^2)dr. and this is 1/2
</aside>
</section>
<section data-vertical-align-top>
<h3>Hypersphere Volume</h3>
<ul style="list-style-type: none; font-size: 34px;">
<li class="fragment roll-in"> $\mbox{volume} = C_n r^n$
<li class="fragment roll-in"> $C_1 = 2$, $C_2 = \pi$, $C_3 = \frac{4\pi}{3}$
<li class="fragment roll-in"> $\mbox{surface area} = \frac{dV_n(r)}{dr} = n C_n r^{n-1}$
<li class="fragment roll-in"> $\left(\frac{dV_n(r)}{dr}\right)dr = n C_n r^{n-1}dr$
<li class="fragment roll-in" style="font-size: 32px;">
\begin{align}
\Gamma^n\left(\frac{1}{2}\right) = \pi^{n/2} &= \displaystyle \int_0^{\infty} e^{-r^2} \left(\frac{dV_n(r)}{dr}\right)dr \\
& = \frac{nC_n}{2} \displaystyle \int_0^{\infty} e^{-t} t^{n/2-1}dt \\
& = \frac{nC_n}{2} \Gamma\left(\frac{n}{2}\right)\\
& = C_n \Gamma\left(\frac{n}{2} + 1\right)
\end{align}
</ul>
<aside class="notes">
1. substitute r^2 with t in Г(1/2)<br>
</aside>
</section>
<section>
<h3>hyperSphere Volume</h3>
<row>
<col40 style="font-size: 28px;">
\begin{align}
C_n &= \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}\\
C_nr^n & = \frac{(\pi r^2)^k}{k!} \to 0 \\
& \mbox{ as } k\to\infty
\end{align}
</col40>
<col60>
<table style="font-size: 28px;">
<thead>
<tr>
<th>Dimension $n$</th>
<th>Coefficient $C_n$</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>2</td>
<td>= 2.0000...</td>
</tr>
<tr>
<td>2</td>
<td>$\pi$</td>
<td>= 3.14159...</td>
</tr>
<tr>
<td>3</td>
<td>$4\pi/3$</td>
<td>= 4.18879...</td>
</tr>
<tr>
<td>4</td>
<td>$\pi^2/2$</td>
<td>=4.93480...</td>
</tr>
<tr>
<td>5</td>
<td>$8\pi^2/15$</td>
<td>=5.26379...</td>
</tr>
<tr>
<td>6</td>
<td>$\pi^3/6$</td>
<td>=5.16771...</td>
</tr>
<tr>
<td>7</td>
<td>$16\pi^3/105$</td>
<td>=4.72477...</td>
</tr>
<tr>
<td>8</td>
<td>$\pi^4/24$</td>
<td>=4.05871...</td>
</tr>
<tr>
<td>9</td>
<td>$32\pi^4/945$</td>
<td>=3.29851...</td>
</tr>
<tr>
<td>10</td>
<td>$\pi^5/120$</td>
<td>=2.55016...</td>
</tr>
<tr>
<td>$2k$</td>
<td>$\pi^k/k!$</td>
<td>$\to 0$</td>
</tr>
</tbody>
</table>
</col60>
</row>
</section>
<section>
<h3>In a general hyper<del>watermelon</del>sphere</h3>
<div class="fragment" data-fragment-index="0" style="font-size: 32px;">
\begin{align}
\frac{C_nr^n - C_nr^n(1-\epsilon)^n}{C_nr^n} &= 1 - (1-\epsilon)^n\\
\frac{V_{rind}}{V_{total}} &= 1 - (1 - r)^d
\end{align}
</div>
<div class="fragment" data-fragment-index="1" >
<object id="rind_plot" data="figures/curse_rind.svg#1_4" width="600"></object>
</div>
</section>
<section>
<h3>Fraction of volume</h3>
<div class="row">
<div class="col_left4">
\[
\frac{V_{rind}}{V_{total}} = 1 - (1 - \epsilon)^D
\]
<blockquote>
What volume fraction of a hyper-melon does the rind occupy if it takes up an $\epsilon$ fraction of its radius?
</blockquote>
</div>
<div class="col_right">
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 255);" width="700"
src="figures/curse_fraction.png" alt="fraction">
</div>
</div>
</section>
<section data-vertical-align-top>
<h3>Gaussian distribution</h3>
<div class="fragment" data-fragment-index="0" style="font-size: 30px;">
\[
G({\bf x}) = \frac{1}{(2\pi)^{K/2} |{\bf C}|^{1/2}}
e^{-\frac{1}{2}{\bf x}^{\;{\rm
T}} {\bf C}^{-1} {\bf x}}
\]
</div>
<div class="fragment" data-fragment-index="1" style="font-size:22px">
What's the probability of a random sample falling on a radius $r$ "shell" around the mean?
<img width="650"
src="figures/curse_normal.png" alt="density estimation">
</div>
<aside class="notes">
Talk about how to integrate probabilities.<br>
Probability of being in a shell around the radius.
</aside>
</section>
<section>
<h3>What if the flesh isn't in the center?</h3>
<object id="hypermelon" data="figures/watermelon_target.svg#1_0" ></object>
<div class="fragment" data-fragment-index="10" style="font-size: 30px;">
<em>Question:</em> What is the probability of randomly hitting the flesh?
</div>
</section>
<section>
<h3>Local minima and multiple starts</h3>
blackboard
</section>
<section>
<div id="header-right" style="right: -20%; top: -30%;">
<img style="margin-bottom: -5%" width="300" src="figures/CD_angleCosineRule.png" alt="Cosines Law">
<br>
<small>Law of cosines</small>
</div>
<h3>Angles between vectors</h3>
<ul style="list-style-type: none;">
<li class="fragment roll-in"> $\cos\theta = \frac{1}{\sqrt{n}} \to 0$ and $\theta \to \frac{\pi}{2}$
<li class="fragment roll-in"> $\cos\theta = \frac{\sum_{k=1}^n x_k y_k}{XY}$
<li class="fragment roll-in"> Draw vectors to two random points $(\pm 1, \pm 1, \pm 1. \dots, \pm 1)$
<li class="fragment roll-in"> $\cos\theta = \frac{\sum_{k=1}^n (\pm 1)}{n} \stackrel{a.s.}\to 0$, as $n \to \infty$
</section>
<section>
<h3>Angles between vectors</h3>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="700"
src="figures/curse_angles.png" alt="angles">
</section>
<section>
<h3>Probability of orthogonality</h3>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="800"
src="figures/prob_orthogonal.svg" alt="angles orth">
</section>
<section>
<h3>Estimating probability</h3>
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="800"
src="figures/curse_density.png" alt="density estimation">
<div class="fragment" data-fragment-index="0" >
<em>Question:</em> What's the minimum number of samples we need to estimate a $d$ dimensional density in a cube?
</div>
</section>
<section data-fullscreen>
<h3>4x4 paradox</h3>
<img width="50%" src="figures/CD_paradoxSphere.png" alt="paradox">
</section>
<section>
<h3>Alternative distances</h3>
<row>
<col50>
$L_1$
<img width="700" src="figures/CD_L1_circle.png" alt="L1">
</col50>
<col50>
$L_{\infty}$
<Img width="700" src="figures/CD_LinfCircle.png" alt="Linf">
</col50>
</row>
</section>
<section>
<h3>Conditions on a metric</h3>
<ol>
<li class="fragment roll-in"> $D(x,y) \ge 0$ (non-negative)
<li class="fragment roll-in"> $D(x,y) = 0$ iff $x=y$ (identity)
<li class="fragment roll-in"> $D(x,y) = D(y,x)$ (symmetry)
<li class="fragment roll-in"> $D(x,y) + D(y,z) \ge D(x,z)$ (triangle inequality)
</ol>
</section>
<section>
<h3>Case study: an average human</h3>
<img width="750" src="figures/ww2_pilots.jpg" alt="average human">
<aside class="notes">
1950s - too many incidents and accidents outside warfight<br>
250 harward male hands (white and rich) - no single typical<br>
measured 10-features of 4063 pilots including height, chest circumference, sleeve length<br>
find people within 30% of each parameter range<br>
How many were these?<br>
</aside>
</section>
<section>
<h2>Reading list</h2>
<ul>
<li style="list-style-image: url('figures/Bishop_PR.jpg');">p. 33 of "Pattern recognition and Machine Learning" by C. Bishop
<li style="list-style-image: url('figures/Hastie_Elements.jpg');">p. 22 of "The Elements of Statistical Learning" by T. Hastie, R. Tibshirani, and J. Friedman
<li style="list-style-image: url('figures/H.png');">Chapter 9 of "The Art of Doing Science and Engineering: Learning to Learn" by R. Hamming
</ul>
</section>
</section>
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