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\begin{document}

\title{Introduction to Machine Learning\\
Unit 8 Problems:  Support Vector Machines}
\author{Prof. Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}

\item Consider the data set for four points with
features $\xbf_i=(x_{i1},x_{i2})$ and binary class
labels $y_i=\pm 1$.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|} \hline
$x_{i1}$ & 0 & 1 & 1 & 2 \\ \hline
$x_{i2}$ & 0 & 0.3 & 0.7 & 1 \\ \hline
$y_i$ & -1 & -1 & 1 & 1 \\ \hline
\end{tabular}
\end{center}

\begin{enumerate}[(a)]
\item Find a linear classifier that separates the two classes.
Your classifier should be of the form
\[
    \hat{y} = \begin{cases}
        1 & \mbox{if } b + w_1 x_1 + w_2 x_2 > 0 \\
        -1 & \mbox{if } b + w_1 x_1 + w_2 x_2 < 0
    \end{cases}
\]
State the intercept $b$ and weights $w_1$ and $w_2$ for your classifier.
Note there is no unique answer as there are multiple linear classifiers
that could separate the classes.

\item Find the maximum $\gamma$ such that
\[
    y_i(b+w_1x_{i1} + w_{i2}x_{i2}) \geq \gamma, \mbox{ for all } i,
\]
for the classifier in part (a)?

\item Compute the margin of the classifier
\[
    m = \frac{\gamma}{\|\wbf\|}, \quad \|\wbf\|= \sqrt{ w_1^2 + w_2^2 }.
\]

\item Which samples $i$ are on the margin for your classifier?
\end{enumerate}

\item  Consider the data set with scalar features $x_i$
and binary class labels $y_i=\pm 1$.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
$x_i$ & 0 & 1.3 & 2.1 & 2.8 & 4.2 & 5.7 \\ \hline
$y_i$ & -1 & -1 & -1 & 1 & -1 & 1 \\ \hline
\end{tabular}
\end{center}

Consider a linear classifier for this data of the form,
\[
    \hat{y} = \begin{cases}
        1  & z > 0 \\
        -1 & z < 0,
        \end{cases}
    \quad
    z = x-t,
\]
where $t$ is a threshold.  For each threshold $t$, let $J(t)$
denote the sum hinge loss,
\[
    J(t) = \sum_i \epsilon_i, \quad \epsilon_i = \max(0, 1-y_iz_i).
\]

\begin{enumerate}[(a)]
\item Write a short python program to plot $J(t)$ vs.\ $t$ for
100 values of $t$ in the interval $t \in [0,5]$.

\item Based on the plot, what is one value of $t$ that minimizes
$J(t)$.

\item For the value of $t$ in part (b), find the corresponding
slack variables $\epsilon_i$.

\item Which samples $i$ violate the margin ($\epsilon_i > 0$)
and which samples $i$ are misclassified ($\epsilon_i > 1$).

\end{enumerate}

\item
Consider an image recognition problem, where an image $\Xbf$
and filter $\Wbf$ are $4 \x 4$ matrices:
\[
    \Xbf = \left[
        \begin{array}{cccc}
        0 & 0 & 0 & 0 \\
        0 & 0 & 1 & 0 \\
        0 & 0 & 1 & 0 \\
        0 & 0 & 1 & 0
        \end{array} \right], \quad
    \Wbf = \left[
        \begin{array}{cccc}
        0 & 0 & 0 & 0 \\
        0 & 1 & 1 & 0 \\
        0 & 1 & 1 & 0 \\
        0 & 0 & 0 & 0
        \end{array} \right].
\]
\begin{enumerate}[(a)]
\item Recall that in linear classification, the $4 \x 4$
image matrices $\Xbf$ and $\Wbf$
can be represented as 16-dimensional vectors, $\xbf = \mathrm{vec}(\Xbf)$
and $\wbf = \mathrm{vec}(\Wbf)$ by stacking the columns of the matrices
vertically. What are $\xbf$ and $\wbf$ for the matrices above.

\item What is the inner product $z = \wbf\tran \xbf$.

\item What is the inner product $z = \wbf\tran \xbf_{\rm right}$ where
$\xbf_{\rm right}$ is
the vector corresponding to the matrix $\Xbf$ right shifted by one pixel
with the left column filled with zeros.

\item What is the inner product $z = \wbf\tran \xbf_{\rm left}$ where
$\xbf_{\rm left}$ is
the vector corresponding to the matrix $\Xbf$ left shifted by one pixel
with the right column filled with zeros.

\item Write the python command that can covert a $4 \x 4$ image matrix, \pycode{Xmat}
to the 16-dimensional vector, \pycode{x}.  What is the python command
to go from \pycode{x} to \pycode{Xmat}.
\end{enumerate}

\item
Consider the data set with scalar features $x_i$
and binary class labels $y_i=\pm 1$.

\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
$x_i$ & 0 & 1 & 2 & 3 \\ \hline
$y_i$ & 1 & -1 & 1 & -1 \\ \hline
\end{tabular}
\end{center}

A support vector classifier is of the form
\[
    \hat{y} = \begin{cases}
        1  & z > 0 \\
        -1 & z < 0,
        \end{cases}
    \quad
    z = \sum_i \alpha_i y_i K(x_i,x),
\]
where $K(x,x')$ is the radial basis function, $K(x,x') = e^{-\gamma(x-x')^2}$, and
$\gamma > 0$ and $\alphabf = [\alpha_1,\ldots,\alpha_4]$ are parameters of the classifier.

\begin{enumerate}[(a)]
\item Use python to plot $z$ vs.\ $x$ and
$\hat{y}$ vs.\ $x$ when $\gamma = 3$ and $\alphabf = [0,0,1,1]$.
\item Repeat (a) with $\gamma = 0.3$ and $\alphabf = [1,1,1,1]$.
\item Which classifier makes more errors on the training data.
\end{enumerate}
\end{enumerate}
\end{document}