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\fancyhead[C]{COMP 136 - 2021s - HW4}
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\begin{document}
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\Large{\bf Student Name:}

\Large{\bf Collaboration Statement:}

Total hours spent: TODO

I consulted the following resources:
\begin{itemize}
\item TODO
\item TODO
\end{itemize}

\tableofcontents


\newpage
\officialdirections{
\subsection*{1a: Problem Statement}
Prove that the mean of vector $x$ under the mixture distribution is given by:
\begin{align}
\mathbb{E}_{p^{\text{mix}}(x)}[x] = \sum_{k=1}^K \pi_k \mu_k
\end{align}
}

\subsection{1a: Solution}
TODO

\newpage
\officialdirections{
\subsection*{1b: Problem Statement}
Given: $m = \mathbb{E}_{p^{\text{mix}(x)}}[x]$ (as in 1a).
Prove that the covariance of vector $x$ is:
\begin{align}
\text{Cov}_{p^{\text{mix}}(x)}[x] = \sum_{k=1}^K \pi_k (\Sigma_k + \mu_k \mu_k^T ) - m m^T
\end{align}
}
\subsection{1b: Solution}
TODO
 
 
\newpage
\officialdirections{
\subsection*{2a: Problem Statement}
Find optimal one-hot assignments $r^1$, and report the value of the cost function.
}
\subsection{2a: Solution}

TODO fill in the $r^1$ values below with the one-hot assignments the algorithm produces

\begin{tabular}{p{5cm} | p{5cm} | r}
$\mu^0$ & $r^1$ & $J(x_{1:N}, r^1, \mu^0)$
\\
\midrule
\begin{verbatim}
[[-3.  -2. ]
 [ 1.5  3. ]
 [ 2.   2. ]]
\end{verbatim}
&	
\begin{verbatim}
[[0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]]
\end{verbatim}
&
 ??.?????
\end{tabular}

\officialdirections{
\subsection*{2b: Problem Statement}
Find optimal locations $\mu^1$, and report the value of the cost function.
}
\subsection{2b: Solution}

\begin{tabular}{p{5cm} | p{5cm} | r}
$\mu^1$ & $r^1$ & $J(x_{1:N}, r^1, \mu^1)$
\\
\midrule
\begin{verbatim}
[[ ?.???  ?.???]
 [ ?.???  ?.???]
 [ ?.???  ?.???]]
\end{verbatim}
&	
\begin{verbatim}
[[0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]]
\end{verbatim}
&
 ??.?????
\end{tabular}

\newpage
\officialdirections{
\subsection*{2c: Problem Statement}
Find optimal one-hot assignments $r^2$, and report the value of the cost function.
}
\subsection{2c: Solution}

\begin{tabular}{p{5cm} | p{5cm} | r}
$\mu^1$ & $r^2$ & $J(x_{1:N}, r^2, \mu^1)$
\\
\midrule
\begin{verbatim}
[[ ?.???  ?.???]
 [ ?.???  ?.???]
 [ ?.???  ?.???]]
\end{verbatim}
&	
\begin{verbatim}
[[0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]]
\end{verbatim}
&
 ??.?????
\end{tabular}

\officialdirections{
\subsection*{2d: Problem Statement}
Find optimal locations $\mu^2$, and report the value of the cost function.
}
\subsection{2c: Solution}

\begin{tabular}{p{5cm} | p{5cm} | r}
$\mu^2$ & $r^2$ & $J(x_{1:N}, r^2, \mu^2)$
\\
\midrule
\begin{verbatim}
[[ ?.???  ?.???]
 [ ?.???  ?.???]
 [ ?.???  ?.???]]
\end{verbatim}
&	
\begin{verbatim}
[[0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]]
\end{verbatim}
&
 ??.?????
\end{tabular}

\newpage
\officialdirections{
\subsection*{2e: Problem Statement}
What interesting phenomenon do you see happening in this example regarding cluster 2? What could you do about cluster 2's location from 2d to better fulfill the goals of K-means (find K clusters that reduce cost the most)?
}
\subsection{2e: Solution}

TODO



\newpage
\officialdirections{
\subsection*{3a: Problem Statement}
Show (with math) that using the parameter settings defined above, the general formula for $\gamma_{nk}$ will simplify to the following (inspired by PRML Eq. 9.42):

\begin{align}
	\gamma_{nk} = \frac
	{  e^{ -\frac{1}{2} \frac{1}{\epsilon} || x_n - \mu_k ||^2 } }
	{ \sum_{j=1}^K e^{ -\frac{1}{2} \frac{1}{\epsilon} || x_n - \mu_j ||^2 }}
\end{align}
}

\subsection{3a: Solution}

TODO



\newpage
\officialdirections{
\subsection*{3b: Problem Statement}
}

\subsection{3b: Solution}

Using $\epsilon = 1.0000$, we obtain $\gamma$ as:
\begin{verbatim}
[[?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]]
\end{verbatim}

\officialdirections{
\subsection*{3c: Problem Statement}
}

\subsection{3c: Solution}

Using $\epsilon = 0.1$, we see the $\gamma$ values get more extreme:
\begin{verbatim}
[[?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]
 [?.??? ?.??? ?.???]]
\end{verbatim}

\newpage
\officialdirections{
\subsection*{3d: Problem Statement}
What will happen to the value of $\gamma$ as $\epsilon \rightarrow 0$? How is this related to the K-means results from Problem 2?
}

\subsection{3d: Solution}

TODO

\newpage
\officialdirections{
\subsection*{4a: Problem Statement}
}

\subsection{4a: Solution}

TODO

\end{document}