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[IML] Examples

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semester6/iml/parts/08_unsupervised.tex
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@@ -95,6 +95,97 @@ This is an iterative method to find the cluster centers.
\remark Each iteration is in $\mathcal{O}\bigl( nkd \bigr)$.
}
{\footnotesize
\textbf{Example}: Consider $x_{1,2,3} = \begin{bmatrix}
2 \\
1
\end{bmatrix},
\begin{bmatrix}
-1 \\
-1
\end{bmatrix},
\begin{bmatrix}
-2 \\
-4
\end{bmatrix}$ for which we'd like $k=2$ clusters. We choose:
$$\mu_1^{(0)}= \begin{bmatrix}
1 \\
1
\end{bmatrix} \qquad \mu_2^{(0)} = \begin{bmatrix}
-1 \\
-1
\end{bmatrix}
$$
For Lloyd's Heuristic, the initial cluster assignments $z^{(1)}_{1,2,3}$:
$$
z_1^{(1)} = \underset{j\in\{1,2\}}{\text{arg min}} \Biggl( \Biggl\Vert \begin{bmatrix}
2 \\
1
\end{bmatrix} - \begin{bmatrix}
1 \\
1
\end{bmatrix} \Biggr\Vert, \Biggl\Vert \begin{bmatrix}
2 \\
1
\end{bmatrix} - \begin{bmatrix}
-1 \\
-1
\end{bmatrix} \Biggr\Vert \Biggr) = 1
$$
$$
z_2^{(1)} = \underset{j\in\{1,2\}}{\text{arg min}} \Biggl( \Biggl\Vert \begin{bmatrix}
-1 \\
-1
\end{bmatrix} - \begin{bmatrix}
1 \\
1
\end{bmatrix} \Biggr\Vert, \Biggl\Vert \begin{bmatrix}
-1 \\
-1
\end{bmatrix} - \begin{bmatrix}
-1 \\
-1
\end{bmatrix} \Biggr\Vert \Biggr) = 2
$$
$$
z_2^{(1)} = \underset{j\in\{1,2\}}{\text{arg min}} \Biggl( \Biggl\Vert \begin{bmatrix}
-2 \\
-4
\end{bmatrix} - \begin{bmatrix}
1 \\
1
\end{bmatrix} \Biggr\Vert, \Biggl\Vert \begin{bmatrix}
-2 \\
-4
\end{bmatrix} - \begin{bmatrix}
-1 \\
-1
\end{bmatrix} \Biggr\Vert \Biggr) = 2
$$
And then the updated cluster centers:
$$
\mu_1^{(1)} = \frac{1}{1}\Biggl( \begin{bmatrix}
-1 \\
-1
\end{bmatrix} \Biggr) = \begin{bmatrix}
-1 \\
-1
\end{bmatrix}
$$
$$
\mu_2^{(1)} = \frac{1}{2}\Biggl( \begin{bmatrix}
-1 \\
-1
\end{bmatrix} + \begin{bmatrix}
-2 \\
-4
\end{bmatrix} \Biggr) = \frac{1}{2}\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
$$
}
% Continue with convergence analysis, k-means++
\subsubsection{Convergence}