$(2^($pt$))$ Consider the two$-$moon data set of size $500\backslash$times $2$ which can be downloaded via

from sklearn.datasets import make$\_($m$)$oons

x$,$y$=$ make$\_($m$)$oons $,$ noise $=\mathrm{.}07,$ random$\_($s$)$tate $=20$ k$-$means on the two$-$moon data set with k$=2,$ i$.$e$.,$ using the

following steps to classify the data set:

$($a$)$ Create a fully connected graph with the weight between the i th data point x$\_($i$)$ and the j th data point x$\_($j$)$

defined by W$($i$,$j$)=$exp$(-||$x$\_($i$)-$x$\_($j$)|(|^(2))/(2\backslash$sigma $^(2)))$ with $\backslash$sigma $=0\mathrm{.}1.$

$($b$)$ Compute the symmetric normalized graph Laplacian L$\_($sym$).$

$($c$)$ Find the two eigenvectors corresponding to the two smallest eigenvalues of L$\_($sym$)$ and get a matrix

UinR$^($n$\backslash$times $2).$

$($d$)$ Apply k$-$means clustering to the matrix U to get $2$ clusters, and visualize the clustering.

$($e$)$ Compare your spectral clustering result with the k$-$means one. Comment on your observations.

Solve using python