In $[24]$: def calculate$\_$S$($X$,$ n$,$ m$)$:

$,$

function calculate$\_$S$($X$)$

Computes the S matrix.

Syntax:

S$=$calculate$\_$S$($X$)$

Input:

X: nxd data matrix with n vectors $($rows$)$ of dimensionality d

n : number of rows in X

m : output number of columns in S

Output:

Matrix S of size nxm

S$[$i$,$j$]$ is the inner$-$product between vectors $\backslash (\backslash$mathrm$\{$X$\}[\backslash$mathrm$\{$i$\},$:$]\backslash )$ and $\backslash (\backslash$mathrm$\{$X$\}[\backslash$mathrm$\{$i$\},$:$]\backslash )$

assert $\backslash (\backslash$mathrm$\{$n$\}==\backslash$mathrm$\{$X$\}\backslash ).$shape$[0]$

$\backslash$# YOUR CODE HERE

raise NotImplementedError$()$

In $[25]$: def calculate$\_$R$($Z$,$ n$,$ m$)$:

function calculate$\_$R$($Z$)$

Computes the R matrix.

Syntax:

R$=$calculate$\_$R$($Z$)$

Input:

Z: mxd data matrix with m vectors $($rows$)$ of dimensionality d

n : output number of rows in Z

m : number of rows in Z

Output:

Matrix R of size nxm

$\backslash ($ R$[$i$,$ j$]\backslash )$ is the inner$-$product between vectors $\backslash ($ Z$[$j$,$:$]\backslash )$ and $\backslash ($ Z$[$j$,$:$]\backslash )$

assert m $==$ Z$.$shape$[0]$

$\backslash$# YOUR CODE HERE