Limitations of the K$-$Means Algorithm I

$1$ point possible $($graded$)$

Remember that the k$-$Means Algorithm is given as below:

Randomly select $z_1,$dots,$z_K$

Iterate

Given $z_1,$dots,$z_K,$ assign each data point $x^{(i)}$ to the closest $z_j,$ so that

Cost$(z_1,dots{z}_K)={\displaystyle \underset{i=1}{\overset{n}{}}}mi{n}_j=1,$dots,$k||x^{(i)}-z_j|{|}^2$

Given $C_1,$dots,$C_K$ find the best representatives $z_1,$dots,$z_K,$ i$.$e$.$ find $z_1,$dots,$z_K$ such that

$z_j=argmi{n}_z{\displaystyle \underset{iin{C}_j}{\overset{?}{}}}||x^{(i)}-z|{|}^2=\frac{{\displaystyle \underset{iin{C}_j}{\overset{?}{}}}{x}^{(i)}}{|C_j|}$

where $|C_j|$ is the number of points in $C_j.$

Which of the following are false about K$-$Means Algorithm? Please choose all those apply.

$C_1,$dots,$C_K$ foind by the algorithm is always a partition of $\{x_1,$dots,$x_n\}$

It is always guaranteed that the $K$ representatives $z_1,$dots,$z_{K}in\{x_1,$dots,$x_n\}.$

The algorithm may output different $C_1,$dots,$C_K$ and $z_1,$dots,$z_K$ depending on the initialization of line $1$

Line $2\mathrm{.}2$ of the algorithm$($Given $C_1,$dots,$C_K$ find the best representatives $z_1,$dots,$z_K$dots $)$ finds the cost$-$minimizing representatives $z_1,dots{z}_K.$

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