Consider applying the perceptron algorithm through the origin based on a small

training set containing three points:

$x^{(1)}=[-1,-1].,y^{(1)}=1$

$x^{(3)}=(1,0).,y^{(3)}=-1$

$x^{(3)}=(-1,1\mathrm{.}5).,y^{(3)}=1$

Given that the algorithm starts with ${}^{(0)}=0,$ the first point that the algorithm

sees is always considered a mistake. The algorithm starts with some data point

and then cycles through the data $($in order$)$ until it makes no further mistakes.

$($a$)$

$4$ points possible $($graded$)$

How many mistakes does the algorithm make until convergence if the algorithm

starts with data point $x^{(1)}?$ How many mistakes does the algorithm make if it

starts with data point $x^{(2)}?$

Also provide the progression of the separating plane as the algorithm cycles in

the following list format: $[[{}_1^{(1)},{}_2^{(1)}],$dots,$[{}_1^{(N)},{}_2^{(N)}]],$ where the superscript

denotes different $$ as the separating plane progresses. For example, if $$

progress from $0,0($initialization$)$ to $1,2$ to $3,-2,$ you should enter

$\left[\begin{array}{cc}1& 2\\ 3& -2\end{array}\right]$

Please enter the number of mistakes of Perceptron algorithm if the algorithm

starts with $x^{(1)}.$

Please enter the progression of the separating hyperplane $(,$ in the list format

described above$)$ of Perceptron algorithm if the algorithm starts with $x^{(1)}.$

$0$

Please enter the number of mistakes of Perceptron algorithm if the algorithm

starts with $x^{(2)}.$

Please enter the progression of the separating hyperplane $(,$ in the list format

described above$)$ of Perceptron algorithm if the algorithm starts with $x^{(3)}.$