In this problem we will build our intuition for constructing optimal decision boundaries.

Consider a natural language processing $($NLP$)$ task where we are trying to determine whether an Amazon

book review is positive or negative. We have checked each review to see if they contain the words $$def$-$

initely$,$great$$ and $$best$($disregarding capitalization$),$ which we encoded into a feature vector in the

following way: $$x $=[$x$1,$ x$2,$ x$3]$T $,$ where xi is $1$ if the word exists in a review and $0$ otherwise $($in the order

$$definitely$,$great$$ and $$best$).$ This is a form of text representation known as a $$bag of words$$ model.

Our training dataset consists of the following reviews and corresponding ratings $($y$($i$)$ represents the label

for review i$.$ y$($i$)$ is $1$ for positive reviews and $1$ for negative reviews$)$:

Review y$($i$)$

$$My Professor said this book was great, but I was definitely disappointed$-1$

$$My sister Radhika loved this book! So great! It$$s definitely the best gift.$1$

$$This author is the best. Definitely my favorite.$1$

$$Just bad. So so bad. Will definitely never read again.$-1$

Table $1$: Dataset for Question $2$

For example, the review $$My sister Radhika loved this book! So great! It$$s the best gift.$$ would be encoded

in the bag of words model as $$x $=[0,1,1]$T $.$ We aim to find parameters $\backslash$theta $,$ b such that for all $$x$($i$)$ in our

training set sign$(\backslash$theta $$x$($i$))=$ y$($i$).$

$($a$)(1$pt$)$ Provide a mathematical representation of the feature space $($i$.$e$.$ all possible feature vectors$).$

$($b$)(1$pt$)$ Provide a mathematical representation of the label space $($i$.$e$.$ all possible output classes$)$ for this

classification problem.

$($c$)(3$pts$)$ If b $=0($i$.$e$.,$ no explicit offset$),$ would it be possible to learn such a $\backslash$theta $?$ Provide an example of

such $\backslash$theta or explain why it$$s not possible.