# HOMEWORK 4

## Background

The following depicts a simple, idealized two-dimensional
vector-space model in which the x-axis captures emphasis/attenuation
and the y-axis captures positive/negative polarity. Each word or
phrase is annotated with its vector representation (for example, the
vector for *good* is [0,2]).

## Question 1 [2 points]

In a sentence or two, describe in informal terms
how *extremely* acts on its argument with respect to the
dimensions of emphasis/attenuation and positive/negative.

## Question 2 [2 points]

In a sentence or two, describe in informal terms how *fairly*
acts on its argument with respect to the dimensions of
emphasis/attenuation and positive/negative.

## Question 3 [2 points]

Find matrices \( A \) and \( B \) such that the following
composition function perfectly predicts the vectors for *extremely
good* and *extremely awful*:

\[
p = A a + B b
\]
Here, \(a\) is the vector representations of the left word, \(b\)
is the vector representations of the right word, and \(p\) is the
predicted parent vector. (Also: \(Xx\) is standard matrix
multiplication of \(X\) and \(x\); and, where \(x\) and \(y\) are
vectors of dimension \(n\), \(x + y\) is the vector \( [ x_{i}+y_{i} ,
\ldots, x_{n}+y_{n} ] \).)

## Question 4 [2 points]

Assume the \(A\) and \(B\) you created in the previous problem are
global values used for all adverb–adjective combinations. Apply
them to the task of predicting values for *fairly good*
and *fairly awful*. What values do you obtain? In a sentence or
two, articulate what went wrong, perhaps drawing on the insights you
expressed in questions 1 and 2.

## Question 5 [2 points]

Consider the alternative composition function

\[
p = Ab
\]
where \(A\) is a 2-d matrix representation of the adverb and \(b\)
is a vector representation of its adjectival argument. This approach
has the potential to avoid the problem we saw above with the global
values \(A\) and \(B\). **Your task**: find a matrix
for *extremely* that perfectly models *extremely good*
and *extremely awful*.