PROBLEM $2$

This exercise is about topic analysis. A topic is a latent variable that

represents or summarizes important concepts of a text, such as its meaning

or main ideas. A topic is made up of several words semantically related to

each other according to a certain context. In the area of natural language

processing $($NLP$),$ it is part of a general task called information retrieval

$($IR$).$ For us$,$ from a machine learning perspective, we will consider it as an

unsupervised learning task based on a particular vector representation of

the texts. Consider a document$-$term representation like the ones we saw

in class. A simple way to extract latent structures between documents and

terms is using latent semantic analysis $($LSA$),$ which is based on appropriate

factorizations of that matrix. Let $A_{mn}$ be the TF$-$IDF matrix of rank $r,$

with $m$ rows $($documents$)$ and $n$ columns $($terms$).$ A rank approximation $k$

of this matrix is given by the SVD factorization $A$~~$A^{(k)}=U^{(k)}{}^{(k)}{V}^{(k)\text{'}},$

where ${}^{(k)}$ is diagonal ${?}^1$ with the $k$ largest eigenvalues of A and $U^{(k)},V^{(k)}$

contain the corresponding

${?}^1$ We are considering the reduced representation of SVD$,$ where all zero

entries have been removed from $,$ and the corresponding columns from $U$

and $V,$ and even more, the smallest $k-r$ eigenvalues have been replaced

by zeros.

Left and right eigenvectors that define an orthonormal basis for the column

and row spaces, respectively. By applying this factorization in document$-$

term matrices, we can extract the semantic and conceptual relationships

between documents and terms expressed in a set of components $($or to$-$

pics$)k,$ through dense and low$-$dimensional representations, where $V_{nk}^{(k)}$

and $U_{mk}^{(k)}$ provide us with a representation of the terms and documents,

respectively in terms of the $k$ topics, and ${}^{(k)}$ gives us the importance of each

topic. In python, you can use the sklearn.decomposition. TruncatedSVD im$-$

plementation. In this exercise, you will perform an analysis of topics in the

transcripts of the morning conferences of the presidency of Mexico, which

you can access in this repository ${?}^2.$ To build your topic model, consider

the texts of the conferences per week during the years $2019$ to $2023,$ using

the transcripts that correspond to the president, contained in the archives

"PRESIDENT ANDRES MANUEL LOPEZ OBRADOR.csv$"3.$ a$)$ Obtain

a TF$-$IDF representation of the texts. Define the size of the vocabulary

and carry out the preprocessing that you consider necessary in the texts,

considering that for a topic analysis, it is not recommended that the voca$-$

bulary be so large, and it is better to conserve words whose use within the

text can be associated with topics. Document and justify your settings. b$)$

Obtain $k$ topics using SVD decomposition. Choose a suitable $k$ and justify

it$.$ Represents each topic through a word cloud ${?}^4$ of the terms that make

up each topic according to the importance expressed in the magnitudes of

the rows of $V^{(k)}.$ Can you assign a representative "name" for each topic? c$)$

Using the topic model adjusted in the previous step, obtain the correspon$-$

ding representation of each of the president's conferences during the years of

the study, calculating the document$-$topic matrix using the product $x{V}^{(k)}$

$($or with TruncatedSVD's transform method$).$ Assign each conference to its

corresponding topic using the maximum value of each row of the matrix as a criterion. Use low$-$dimensional visualizations based on PCA, Kernel PCA

and t$-$SNE of the topic assignment you obtained. Do you see interesting pat$-$

terns? Briefly describe your findings. d$)$ A problem that arises when using

SVD is the lack of interpretability, since it is not clear how negative values

in the matrices $U$ and $V$ can be considered. One way to solve this problem

is to use a non$-$factorization. $-$Negative Matrix $($NMF$),$ which is suitable for

matrices with non$-$negative entries, such as TF$-$IDFs. For a matrix A of rank

$r$ with non$-$negative entries, NMF computes an approximation of rank