In the course, you learned about different techniques for dimensionality reductions that embed high dimensional data into a low dimensional space.

Explain in at most $50$ words, how PCA and tSNE approach dimensionality reduction respectively. Name one disadvantage for each of the methods, again using a maximum of $50$ words.

Why is it important to normalize the data before applying PCA?

Given a dataset of $n$ observations $xin{R}^{np}$ and a column wise mean of zero, the first principal component is the direction of maximum variance in the data, i$.$e$.$

$argma{x}_{win{R}^p,||w||}=1\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}({\displaystyle \underset{j=1}{\overset{p}{}}}{w}_j{x}_{i,j}{)}^2.$

Show that the first principal component also corresponds to the eigenvector with the largest eigenvalue of the covariance matrix $x^{T}x$ of the data.