I am reading a text book that basically says the following: Given a matrix A where A is USERS x ITEMS we can use SVD to decompose the matrix into:
$$A = U \times \Sigma \times V^T$$
Then we can take the first $n$ columns of these matrices to get: $$A \approx U_n \times \Sigma_n \times V_n^T$$
Then say we have a USER vector, $u$, that has ratings for items, we find out where $u$ is position in the $n$-dimensional space:
$$u_{nd} = u \times U_n \times\Sigma_{n}^{-1}$$
And we can use the position of $u_{nd}$ to discover what items and users are similar to $u_{nd}$ by using a method like cosine similarity.
So this all makes sense to me (I think). Pretty straight forward.
However if I go to this tutorial, and go to the SVD section, she writes that you can provide predictions by just getting a low-rank approximation of a ratings matrix?
Moreover, if you have: $$A = U \times \Sigma \times V^T$$ We can simply make predictions by: $$A \approx U_n \times \Sigma_n \times V_n^T$$ where $n$ is the $n$ lower-rank approximation.
Why do both of these methods work? What is the point of the first method if I can just use the second method to make predictions?
FURTHER EDIT: Why does the first method use the inverse of matrix $\Sigma$ whereas the second does not?
