Assume you have the ratings of $n$ users for $m$ movies in a matrix $R \in \mathbb{R}^{n \times m}$. You compute a representation

$$R = U \times \Sigma \times V$$

by initializing $u_i, v_j \forall i \in 1, \dots, n \forall j \in 1, \dots, m$ randomly and optimizing the following expression through gradient descent:

$$\min_{u_i, v_i} \sum_{p_{ij}} \left ( p_{ij} - u_i \cdot v_j \right )^2 \text{ with } u_i \in \mathbb{R}^{1 \times r}, v_j \in \mathbb{R}^{r \times 1}$$

This is how I understand how Simon Funk did it.

But how would you deal with a new user? How would you tell what that user likes?

(Or similarly, with a new movie?)