Explaining a model that obtain matrice A and B from M by solving optimization problem

Question

I'm supposed to explain the model $$\operatorname*{argmin}_{A\in R^{6 x 2}, B \in R^{2 x 10}}||I\odot(M-AB)^2||$$ where I is a 6x10 binary matrix, which is 1 where $m_{ij} \neq 0$, and else 0. A is 6 x 2 matrix and B is a 2 X 10 matrix where $A x B \approx M$, where $m$ consist of random numbers.

As I understand it, we say the $M=I_{ij}*(M_{ij}-(a_{i1}b_{i1}+a_{j2}b_{j2})^2$ and then summing up from $i=1$ to $6$, and $j=1$ to $10$, however here I don't make use of the argmin, so I probably misunderstood something

So can anyone shortly explain to me, what this model actually does, and how to interpret (if we can) matrices A and B?

Answer 1

If the question were just $$\operatorname*{argmin}_{A\in R^{6 x 2}, B \in R^{2 x 10}}||(M-AB)^2||$$ then this would be finding the best rank-2 approximation to $M$.

By adding $I\odot$, you are saying that you only care about how well $AB$ matches $M$ in the places where $M$ is nonzero.

Typically this problem means that

• $M$ is some set of data points naturally indexed by two variables (e.g., $M_{ij}$ is how much person $i$ likes movie $j$).
• You think that $M$ probably has a good low-rank approximation (because there is a correlation between what kind of movies people like).
• You don't have a complete set of data for $M$ (so you are putting $0$s in the spots where you know nothing).
• You would like to guess what the missing data is (using your low-rank approximation $AB$).

This is known as the matrix completion problem.