I've recently been reading a lot of papers and watching a lot of videos on both subspace learning, and matrix factorization. One thing is particularly eluding me though - how does any of this get tested?
Let's take, straight from Wikipedia, Matrix Factorization Non-Negative.
$$V = WH$$
So, you have a data matrix $V$. Your goal is to learn components $W$ and $H$, which when multiplied together, give a good approximation of $V$. This can be done by minimizing over $W$, $H$
$$\| V - WH \|$$
That seems fine so far. My problem, theoretically, is understanding when we want to apply this to a problem, like say Regression.
If you wanted to minimize:
$$Y - WH*B$$
How do you do this with a test point? I get confused here, because if we had, say a 100-user test set with 10 features. Then we do a 90/10 split, we get a size of $W*H$ that is different than the size of our test data.
Do people just plug the test data in directly when testing, in place of $W*H$, and just rely on those learned weights $B$?