Focusing on following least squares problem: $$\min\limits_{V} \lVert Z - WV \rVert _{_F}^2$$

$$Z∈{R}^{m*n},\quad W∈{R}^{m*k},\quad V∈{R}^{k*n},\quad k\lt m\lt n $$ This problem can be easily solved given $Z$ and $W$. I assume the solution of this problem as $\overline{V}$.

My question is that, how can we recover $W$ by $\overline{V}$ and $Z$?

I found something called Inverse Optimization, but cannot figure out the relationship between them.

EDIT: It is acceptable to have a set of $W$ or one feasible $W$ if $W$ is not unique.

($\overline{V}$ and $Z$ are both nonzero matrices, and $Z$ is not promised to be full rank).

1more comment