# How to solve the inverse problem of least-squares?

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).

• Whatever your W was fixed at in the least squares problem in which you solved for V. Of course, this combination of V and W won't even be a local minimum, let alone a global minimum, if convergence of the overall algorithm has not yet occurred. scicomp.stackexchange.com is probably a better location for this question. Commented Jun 5, 2018 at 1:52
• @MarkL.Stone Thank you. You are right. What I concern is an intermediate result before convergence.
– lee
Commented Jun 5, 2018 at 2:05
• Well, intermediate results may be lousy. If it hasn't converged to a global minimum,. even the converged result could be lousy. Commented Jun 5, 2018 at 2:09
• cross-posted Commented Jun 5, 2018 at 13:12
• In general the answer is not unique. Let's say $Z=\bar V=0$. Then $W$ can be absolutely anything. You need to assume some non-degeneracy to make it meaningful. So, what are the assumptions you are operating under? Commented Jun 5, 2018 at 15:37