Assume I have the following system of equations:
$$y = \operatorname{Diag}(h) x$$
where $x,y, h \in {\bf R}^N$ and suppose that I know them.
I also know that $h$ can be factorized as: $$h=Bz$$
where $B \in {\bf R}^{N \times K}$ is a tall full column-rank matrix ($N >K$) and $z \in {\bf R}^K$.
Goal: I would like to estimate both $B$ and $z$.
Questions
- Does it make any difference to approach the problem by trying to solve the first system of equations as $y=\operatorname{Diag}(Bz) x $ instead of trying to solve directly the second?
- I can also have multiple observation vectors $\{x_i, y_i\} $ and have a bigger system $Y= \operatorname{Diag}(h) X $ if that helps.
I have tried to think the ways of approaching the problem, for instance an alternating minimization approach on the variables $B$ and $z$, though I am worried it might result in ambiguities. Or trying to reformulate the problem by rewriting
$$y=\operatorname{Diag}(Bz) x = \operatorname{Diag}(x) Bz $$ and trying to make arguments on column space of $B$ or other constraints to plug in into the optimization algorithm. Any help is appreciated.