Consider a model $Y=XW$ of consistent dimensions with $X$ full column rank and $W$ full row rank. I would like to estimate $X$ and $W$, so this is related to blind source separation, deconvolution problems. Consider also the economy-size SVD of $Y= U\Sigma V^\top$.
Since both $X$ and $U$ are bases for the column space of $Y$, there exists a linear transformation $L$ for which $X=UL$. A typical subspace approach would solve, in place of $\|Y-XW\|_F^2$, the problem.
$$\min_{X,L} \|X-UL\|_F^2$$ Because $L= U^\dagger X$, the problem can be rewritten as: $$\min_{X} \|X-UU^\dagger X\|_F^2 = \min_{X} \|(I-UU^\dagger) X\|_F^2 $$
I would like to understand if a solution of this problem is necessarily also a solution of for the original problem with $Y=XW$, and the relationships among the ambiguities of the two problems. Namely, the model $Y=XW$ might hold also for $Y=XTT^{-1}W= X^\prime W^\prime$ for an ambiguity $T$. Are the ambiguities of this original problem the same as in the subspace problem? Or there might be different?