Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.

The goal is to return $\widehat A, \widehat B$ such that $\widehat A \widehat B\approx C $.

Is it possble to efficiently find $\widehat A, \widehat B$ such that $\widehat A \widehat B= C $ (assuming such $A,B$ exist)?

If not, can we find $$\text{minarg}_{\widehat A, \widehat B}||C-\widehat A \widehat B||_F$$?

this seems doable by writing all $mn$ constraints and running least squares, but I'm wondering if it has a nice close form.

Assuming such $A,B$ exist, are $\widehat A, \widehat B$ unique (up to rotation/scaling/permutation)?