# Possible analytical way to solve or approximate a specific optimization problem's solution

In my research on linear algebra and optimization, I have come across the following problem repeatedly:

Given constant matrices $$C\in\mathbb{R}^{k \times k}$$ and $$X\in\mathbb{R}^{n \times n}$$, $$\min_{A\in\mathbb{R}^{n\times k}, B\in\mathbb{R}^{k \times n}} \| X - A C B X \|_F$$

where $$C$$ may be singular, as $$k \leq n$$ ($$k,n$$ are constant). We minimize the norm over $$A, B$$ of fixed dimensions (maybe rectangular) with no additional constraints.

If $$C$$ were absent (replaced with the unit matrix), this could be solved analytically via low-rank matrix approximation ($$AB$$ can be viewed as the rank factorization of the approximating matrix), but can anyone tell me if an analytical solution is available in the presence of singular/rectangular $$C$$ matrices? Perhaps good approximations with appropriate bounds on the error? The relation to the problem without $$C$$?

I am unable to find an analytical solution and a numerical/iterative solution would not be very informative on the role of the $$C$$ in the solution for the optimal $$A, B$$, which is my goal. I thank all helpers and appreciate all assistance.

$ACB$ ranges over all the matrices with rank smaller or equal to the rank of $C$, so this is equivalent to a problem with $C=I$ (and possibly with a smaller $k$). That said, it is not clear to me how you planned to solve the problem with $C=I$. It's not the standard setup of low-rank approximation with SVD (Eckart-Young theorem).