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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.

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$ACB$ 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).

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