Given a full rank matrix $W \in \mathbb{R}^{n\times n}$, and matrices $X, Y \in \mathbb{R}^{n \times k}$, $k < n$, I want to find matrices $A, B \in \mathbb{R}^{n \times k}$ which optimize:
$$ \min\limits_{A, B}\;\lVert W - (BX^T + YA^T + BA^T)\rVert_F $$
That is, we're trying to approximate $W$ by $BX^T + YA^T + BA^T$
If $X$ and $Y$ were zero, this is solved by taking the SVD of $W$, but I don't know how to approach it otherwise. Is this a well-known kind of problem? I'd also be interested in minimizing this under the $L^2$ norm.