I have two [$n \times k$] matrices $X$ and $Y$, and a symmetric [$m \times m$] matrix $B$.
At a high-level, my goal is to find a low-rank subspace, $A$, in which $X$ and $Y$ are related by the linear mapping $B$. To find $A$, I've come up with the following optimization problem:
$$ \min_{A \in \mathbb{R}^{k \times m}\mbox{, }A^TA = I} \| YA - XAB \|_F^2 $$
where $\|C\|_F^2 = \sum_{i,j} (C_{i,j})^2$ is the Frobenius norm.
I have a couple of questions:
Is this problem tractable? Any ideas for how I might decompose this somehow into a more convenient form?
Is this objective convex? (I realize that my domain, the set of rank-m projection matrices, isn't a convex set. But I'm curious to know if the objective function itself is convex.)
Thanks for any advice!