I have two matrices with orthonormal columns, $A$ $(k\times n)$ and $B$ $(k\times m)$, with $k \ge m \ge n$. I would like to find the optimal orthogonal $(n\times n)$ matrix, $T$, that transforms $A$ into $A' = AT$, such that $A'$ is as close as possible to some subset of $n$ columns of $B$ ($B'$).
If $m = n$, or if I know which subset of $B$ I want to target, this is the standard orthogonal Procrustes problem, right? But the trick is in this case I don't know which subset of $B$ to use, it should be the one that allows $A'$ to be closest to $B'$. I guess I could just cycle through all possible $B'$ and select the best of the corresponding $A'$ (and $T$). But with large $n$ and $m-n$ this becomes unmanageable. Is there some more direct way to solve this problem?