Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that 
\begin{equation}
UV^\top :=\arg\min\_{R \in \mathcal{O}^{r\times r}}\|AR-B\|\_\mathrm{F},
\end{equation}
where $USV^\top$ is the SVD of $A^\top B $ and $\mathcal{O}^{r\times r}$ means the set of $r\times r$ orthonormal matrices. 


However, if I change the metric from Frobenius norm to operator norm, what is the best orthonormal matrix? 


In other words, what's $R$ that attains the minimum of the following?
\begin{equation}
\min\_{R \in \mathcal{O}^{r\times r}}\|AR-B\|\_\mathrm{op}.
\end{equation}


It seems that the two rotation matrices are not the same (for Frobenius and for operator norm). If this is true, what can we say about
\begin{equation}
\|AUV^\top-B\|\_\mathrm{op}
\end{equation}
and how worse is it compared with the optimal one?



What's the best orthonormal matrix to align two matrices in the operator norm sense?
The operator norm version of this problem is considered in: *[The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms](https://link.springer.com/article/10.1007%2FBF02521606?LI=true),* by G. A. Watson, Advances in Computational Mathematics, **2**(4), pp 393–405, 1994, which actually looks at solving this "Procrustes problem" in Schatten-p norms.


