Let $\{{X}_{1},\dots ,{X}_{K}\}$ is a set of random matrices, where ${X}_{k}\in {\mathbb{R}}^{M\times N},k=1,\dots ,K$, and $U\in {\mathbb{R}}^{M\times r}$ and $V\in {\mathbb{R}}^{N\times r}$ are two matrices containing orthogonal columns (i.e., ${U}^{\mathrm{\top}}U=I,{V}^{\mathrm{\top}}V=I$). I was wondering, if the following question has a analytical solution:

$\underset{U,V}{max}\sum _{k=1}^{K}\Vert {U}^{\mathrm{\top}}{X}_{k}V{\Vert}_{F}^{2}$

If not, how should I solve it? Alternating optimization?

(At first, I thought it may be related to the SVD of the sum of the matrices $\{{X}_{k}\}$, but so far I have no hint to prove it.)

$\underset{U,V}{max}\sum _{k=1}^{K}\Vert {U}^{\mathrm{\top}}{X}_{k}V{\Vert}_{F}^{2}$

If not, how should I solve it? Alternating optimization?

(At first, I thought it may be related to the SVD of the sum of the matrices $\{{X}_{k}\}$, but so far I have no hint to prove it.)