Given $M \in \mathbb R^{n \times n}$, I would like to find its $k$-dimensional "leading singular vector space". $M$ is not necessarily symmetric. In contrast with the standard SVD problem, the left and right $k$-dimensional "leading singular vector spaces" must be the same.
Formally, the optimization problem in $U \in \mathbb R^{n \times k}$ and $Q \in \mathbb R^{k \times k}$ is as follows
$$\hat{U} = \arg\max_{U^T U = I_k} \,\max_{Q^T Q = I_k} \mbox{tr} \left( U^T M U Q \right)$$
An equivalent problem is to solve
$$\arg \max_{U^T U = I_k} \|U^T M U\|_*$$
Here $\|\cdot\|_*$ denotes the nuclear norm, i.e., the sum of singular values.
Remark: I understand that the slightly different problem
$$\arg\max_{U^T U = I,V^T V = I} \|U^T M V\|_*$$
will be solved by finding the spaces spanned by the leading $k$ left and right singular vectors of $M$. However, I could not figure out how to solve the new problem.