Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising $$ \left\lVert \, |O| - T \right\lVert_F,$$ where $\lVert \cdot \lVert_F$ denotes the Frobenius norm, and $| \cdot |$ denotes the entry wise absolute value. If $|O|$ is replaced with $O$, then this is the classic Orthogonal Procrustes Problem. I don't expect a closed form solution, but rather a way to reduce the dimensionality of the problem and tackle the problem numerically.