I want to find an orthogonal matrix $O\in SO(n)$ such that $\|Y - OX \|_1$ is minimized, where X and Y are matrices (of appropriate sizes).
I know that there is a solution to this problem using SVD for $L^2$ for example, https://igl.ethz.ch/projects/ARAP/svd_rot.pdf.
But for my problem, I have $L^1$ norm and the procedure described in the above paper does not seem to be applicable to $L^1$.
If there is no close form then how do I go about a numerical optimization? Any help on this would be appreciated.