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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.

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    Although the objective function is convex on the space of square matrices, the set of orthogonal matrices isn't convex. So numerically that is rather nasty. – 2017-01-11
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    Heck, it's not convex for $\ell_2$ either, but it just happens to have a closed-form solution anyway. – 2017-01-11

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