pairwise distance for stiefel manifold

[Kisung]

Dear All,

From the stiefelfactory, I was looking for some (not necessarily canonical) geodesic distance measures for Stiefel manifold.

In Grassmanian, Manopt has a member function to do so, but for stiefel It's written that it's not implemented yet.

I've tried to find explicit computational formula for Stiefel's natural geodesic distance, but all I've been able to find was the exact form of geodesic equation.

Any help or comments would I truly appreciate.

Regards,

Kisung

[Nicolas Boumal]

Hello Kisung,

Unfortunately, I am not aware of explicit formulas for the distance between two points on the Stiefel manifold, considered as a Riemannian submanifold of orthonormal matrices in R^nxp (as done in Manopt).

Perhaps this could help: there is a SIMAX paper where the authors develop two numerical schemes to approximate endpoint geodesics on Stiefel, with the classical Riemannian submanifold geometry: http://epubs.siam.org/doi/abs/10.1137/16M1103099.

Essentially, this is a numerical scheme (not an explicit expression) to compute the logarithm (the inverse of the geodesic map) -- the length of the resulting vector is the geodesic distance.

Best,

Nicolas

[Kisung]

Hi Nicolas,

Thanks for the kind moment.

I took a look at your suggested reading. Also, I've found some interesting article of finding log map under canonical metric : https://arxiv.org/pdf/1604.05054.pdf

This article also contains matlab code, which manopt package could probably take advantage of.

Regards,

Kisung

[Nicolas Boumal]

Hello Kisung,

Thanks for the link to that paper, I did not know about it. I agree that it would be nice to include Stiefel with canonical metric, and maybe also an implementation of the Log map in that paper, in Manopt at some point.

Best,

Nicolas