Computation and differentiation of the geodesic distance on the unitary matrix manifold

[Lorenzo Audibert]

Dear Manopt Team,

I think I have found how to compute the geodesic distance over the unitary matrix manifold but I would like to compute it's derivatives.

My problem look like this:

a function f(x) outputs a unitary matrix. I would like to compare f(x) with some data b that is also a unitary matrix using the geodesic distance and at some point differentiate with respect to x the quantity  x-> d_geodesic(f(x),b).

Thanks for your help in pointing me with the correct material.

Regards,

Lorenzo

[Nicolas Boumal]

Hello Lorenzo,

The gradient of the Riemannian distance function is given by the so-called logarithmic map (the inverse of of the exponential map, that generates geodesics).

You can read more here for example: https://math.stackexchange.com/questions/2977849/gradient-of-distance-squared-on-riemannian-manifold