Gradient descent generalises naturally to Riemannian manifolds, and tohyperbolic $n$-space, in particular. Namely, having calculated the gradient atthe point on the manifold representing the model parameters, the updated pointis obtained by travelling along the geodesic passing in the direction of thegradient. Some recent works employing optimisation in hyperbolic space have notattempted this procedure, however, employing instead various approximations toavoid a calculation that was considered to be too complicated. In thistutorial, we demonstrate that in the hyperboloid model of hyperbolic space, thenecessary calculations to perform gradient descent are in factstraight-forward. The advantages of the approach are then both illustrated andquantified for the optimisation problem of computing the Fr\'echet mean (i.e.barycentre) of points in hyperbolic space.

Gradient descent in hyperbolic space

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