Hyperbolic space in the Poincar upper half space model looks like ordinary $\Bbb R^n$ but with the notion of angle and distance distorted in a relatively simple way. In Euclidean space I can sample a random point unformly in a ball in several ways, e.g. by generating $n$ independent Gaussian samples to obtain a direction, and separately uniformly sample a radius (is this a good way?). In the hyperbolic upper half plane a sphere happens to still be a sphere, only its centre will not be the centre in the Euclidean metric, so we could do the same. If we want to sample according to a non-uniform distribution, but still in an isotropic way, e.g. a Gaussian distribution, this doesn't seem so easy. In Euclidean space we could just generated a Gaussian sample for each coordinate (this only works for the Gaussian distribution), or equivalently generate a multidimensional Gaussian sample. Is there a direct way to convert this sample to a sample in hyperbolic space? An alternative approach could be to first generate a direction uniformly distributed direction (e.g. from $n$ Gaussian samples) then a Gaussian sample for the radial component, and finally generate the image under the exponential map in the specified direction for the specified length. A variation would be to just take the Euclidean Gaussian sample and map it under the exponential map.My questions:what would be a good and efficient way to obtain a Gaussian sample with given mean and standard deviation in hyperbolic space?do the ways I describe above provide the desired sampling?did anyone work out the formula's already?how does this generalize to other metrics and other probability distributions?Thanks in advance.EDITI just realized that even in the case of uniform sampling these questions remain; even though a sphere is a sphere, a uniform distribution would not be described by a constant function on a ball.
How to sample points in hyperbolic space?
computational geometry;monte carlo;geometry;random sampling