How to sample from a multivariate normal distribution in hyperbolic space?

Question

I want to generate samples from a multivariate normal distribution on the hyperbolic space, such that $\forall x \in \mathbb{H},$

$$ \text{pdf}(x) = \frac{1}{c} e^{-\frac{d^2_{\mathbb{H}}(x,\mu)}{2\sigma^2}}, $$

where, $c$ is the normalizing constant, $\mu \in \mathbb{H}$ and $\sigma$ are the mean and std, respectively, and $d_{\mathbb{H}}(\cdot, \cdot)$ is the distance function in the hyperbolic space. I am looking for a procedure that can transform a uniform sample in the Euclidean space to a sample from the above mentioned distribution.

I am currently working with the Poincare disk model for the hyperbolic space, but I guess the sampling procedure should work for Klein/Hyperboloid etc. models and vice-cersa as well.

Some of my initial guesses:

• Can we sample a point $y$ from a normal distribution in Euclidean space, with same mean and std, then find a point $x$ such that $d^2_{\mathbb{H}}(x,\mu) = d^2_{\mathbb{E}}(y,\mu)$, where $d_{\mathbb{E}}$ is the Euclidean distance?

• Is there a Box-Muller equivalent for Hyperbolic spaces?

• (preferably not) Rejection sampling based approach, but what would be the base distribution we sample the points from?

Also, a related question - https://scicomp.stackexchange.com/questions/19685/how-to-sample-points-in-hyperbolic-space

Answer 1

I'm in the middle of doing this for myself. I think the most appropriate analogue to the Gaussian would be the heat kernel in hyperbolic space. Fortunately, this has been figured out before: https://www.math.uni-bielefeld.de/~grigor/nog.pdf

If you use the standard decay ($e^{-dist^2/constant}$), I expect that the total mass will be larger than 1, due to exponential increase in volume with radius for hyperbolic space.

To sample uniformly on a given ball (or other compact set), one could do rejection sampling with the volume form: $$\left(\frac{2}{1-||x||^2} \right)^n \, dx_1 \, \ldots \, dx_n$$

Here's a uniform sample for the ball of radius 3 centered at the origin:

If desired, I'd be glad to say more. I just thought I'd put this up, since there was clearly some interest in this, at least in the past.