Inner product in hyperbolic space

Question

I am trying to learn about Hyperbolic Spaces. I can't find information about the inner product in hyperbolic spaces. So the paper "Multi-relational Poincaré Graph Embeddings" from Balaževic et al. states: "... rely on inner product as a similarity measure and there is no clear correspondence to the Euclidean inner product in hyperbolic space". Whereas the paper "Neural Embeddings of Graphs in Hyperbolic Space" from Chamberlain et al. gives the following definition of an inner product of two vectors in polar coordinates in the Poincaré disk model: $$x = (r_x,\theta)\\y=(r_y,\theta)\\ \langle x,y \rangle = \lVert x \rVert \lVert y \rVert cos(\theta_x - \theta_y) = 4 \text{arctanh } r_x \text{arctanh } r_y \cos(\theta_x - \theta_y) $$

I am confused whether there exists an inner product in hyperbolic space and if so, how is it defined for the Poincaré ball? I already tried to read introductions in Hyperbolic space but I did not find an answer. I would really appreciate your help! I am studying CS, so an intuitive explanation would be nice ;)

Thank you!

Answer 1

The answer is "yes" and "no."

1. Why "no": The notion of an inner product is reserved for vector spaces and hyperbolic plane
   ${\mathbb H}^2$ does not have a natural vector space structure. This is especially clear if one uses an axiomatic definition of the hyperbolic plane, which you can find, for instance, in Greenberg's book "Euclidean and Non-Euclidean Geometries."

2. Why "yes": Nevertheless, one can construct an "identification" (a mathematician would call it a "diffeomorphism") between the hyperbolic plane and a 2-dimensional vector space $V$. Such an identification requires a choice of the "origin" $o$ in the hyperbolic plane (which will correspond to zero in the vector space). In the case of the unit disk model, the convenient choice of the origin is the Euclidean center of the disk. In the paper you are reading, the identification is given by $$ f: z= (x,y)\mapsto (d(o, z), \theta), $$ where $d$ is the hyperbolic distance, $\theta$ is the angle and $(\rho, \theta)$ is understood as the polar coordinates of a vector. Now, the vector space $V$ has a natural inner product (called the "Riemannian metric" in differential geometry) and this is the one described in the paper you are reading (except they do it via an ad hoc method which obscures the procedure). A Riemannian geometer would call the inverse $\exp_o$ of the map $f: {\mathbb H}^2\to V$ the exponential map (the map $f$ itself is sometimes called the Riemannian logarithmic map).

Using $f$ one then "transfers" the inner product $V$ to ${\mathbb H}^2$: $$ \langle \exp_o(v), \exp_o(w)\rangle := \langle v, w\rangle. $$ This is what the authors of the paper are doing.

3. What is this good for? I do not know, maybe the authors of the paper you are reading found some use for this construction.

4. In the literature you will likely encounter other notions of "products" on hyperbolic spaces, most importantly, Gromov product $$ (z, w)_p= \frac{1}{2}(d(p,z)+ d(p,w)- d(z,w)). $$ This will be also different from the one defined in your question.