Poincaré ball and Lorentz model of the hyperbolic space

The Poincaré ball model represents the hyperbolic space as the interior of a unit ball in the Euclidean space: P=z∈RM+1∣∥z∥<1,z0=0,M∈Z+, where z=(z0,…,zM)T. The distance between two points z1,z2\inP is defined as:

where cosh−1(z)=ln(z+z2−1) is the inverse hyperbolic cosine function, which is monotonically increasing for z≥1. The symbol ∥⋅∥ represents the Euclidean norm. Notice that cosh−1(1+z)=ln(1+z+z2+2z), which approximates 2z when limz→0 and ln(2z) for limz→+∞. When both z1 and z2 are close to the origin with zero norm, d(z1,z2)≈cosh−1(1+2∥z1−z2∥2)≈2∥z1−z2∥. Therefore, the Poincaré ball model resembles Euclidean geometry near the center of the unit hyperball. The induced norm of a point z\inP is

As z moves aways from the origin and approaches the border with ∥z∥≈1, the induced norm ∥z∥P grows exponentially. Hyperbolic geometry is useful to represent data with an underlying approximate hierarchical structure.

The Lorentz model is a model of the hyperbolic space and points of this model satisfy HM={z∈RM+1∣z0>0,⟨z,z⟩H=−1}, where ⟨z,z′⟩H=−z0z0′+∑i=1Mzizi′ is the Lorentzian inner product (or Minkowski inner product when z∈R4). The special one-hot vector 𝛍0=(1,0,…,0)T is the origin of the hyperbolic space. The distance between two points of the Lorentz model is defined as:

The tangent space of HM at point 𝛍∈HM is defined as T𝛍HM:={z∣⟨𝛍,z⟩H=0}, i.e., all the vectors that pass point 𝛍 and are orthogonal to vector 𝛍 based on the Lorentzian inner product. A point (z0,z1,…,zM)T in the Lorentz model can be conveniently mapped to the Poincaré ball²¹ for visualization:

We discard the first element as it is a constant of zero.

We used wrapped normal priors and wrapped normal posteriors defined in the Lorentz model to embed cells to a hyperbolic space^25,34,79. A wrapped normal distribution in HM is constructed by first defining a normal distribution on the tangent space T𝛍0HM (a Euclidean subspace in RM+1) at the origin 𝛍0=(1,0,…,0)T of the hyperbolic space. Samples from a normal distribution on the tangent space are parallel-transported to desired locations and further projected onto the final hyperbolic space²⁵.

We used a set of invertible functions to transform samples from a normal distribution N(z∣0,IMσ) in RM to samples from a wrapped normal distribution in HM with mean of 𝛍, where σ∈RM is the standard deviation of components z1 to zM, respectively, and IM is the identity matrix in RM^25,55. First, let z0=(0,z0′)T, which can be considered as a sample vector from T𝛍0HM, where z0′ is sampled from N(z∣0,IMσ). Next, z0 is parallel-transported to vector z1 in the tangent space T𝛍HM at 𝛍, in a parallel manner (i.e., z1 and z0 pointing in the same direction relative to the geodesic between 𝛍0 and 𝛍) and vector norm preserving (i.e., ⟨z0,z0⟩H=⟨z1,z1⟩H)^25,80:

with α=−⟨𝛍0,𝛍⟩H.

Finally, the exponential map^24,25,79 projects vector z1 in the tangent space T𝛍HM back to the hyperbolic space by:

such that the vector norm is preserved: ∥z1∥H=⟨z1,z1⟩H=dH(𝛍,z).

The likelihood after the invertible transformations can be calculated by

The encoder outputs a vector h in the tangent space at the origin (T𝛍0HM, so ∥h∥H=∥h∥2) and can be mapped to HM using the exponential map (the first zero element of h is omitted) to get 𝛍:

Given a sample z from the wrapped normal distribution, we need to evaluate its density logpz for calculating the KL-divergence term of the ELBO. We can use the inverse exponential map and the inverse parallel transport to compute the corresponding z1 and z0, respectively, for evaluating the density:

where β=−⟨𝛍,z⟩H and α=−⟨𝛍0,𝛍⟩H. We now have all the ingredients to compute Eq. (¹) for each training point.