Isometric Invariant Quantification of Gaussian Divergence over Poincare Disc

The objective of this note is to present a geometric relationship between the spherical squared-Hellinger distance -commonly used in information theory, machine learning and artificial intelligence -and a hyperbolic isometric invariant of the Poincaré disc under the action of the general Möbius group. This spherical-hyperbolic duality motivates us to propose the L 2 -embedded hyperbolic isometric invariant as an alternative way to quantify divergence between two Gaussian probability measures. We shall keep this note reasonably concise towards communicating a geometric connection as a contribution to information theory.

For the rest of this paper, consider a measurable space (Ω, F) endowed with probability measures P and Q. For our purposes, P and Q are Gaussian measures hereafter. Using the language of measure theory, the squared-Hellinger distance between P and Q (see [5]), which we shall denote as Φ(P, Q), is given by the following:

We are now in position to present our main statement before providing the details in the next section.

Proposal 1.1. Define Ψ(P, Q) as follows:

(

Then, Φ in (1) and Ψ in (2) form a spherical-hyperbolic dual in a geometric sense.

In the next section, we shall unpack Proposal 1.1 in detail and provide the mathematical arguments leading to it, which will clarify the statement for the reader. We shall also provide a closed-form solution to (2) in order to enable its implementation more broadly in machine learning and artificial intelligence applications. The closed-form equation will also allow us to quantify a distributional distance between two Gaussian processes as they progress in time. Finally, we shall provide the parametric asymptotics of (2) across relevant limits, which shed further light on how it behaves.

For the rest of this paper, let P and Q be measures on R, where p : R → R + and q : R → R + are the corresponding Gaussian probability density functions. The choice of the state-space R here is for parsimony and can be generalized to R n for n ≥ 1. We identify the parameters of p and q as Θ p = {µ p , σ 2 p } and Θ q = {µ q , σ 2 q }, respectively, where µ p , µ q ∈ (-∞, ∞) and σ p , σ q ∈ (0, ∞). Denoting ||.|| L 2 as the L 2 -norm and using (1), we have the following:

Since p and q are non-negative functions and || √ p|| L 2 = || √ q|| L 2 = 1, both √ p and √ q determine points on the positive-orthant of the unit-sphere S + ⊂ L 2 .

Remark 2.1. Note that (S, ⟨.⟩) is a Riemannian manifold with constant curvature K = +1, given that the inner product ⟨.⟩ on L 2 is the Riemannian metric on S.

The geodesics between any two points on S are the great-circles, and accordingly, an angle using the L 2 -inner product can be defined via

where d S (P, Q) is the Bhattacharyya angle between P and Q -the angle from the center of S subtended to the endpoints on S + -which is equivalent to the spherical distance on S + with values in [0, π/2]. Therefore, using (3), the map Φ can be rewritten in terms of the cosine of the spherical distance on S + as follows:

In information geometry, it is well-known that the parameter space of Gaussian measures forms a 2-dimensional differentiable manifold locally diffeomorphic to R 2 (see, for example, [2]), which is a hyperbolic space with constant curvature K = -1/2. If this manifold, which we shall denote as M, is endowed with the Fisher information metric, then M becomes a Riemannian manifold. Accordingly, it is possible to define a distance between P and Q via this metric by integrating the infinitesimal line element along the geodesic connecting the corresponding points on M -determined by the parameters of p and q, respectively -given by

where ζ is a function of the parameters of p and q:

The expression in (5) holds when µ q ̸ = µ p and σ q ̸ = σ p , and the metric takes alternative forms when µ q = µ p or σ q = σ p (see [4]), which we omit from this note. One can adjust the lengths on M measured in an alternative unit R = -√ -K/K = √ 2, which is analogous to the radian on S. This geometrical perspective on Gaussian measures is what we shall use to establish a connection between the squared-Hellinger distance and a hyperbolic isometric invariant of the Poincaré disc under the action of the general Möbius group. First, recall that the general Möbius group forms a group of transformations of the Riemannian sphere C = C∪{∞} (a topological construction, namely one-point compactification), where geometric quantities (e.g. hyperbolic lengths and hyperbolic angles) are invariant under its action. More specifically, a Möbius transformation is a holomorphic function η * : C → C that satisfies the functional form:

where a, b, c, d ∈ C and ad -bc ̸ = 0. The general Möbius group, which we shall denote as Möb, is generated by the set of Möbius transformations and the set of complex conjugations, such that η ∈ Möb is the composition given by

for some k ≥ 1, where each η * j is a Möbius transformation, C(z) = z for z ∈ C and C(∞) = ∞. We highl

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