On the \(f\)-divergences between hyperboloid and Poincaré distributions (Q6178839)

The authors argue that hyperbolic geometry is becoming ``popular in machine learning due to its capacity to embed discrete hierarchical graph structures with low distortions into continuous spaces for further downstream processing''. In this paper, which summarizes results from [\textit{F. Nielsen} and \textit{K. Okamura}, IEEE Trans. Inf. Theory 69, No. 5, 3150--3171 (2023; Zbl 07820881)] they consider two kinds of hyperbolic distributions, namely Poincaré distributions and hyperboloid distributions, and their statistical \(f\)-divergences. The Poincaré distribution is a special case of distributions introduced by \textit{K. Tojo} and \textit{T. Yoshino}, [Inf. Geom. 4, No. 1, 215--243 (2021; Zbl 1471.62239)]. Given a positive definite, symmetric matrix \(\theta=\left(\begin{array}{cc}a&b\\ b&c\end{array}\right)\) a distribution \(p_\theta\) on the (hyperbolic) upper half-plane \({\mathbb H}\) is defined as \[ p_\theta(x,y)=\frac{1}{\pi}\sqrt{\det(\theta)}\exp(2\sqrt{\det(\theta)})exp(-\frac{1}{y}(a(x^2+y^2)+2bx+c))\frac{1}{y^2}. \] The group \(SL(2,{\mathbb R})\) acts by conjugation on pairs \((\theta,\theta^\prime)\) of positive definte, symmetric matrices. Given a convex function \(f\colon\left(0,\infty\right)\to{\mathbb R}\), its \(f\)-divergence is defined as \[ D_f\left[p_\theta\colon p_{\theta^\prime}\right]=\int_{\mathbb H}p_\theta(x,y)f\left(\frac{p_{\theta^\prime}(x,y)}{p_\theta(X,y)}\right)dxdy \] for Poincaré distributions \(p_\theta,p_{\theta^\prime}\). The authors prove that \(f\)-divergences are \(SL(2,{\mathbb R})\)-invariant and can always be expressed as functions in the three variables \(\det(\theta),\det(\theta^\prime)\) and \(tr(\theta^\prime\theta^{-1})\). They explicitly stated these functions for the Kullback-Leibler divergence, the squared Hellinger divergence and the Neyman chi-squared divergence. Similar results are obtained for two-dimensional hyperboloid distributions. For the entire collection see [Zbl 1528.94003].