Hyperbolic contractivity and the Hilbert metric on probability measures

TL;DR

This paper introduces the Hilbert projective metric on probability measures, explores its properties, and demonstrates its contraction behavior under linear operators, providing new bounds and comparisons with other probability metrics.

Contribution

It offers a comprehensive analysis of the Hilbert metric on probability measures, including dual formulations, geometric insights, and novel bounds relating it to other distances.

Findings

Linear operators contract in the Hilbert metric with hyperbolic tangent scaling

Convergence in Hilbert metric implies convergence in total variation and Wasserstein distances

Derived a new sharp bound for total variation in terms of Hilbert distance

Abstract

This paper gives a self-contained introduction to the Hilbert projective metric $\mathcal{H}$ and its fundamental properties, with a particular focus on the space of probability measures. We start by defining the Hilbert pseudo-metric on convex cones, focusing mainly on dual formulations of $\mathcal{H}$ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the dual formulation of $\mathcal{H}$ in the general setting, and explore the geometry of the probability simplex under $\mathcal{H}$ in the special case of discrete probability measures. Throughout, we compare $\mathcal{H}$ with other distances between probability measures. In particular, we show how convergence in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound for the total variation between two probability measures in terms of their Hilbert distance.