Hyperbolic contractivity and the Hilbert metric on probability measures
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This paper gives a self-contained introduction to the Hilbert projective metric 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 ℋ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of ℋ, which in particular implies Birkhoff's classical contraction result for ℋ. Turning to spaces of probability measures, where ℋ is a metric, we analyse the dual formulation of ℋ in the general setting, and explore the geometry of the probability simplex under ℋ in the special case of discrete probability measures. Throughout, we compare ℋ with other distances between probability measures. In particular, we show how convergence in 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.
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