双曲压缩性与概率测度上的Hilbert度量
Hyperbolic contractivity and the Hilbert metric on probability measures
AI总结：
本文自包含地介绍了Hilbert投影度量$\mathcal{H}$，证明了凸锥上线性算子的双曲正切收缩性，并分析了概率测度空间中$\mathcal{H}$的几何性质，推导出全变分与Hilbert距离间的新颖紧致界限。
摘要：
本文对Hilbert投影度量$\mathcal{H}$及其基本性质进行了自包含的介绍，特别关注概率测度空间。我们首先在凸锥上定义Hilbert伪度量，主要聚焦于$\mathcal{H}$的对偶形式。我们证明了凸锥上的线性算子在由$\mathcal{H}$的双曲正切给定的距离下是收缩的，这特别蕴含了Birkhoff关于$\mathcal{H}$的经典收缩结果。转向$\mathcal{H}$作为度量的概率测度空间，我们在一般设置下分析了$\mathcal{H}$的对偶形式，并在离散概率测度的特殊情况下探讨了概率单纯形在$\mathcal{H}$下的几何性质。在整个过程中，我们将$\mathcal{H}$与概率测度间的其他距离进行了比较。特别地，我们展示了$\mathcal{H}$中的收敛性如何蕴含全变分、$p$-Wasserstein距离以及任意$f$-散度中的收敛性。此外，我们推导出了用两个概率测度间的Hilbert距离表示其全变分的一个新颖的紧致界限。
英文摘要：
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.
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