A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream processing. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincar\'e hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.

翻译：机器学习和计算机视觉的一项关键技术是将离散加权图嵌入到连续空间中以进行后续下游处理。将离散层次结构嵌入双曲几何已被证明非常成功，因为研究表明任何加权树都可以以任意低的失真度嵌入该几何结构中。基于常见双曲几何模型的各种双曲嵌入优化方法已得到广泛研究。本文中，我们考虑标准单形的希尔伯特几何，该几何等距于一个配备变分多面体范数的向量空间。我们通过嵌入图的距离矩阵来研究这种希尔伯特单形几何的表示能力。我们的发现表明，希尔伯特单形几何在嵌入任务中与庞加莱双曲球或欧几里得几何等替代几何具有竞争力，同时其计算速度快且数值稳健。