论文标题
希尔伯特单纯形几何形状中的非线性嵌入
Non-linear Embeddings in Hilbert Simplex Geometry
论文作者
论文摘要
机器学习和计算机视觉的关键技术是将离散的加权图嵌入连续的空间中，以进一步下游处理。双曲线几何形状中嵌入离散的层次结构已被证明非常成功，因为证明任何加权树可以嵌入具有任意低失真的几何形状中。已经研究了基于双曲线几何模型的双曲线嵌入的各种优化方法。在本文中，我们考虑了标准单纯形的Hilbert几何形状，该标准单纯轴与配备有变化多层标准的矢量空间等距。我们通过嵌入图的距离矩阵来研究这种希尔伯特单纯形几何形状的表示能力。我们的发现表明，希尔伯特单纯形几何形状具有替代几何形状的竞争力，例如Poincaré双曲球或Euclidean几何形状，用于嵌入任务，同时快速且数值稳健。
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 Poincaré hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.
