发表机构
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文开创双曲球面度研究，证明其可小于欧氏球面度且依赖半径，并给出半径选择与变化对球面度的影响及振荡构造，对图嵌入有应用价值。
AI 中文摘要
图的球面度是使得图具有等半径d维球交表示的最小维度d。虽然球面度已在欧几里得空间中得到研究，但我们开创了双曲球面度的研究。图的双曲球面度可以显著小于其欧几里得对应值，但与欧几里得情形相反，它强烈依赖于球的半径。我们证明，如果球的半径可以根据图来选择，则双曲球面度以欧几里得球面度为上界。这将先前关于二维双曲空间（即一致圆盘图）的结果推广到任意维度。此外，我们的证明显著更简单。如果我们固定半径，即不使其依赖于图，我们证明双曲球面度可以大于欧几里得球面度，但最多大1。另外，我们研究双曲球面度如何随球半径变化。我们证明选择更大的半径可以大幅降低球面度，同时最多使其增加1。我们还提供了一个图的构造，其中球面度随着半径增加而在不同值之间振荡。除了理论上的趣味性，我们注意到这些结果与机器学习中的图嵌入相关，在机器学习中人们感兴趣的是对图等符号数据的低维数值表示。
英文摘要
The sphericity of a graph is the minimum dimension d such that the graph has an intersection representation of d-dimensional balls of equal radius. While sphericity has been studied in Euclidean space, we initiate the study of hyperbolic sphericity. The hyperbolic sphericity of a graph can be significantly smaller than its Euclidean counterpart, but, contrary to the Euclidean setting, depends strongly on the radius of the balls. We show that, if the radius of the balls can be chosen depending on the graph, the hyperbolic sphericity is upper bounded by the Euclidean sphericity. This extends a previous result for 2-dimensional hyperbolic space, i.e., uniform disk graphs, to arbitrary dimensions. Moreover, our proof is significantly simpler. If we fix the radius, i.e., do not make it dependent on the graph, we show that hyperbolic sphericity can be larger than Euclidean sphericity, but by at most 1. Additionally, we study how hyperbolic sphericity changes with the ball radius. We show that choosing a larger radius can substantially decrease the sphericity while increasing it by at most 1. We also provide a construction of a graph where the sphericity oscillates between different values as the radius increases. Besides being theoretically interesting, we note that these results are relevant for graph embeddings in machine learning, where one is interested in low-dimensional numeric representations of symbolic data like graphs.
