In this work, we propose a novel Knowledge Graph Embedding (KGE) strategy, called MöbiusE, in which the entities and relations are embedded to the surface of a Möbius ring. The proposition of such a strategy is inspired by the classic TorusE, in which the addition of two arbitrary elements is subject to a modulus operation. In this sense, TorusE naturally guarantees the critical boundedness of embedding vectors in KGE. However, the nonlinear property of addition operation on Torus ring is uniquely derived by the modulus operation, which in some extent restricts the expressiveness of TorusE. As a further generalization of TorusE, MöbiusE also uses modulus operation to preserve the closeness of addition on it, but the coordinates on Möbius ring interacts with each other in the following way: any vector attaches to the surface of a Mobius ring becomes its opposite one if it moves along its parametric trace by a cycle. Hence, MöbiusE assumes much more nonlinear representativeness than that of TorusE, and in turn it generates much more precise embedding results. In our experiments, MöbiusE outperforms TorusE and other classic embedding strategies in several key indicators.

在这项工作中，我们提出了一种新的知识图谱嵌入 (KGE) 策略，称为 MöbiusE，其中实体和关系嵌入到 Möbius 环的表面。这种策略的提议受到经典 TorusE 的启发，其中两个任意元素的相加受制于模数运算。从这个意义上说，TorusE 自然地保证了 KGE 中嵌入向量的临界有界性。然而，Torus 环上加法运算的非线性特性是由模运算唯一导出的，这在一定程度上限制了 TorusE 的表达能力。作为 TorusE 的进一步推广，MöbiusE 也使用模运算来保持其上加法的接近性，但 Möbius 环上的坐标以如下方式相互作用：任何附着在莫比乌斯环表面的矢量，如果它沿着它的参数轨迹移动一个周期，就会变成它的相反矢量。因此，MöbiusE 比 TorusE 具有更多的非线性代表性，进而产生更精确的嵌入结果。在我们的实验中，MöbiusE 在几个关键指标上优于 TorusE 和其他经典嵌入策略。