Recently, non-Euclidean spaces became popular for embedding structured data. Following hyperbolic and spherical spaces, more general product spaces have been proposed. However, searching for the best configuration of a product space is a resource-intensive procedure, which reduces the practical applicability of the idea. We introduce a novel concept of overlaying spaces that does not have the problem of configuration search and outperforms the competitors in structured data embedding tasks, when the aim is to preserve all distances. On the other hand, for local loss functions (e.g., for ranking losses), the dot-product similarity, which is often overlooked in graph embedding literature since it cannot be converted to a metric, outperforms all metric spaces. We discuss advantages of the dot product over proper metric spaces.

中文翻译：

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复杂几何的叠加空间与实际适用性

最近，非欧式空间在嵌入结构化数据方面变得流行起来。在双曲和球面空间之后，已经提出了更一般的乘积空间。然而，寻找产品空间的最佳配置是一个资源密集型过程，这降低了该想法的实际适用性。我们引入了一种新的重叠空间概念，它没有配置搜索问题，并且在结构化数据嵌入任务中优于竞争对手，目的是保持所有距离。另一方面，对于局部损失函数（例如，对损失进行排序），点积相似性在图嵌入文献中经常被忽视，因为它无法转换为度量，优于所有度量空间。我们讨论点积相对于适当的度量空间的优势。