Overlaying Spaces and Practical Applicability of Complex Geometries

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.