Exploring the Poincaré Ellipsis

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Exploring different geometries has shown to be useful for exploiting inherent properties of data at hand, becoming attractive to compute embeddings therein. For example, hyperbolic geometry shows superior performance when embedding hierarchical data. This suggests that the ability to explore different geometries for embedding data might be an important consideration, just as the models and algorithms used to perform the embedding. However, utilising non-Euclidean geometries to embed data can sometimes be a laborious task, as the whole process needs to be first analytically derived for each geometry chosen (e.g., spherical, hyperbolic) and then adapted to it. This discourages the exploration of richer spaces to embed data in. Using the framework of Riemannian manifolds, we explore a new choice of geometry. We consider a modified version of the Poincaré disc to accept individual (perdimension) curvatures instead of a single global one, resulting in what we refer to as the Poincaré ellipsis. We experiment with link prediction on graph nodes embedded onto these new spaces, showing the performance implications and highlighting the ease to explore new variants of hyperbolic geometry.
Original languageEnglish
Title of host publication20th Machine Learning on Graphs workshop
Publication date09.2023
Publication statusPublished - 09.2023
Event20th International Workshop on Mining and Learning with Graphs - MLG 2023 - Torino, Italy
Duration: 22.09.202322.09.2023
Conference number: 20