TY - CHAP

T1 - Exploring the Poincaré Ellipsis

AU - Fadel, Samuel

AU - Paulsen, Tino

AU - Brefeld, Ulf

N1 - Conference code: 20

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

UR - https://mlg-europe.github.io/2023/papers/259.pdf

UR - https://mlg-europe.github.io/2023/

M3 - Article in conference proceedings

BT - 20th Machine Learning on Graphs workshop

T2 - 20th International Workshop on Mining and Learning with Graphs - MLG 2023

Y2 - 22 September 2023 through 22 September 2023

ER -