Ultrahyperbolic Neural Networks

Marc T Law

Keywords: differential geometry, pseudo-Riemannian manifolds, hyperbolic geometry, elliptic geometry, ultrahyperbolic geometry, optimization on manifolds, pseudo-Riemannian optimization, quotient manifolds, graph neural networks

TL;DR: A (graph) neural network mapping representations to a pseudo-Riemannian manifold of constant curvature used to classify hierarchical graphs with cycles

Abstract: Riemannian space forms, such as the Euclidean space, sphere and hyperbolic space, are popular and powerful representation spaces in machine learning. For instance, hyperbolic geometry is appropriate to represent graphs without cycles and has been used to extend Graph Neural Networks. Recently, some pseudo-Riemannian space forms that generalize both hyperbolic and spherical geometries have been exploited to learn a specific type of nonparametric embedding called ultrahyperbolic. The lack of geodesic between every pair of ultrahyperbolic points makes the task of learning parametric models (e.g., neural networks) difficult. This paper introduces a method to learn parametric models in ultrahyperbolic space. We experimentally show the relevance of our approach in the tasks of graph and node classification.

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