Ultrahyperbolic Representation Learning

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

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Marc Law, Jos Stam


<p>In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic space which is well suited for tree-like data. In this paper, we propose a representation living on a pseudo-Riemannian manifold of constant nonzero curvature. It is a generalization of hyperbolic and spherical geometries where the non-degenerate metric tensor need not be positive definite. We provide the necessary learning tools in this geometry and extend gradient method optimization techniques. More specifically, we provide closed-form expressions for distances via geodesics and define a descent direction to minimize some objective function. Our novel framework is applied to graph representations. </p>