  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 nondegenerate metric tensor need not be positive definite.
We provide the necessary learning tools in this geometry and extend
gradient-based 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.
