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.

翻译：在机器学习中,数据通常体现在一个(平面)欧几里德空间中,各点之间的距离是直线的。研究人员最近审议了更多外来的(非欧几里德)里曼多元,如极适合树类数据的双曲空间。在本文中,我们建议以常数非零曲线的伪里曼多元为主;它是超双曲和球形地貌的概括化,非半调度度度成分体不一定确定。我们在这种几何学中提供了必要的学习工具,并推广了梯度优化技术。更具体地说,我们为通过大地测量学的距离提供了封闭式表达方式,并确定了一种下降方向,以最大限度地减少某些客观功能。我们的新框架用于图形表达。