In machine learning community, data is usually represented in Euclidean space, while real-world data like scale-free networks and knowledge graphs exhibit a highly non-Euclidean latent anatomy, and suffer from large distortion when embedded in Euclidean space. Riemannian manifolds like hyperbolic and spherical space have demonstrated outstanding performance in embedding tree-like or spherical structures individually but cannot embed the mixed structures. To overcome these issues, we exploit semi-Riemannian manifolds, which could not only cover the hyperbolic and spherical geometries but also their submanifolds which could encode more complex geometries. In this talk, we will show you some basic concepts of non-Euclidean embedding, as well as the motivations to go beyond the Riemannian manifolds. We define some basic operations like matrix-vector multiplications, bias translations in semi-Riemannian manifolds, which are useful for (knowledge) graph embedding. By leveraging these operations, we also extend the graph convolutional networks into semi-Riemannian manifolds.
Speaker: Bo Xiong