Related papers: The analytic continuation of hyperbolic space

Recently, hyperbolic space has risen as a promising alternative for semi-supervised graph representation learning. Many efforts have been made to design hyperbolic versions of neural network operations. However, the inspiring geometric…

Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic…

This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely…

These lecture notes provide an overview of different aspects of de Sitter space and their plausible holographic interpretations. We start with a general description of the classical spacetime. We note the existence of a cosmological horizon…

Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only…

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic…

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary 'critical' region containing a…

We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class…

This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded…

We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…

Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the…

Hyperbolic representations are effective in modeling knowledge graph data which is prevalently used to facilitate multi-hop reasoning. However, a rigorous and detailed comparison of the two spaces for this task is lacking. In this paper,…

We introduce a simple autoencoder based on hyperbolic geometry for solving standard collaborative filtering problem. In contrast to many modern deep learning techniques, we build our solution using only a single hidden layer. Remarkably,…

Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to…

A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter…

Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…

Hyperbolic space is quickly gaining traction as a promising geometry for hierarchical and robust representation learning. A core open challenge is the development of a mathematical formulation of hyperbolic neural networks that is both…

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and…

We introduce hyperbolic attention networks to endow neural networks with enough capacity to match the complexity of data with hierarchical and power-law structure. A few recent approaches have successfully demonstrated the benefits of…

Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning…