Related papers: Numerical Aspects of Hyperbolic Geometry
A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…
Computer vision tasks such as image classification, image retrieval and few-shot learning are currently dominated by Euclidean and spherical embeddings, so that the final decisions about class belongings or the degree of similarity are made…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
Hypergraphs are useful mathematical representations of overlapping and nested subsets of interacting units, including groups of genes or brain regions, economic cartels, political or military coalitions, and groups of products that are…
Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks. One of the most widely known models of this type is the…
We study topological phases in the hyperbolic plane using noncommutative geometry and T-duality, and show that fractional versions of the quantised indices for integer, spin and anomalous quantum Hall effects can result. Generalising models…
We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…
The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…
Complex network topologies and hyperbolic geometry seem specularly connected, and one of the most fascinating and challenging problems of recent complex network theory is to map a given network to its hyperbolic space. The Popularity…
Scene graph representations enable structured visual understanding by modeling objects and their relationships, and have been widely used for multiview and 3D scene reasoning. Existing methods such as MSG learn scene graph embeddings in…
Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and…
Recently there was an increasing interest in applications of graph neural networks in non-Euclidean geometry; however, are non-Euclidean representations always useful for graph learning tasks? For different problems such as node…
Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of…
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic…
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.
In this work some advances in the theory of curvature of two-dimensional probability manifolds corresponding to families of distributions are proposed. It is proved that location-scale distributions are hyperbolic in the Information…
Implicit neural representations have emerged as a powerful tool in learning 3D geometry, offering unparalleled advantages over conventional representations like mesh-based methods. A common type of INR implicitly encodes a shape's boundary…
Metric learning plays a critical role in training image retrieval and classification. It is also a key algorithm in representation learning, e.g., for feature learning and its alignment in metric space. Hyperbolic embedding has been…
HOLONOMY is a virtual environment based on the mathematical concept of hyperbolic geometry. Unlike other environments, HOLONOMY allows users to seamlessly explore an infinite hyperbolic space by physically walking. They use their body as…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…