Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
Random hyperbolic surfaces have low Cheeger constants.
problem Estimating Cheeger constants of random hyperbolic surfaces.
method Modeling random hyperbolic surfaces using ideal triangles and analyzing their Cheeger constants.
result Generic hyperbolic surfaces have Cheeger constants less than 3/2π + ε.
Random quotients preserve hyperbolic properties in groups.
problem Preserving hyperbolic properties in random group quotients.
method Independent random walks, spinning families, projection complexes.
result Random quotients of acylindrical and hierarchical hyperbolic groups remain so.
Simple Euclidean models outperform hyperbolic graph learning models.
problem The effectiveness of hyperbolic graph learning models is questioned.
method Careful analysis of hyperbolic graph representation learning, identifying and addressing issues with baselines, modeling assumptions, and metric usage.
result Simple Euclidean models often outperform hyperbolic graph learning models, even on hyperbolic datasets.
Study properties of hyperbolic Yamabe solitons in submanifolds.
problem Characterize and understand hyperbolic Yamabe solitons in submanifolds.
method Define hyperbolic Yamabe flow, prove properties, and classify solitons.
result Prove that hyperbolic Yamabe solitons are pseudosymmetric or metallic shaped.
By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
Enhances neural networks in hyperbolic space for better data structure capture.
problem Capturing hierarchical data structures efficiently.
method Unified hyperbolic model for neural network components.
result Superior parameter efficiency and outperformance over Euclidean methods.
Upper bound for Hausdorff distance between hyperbolic space and its medianization.
problem Calculating the Hausdorff distance between hyperbolic space and its medianization.
method Using de Sitter space to model finite-dimensional hyperbolic space and its medianization, calculating the Hausdorff distance.
result An upper bound for the Hausdorff distance between hyperbolic space and its medianization is calculated.
We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry f…
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
Study connects hyperbolic geometry to membrane shapes.
problem Understanding the shapes of biological membranes.
method Relates geometry of hyperbolic space to Helfrich model.
result Establishes a connection between membrane shapes and hyperbolic geometry.
Generalizes global hyperbolicity to higher signatures and proves compactness.
problem Global hyperbolicity in higher pseudo-Riemannian signatures.
method Generalization and proof of compactness of causal diamonds.
result Existence of solutions to a Plateau problem and characterization of holonomies.
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bic…
Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic Gaussian}, a Gaussian-like distribution on hyperbolic space whose density can be evalua…
Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network lay…
Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
problem Understanding the generic properties of hyperbolic 3-manifolds.
method Counting model on links and Dehn surgeries.
result Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
Study spectral gaps and bass notes of random hyperbolic 3-orbifolds.
problem Investigate spectral properties of random hyperbolic 3-orbifolds.
method Analyze two models of random hyperbolic 3-orbifolds related to Apollonian and super Apollonian groups.
result Explicit spectral gaps determined for random orbifolds.
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
Study proves unique compactification of hyperbolic space.
problem Proving uniqueness of compactification of hyperbolic space.
method Analyzing one-parameter family of elliptic PDEs on hyperbolic space.
result Euclidean half-plane is the only compactification of hyperbolic space.
A new framework for hyperbolic neural networks using the Klein model is introduced.
problem Previous works focused on Poincaré and hyperboloid models, neglecting the Klein model.
method Formulation of operations using the Klein model, study of the Klein linear layer, and comparison with Poincaré ball model.
result The Klein HNN performs similarly to the Poincaré ball model, offering a third option.
BIGUE algorithm provides credible intervals for hyperbolic network embeddings.
problem Uncertainty in hyperbolic network embeddings.
method Markov chain Monte Carlo (MCMC) algorithm for Bayesian hyperbolic random graph model.
result Samples from the posterior distribution provide credible intervals for hyperbolic coordinates and network properties.
Improves few-shot learning for hierarchical data using hyperbolic space.
problem Few-shot class-incremental learning for hierarchical data.
method Contrastive learning in hyperbolic space, Poincaré ball model, hyperbolic contrastive loss, maximum entropy distribution.
result Effective improvement of coarse and fine class accuracies in few-shot conditions.
Bounds on homology of hyperbolic orbifolds using simplicial models.
problem Bounding homology of hyperbolic orbifolds.
method Efficient simplicial model of the thick part of hyperbolic orbifolds.
result Linear bounds on Betti numbers and torsion homology in terms of volume.
Improves stability in hyperbolic neural networks for complex data generation.
problem Numerical instability in hyperbolic neural networks hinders complex architecture development.
method Proposes a novel hyperbolic AE-GAN architecture with stable layers.
result Demonstrates state-of-the-art performance in generating complex data.
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
A discrete method approximates hyperbolic curvature flow in the plane.
problem Modeling wave phenomena in solid-liquid interfaces.
method Semidiscrete finite difference method for hyperbolic curvature flow.
result Error bounds for natural discrete norms are proven.
The paper connects hyperbolic Dehn surgery and Higgs bundles to construct model objects in representation varieties.
problem Constructing model objects in representation varieties for Higgs bundles.
method Reviewing hyperbolic Dehn surgery and bending procedures, and applying them to Higgs bundles.
result Explicit examples of model objects in representation varieties are produced.
Hyperbolic geometry autoencoder outperforms Euclidean in top-N recommendation tasks.
problem Top-N recommendation performance using hyperbolic geometry.
method Simple autoencoder based on hyperbolic geometry with a single hidden layer.
result Outperforms Euclidean models and state-of-the-art methods.
In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in which both points and isometries are expressed as complex quaternions, analogous…
Graph neural network (GNN) has shown superior performance in dealing with graphs, which has attracted considerable research attention recently. However, most of the existing GNN models are primarily designed for graphs in Euclidean spaces. Recent research has proven that the graph data exhibits non-Euclidean latent ana…
This work improves KG embeddings by integrating hyperbolic and attention mechanisms.
problem Preserving hierarchical and logical patterns in KGs with low-dimensional embeddings.
method Combines hyperbolic reflections/rotations with attention mechanisms to capture complex relational patterns.
result Improves MRR by up to 6.1% on standard benchmarks and new state-of-the-art results in high dimensions.
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
This work tackles manifold regression onto hyperbolic space for tree classification and taxonomy extension.
problem Performing manifold-valued regression onto an hyperbolic space for tree classification and taxonomy extension.
method Formulated as a manifold regression task in hyperbolic space, proposed a parametric deep learning model and a non-parametric kernel method.
result Hyperbolic-based estimators significantly outperform Euclidean space methods in taxonomy expansion.
Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
New hyperbolicity concepts expand manifold study.
problem Studying hyperbolicity on complex manifolds.
method Introducing sG-hyperbolicity, weakly p-Kähler hyperbolic structures, and pluriclosed star split hyperbolic metrics.
result Expands the class of divisorially hyperbolic manifolds.
Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.
problem Eigenvalue rigidity of random hyperbolic surface covers.
method Selberg trace formula and polynomial method.
result Distribution of eigenvalues converges to hyperbolic plane's spectral measure.
APo-VAE generates text in hyperbolic space for better hierarchical representation.
problem Lack of hierarchical structure in Euclidean embeddings for natural language.
method Adversarial Poincare Variational Autoencoder (APo-VAE) in hyperbolic latent space.
result APo-VAE outperforms Euclidean VAEs in capturing latent language hierarchies.
New research shows hyperbolic embeddings are useful for global consistency tasks in graphs.
problem The usefulness of hyperbolic representations in graph learning tasks.
method Computed hyperbolic embeddings for node classification and link prediction tasks, addressing optimization issues at zero curvature.
result Hyperbolic embeddings are more effective for tasks requiring global consistency, while Euclidean models are superior for other tasks.
We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is …
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
The paper defines two types of hyperbolicity for complex manifolds and proves related results.
problem Defining and studying hyperbolicity for a broader class of complex manifolds.
method Introducing SKT hyperbolicity and Gauduchon hyperbolicity, proving results using SKT and Gauduchon metrics.
result Every SKT hyperbolic manifold is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic manifold is divisorially hyperbolic.
Hyperbolic geometry reveals financial network structure and systemic importance.
problem Understanding the structure and importance of financial networks.
method Data from European banking stress tests, hyperbolic geometry analysis.
result Latent dimensions of `popularity' and `similarity' are strongly associated with systemic importance.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.