The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.
problem Existence of small volume hyperbolic 4-manifolds with embedded 3-manifolds.
method Analysis of hyperbolic manifolds and their submanifolds.
result Minimal volume hyperbolic 4-manifolds with embedded 3-manifolds exist.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
Word embeddings in hyperbolic space outperform Euclidean ones.
problem Improving word embeddings for better performance.
method Learning word embeddings in hyperbolic space using skip-gram architecture and hyperbolic distance objective function.
result Hyperbolic word embeddings show potential, especially in low dimensions, but not clear superiority over Euclidean embeddings.
Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.
problem Embedding hyperbolic surfaces into hyperbolic 3-manifolds with specific symmetries.
method Examined orientation-preserving and orientation-reversing actions on surfaces, including nonorientable ones.
result Found conditions for equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds.
Unified framework for hyperbolic embeddings from mixed data types.
problem Computing hyperbolic embeddings from noisy metric and non-metric data.
method Semidefinite programming and spectral factorization methods.
result Efficient computation of hyperbolic embeddings from arbitrary data.
Random subgroups of hyperbolic groups often form free groups and are hyperbolically embedded.
problem Understanding the structure of random subgroups in acylindrically hyperbolic groups.
method Examining a random subgroup generated by independent random walks.
result Random subgroups are often free groups and hyperbolically embedded in the group.
Geodesically embeds simplest arithmetic hyperbolic manifolds into higher dimensions.
problem Embedding arithmetic hyperbolic manifolds.
method Proving embedding into arithmetic hyperbolic (n+1)-manifolds or their universal mod 2 Abelian covers. result Arithmetic hyperbolic manifolds of simplest type can be geodesically embedded.
Paper introduces a hyperbolic approach for community detection on graphs.
problem Detecting communities on graphs efficiently and effectively.
method Combines hyperbolic embeddings with Riemannian K-means or mixture models.
result Demonstrates effectiveness through experiments on real-world social networks.
This work learns low-rank hyperbolic embeddings for tasks with hierarchical structures.
problem Learning hyperbolic embeddings of tasks with hierarchical structures.
method Formulated as manifold optimization problems and proposed computationally efficient algorithms.
result Efficacy of the proposed approach demonstrated through empirical results.
Null geodesics spaces can be embedded into globally hyperbolic spacetimes.
problem Obstructing conformal embeddings of causally simple spacetimes.
method Analyzing null geodesics and conformal embeddings.
result Causally simple spacetimes can be non-conformally embeddable into globally hyperbolic ones.
Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.
problem Embedding geodesic surfaces into hyperbolic 3-manifolds.
method Analyzing finite group actions on surfaces and proving geodesic embeddings for all irreducible cases.
result All quasiplatonic surfaces embed geodesically into hyperbolic 3-manifolds.
Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. 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.
This paper introduces an acceleration structure for hyperbolic embeddings.
problem Efficiently embedding and visualizing high-dimensional data in hyperbolic spaces.
method Building upon a polar quadtree, the paper introduces a new acceleration structure for hyperbolic embeddings.
result The new method computes embeddings in significantly less time compared to existing methods.
HGCN uses hyperbolic geometry to improve graph node embeddings.
problem Distortion in Euclidean embeddings of real-world graphs.
method Derives GCN operations in hyperbolic space and maps Euclidean features to hyperbolic embeddings.
result HGCN achieves up to 63.1% error reduction in ROC AUC for link prediction.
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
We show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group G is Morse, that is to say any quasi-geodesic connecting points in the cyclic group C generated by g stays close to C. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyper…
Hyperbolic embeddings reduce dimensions for hierarchical data with high precision.
problem Embedding hierarchical data structures like synonym or type hierarchies efficiently.
method Combinatorial construction and hyperbolic multidimensional scaling (h-MDS) for metric spaces.
result Hyperbolic embeddings achieve high precision with few dimensions, e.g., 0.989 MAP with only 2 dimensions on WordNet.
Neural embeddings in hyperbolic space improve graph tasks.
problem Graph similarity in complex networks.
method Learning neural embeddings in hyperbolic space.
result Embeddings in hyperbolic space significantly improve graph tasks.
Researchers develop hyperbolic neural networks for improved data embedding.
problem Lack of hyperbolic neural network layers limits the use of hyperbolic embeddings in machine learning.
method Combining Möbius gyrovector spaces and Riemannian geometry of the Poincaré model to derive hyperbolic neural network layers.
result Hyperbolic sentence embeddings outperform or match Euclidean variants on textual entailment and noisy-prefix recognition tasks.
MuRP embeds multi-relational graphs in hyperbolic space for better hierarchical representation.
problem Current hyperbolic models struggle with multi-relational knowledge graphs that exhibit multiple hierarchies.
method MuRP embeds multi-relational graph data in the Poincaré ball model of hyperbolic space, learning relation-specific parameters for entity embeddings.
result MuRP embeddings outperform Euclidean counterparts and other methods on link prediction tasks, especially at lower dimensions.
The paper proves smooth minimal surfaces in hyperbolic 3-manifolds.
problem Existence of smoothly embedded minimal surfaces in hyperbolic 3-manifolds.
method Analytical proof in hyperbolic geometry.
result Existence of smoothly embedded closed minimal surfaces in hyperbolic 3-manifolds.
Unified framework for hyperbolic network embedding considering multiplex interactions.
problem Misleading results from ignoring multiplex interactions in real-world networks.
method Combines multiplex network hyperbolic embedding and community detection using random walk.
result Effective and efficient node embedding across multiplex channels compared to state-of-the-art.
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.
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
Study of curves in hyperbolic plane with variable curvature.
problem Finding curves with prescribed almost constant curvature in hyperbolic plane.
method Analyzing closed and embedded curves with geodesic curvature.
result Existence of curves with specified curvature variations.
Essential embeddings of metric graphs on hyperbolic surfaces are constructed and studied.
problem Embedding metric graphs on hyperbolic surfaces with negative Euler characteristic.
method Construction of essential embeddings and study of minimal embeddings.
result Formula to compute essential genus and method for explicit essential embedding.
New construction of minimal surfaces in hyperbolic space.
problem Creating minimal surfaces with specific properties in hyperbolic space.
method Presented a new construction method.
result Embedded minimal surfaces with 3 asymptotically totally geodesic ends and arbitrary finite genus.
Efficiently learns high-quality hierarchical embeddings in hyperbolic space.
problem Discovering hierarchical relationships from large-scale similarity scores.
method Used the Lorentz model of hyperbolic geometry to learn embeddings efficiently.
result The proposed approach yields high-quality embeddings that improve over Poincaré embeddings, especially in low dimensions.
Vortices on conical surfaces embedded in hyperbolic space.
problem Embedding Abelian vortices on conical surfaces in hyperbolic space.
method Analyzing elliptic sinh-Gordon and Tzitzeica equations, asymptotic analysis of Painleve III ODE.
result Radial solutions can be globally embedded in hyperbolic space.
Paper presents a new method for learning hyperbolic representations using tree structures.
problem Learning faithful low-dimensional hyperbolic embeddings of data.
method Metric-first approach to learn tree structure, then embed into hyperbolic manifold.
result Novel fast algorithm TreeRep learns tree approximating original metric.
We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.
Constructs special surfaces in hyperbolic space with constant mean curvature.
problem Creating surfaces with specific geometric properties in hyperbolic space.
method Using the DPW method to construct surfaces with constant mean curvature.
result Constructs surfaces with constant mean curvature in hyperbolic space.
Tight embedding proves Gromov norm for quaternionic Kähler class.
problem Calculating the Gromov norm for quaternionic Kähler class.
method Embedding of quaternionic hyperbolic disc into quaternionic hyperbolic n-space.
result Value of the Gromov norm of the quaternionic Kähler class determined.
New result on embedding cohomology of hyperbolic groups.
problem Embedding cohomology of hyperbolic groups into their virtually free subgroups.
method Probabilistic argument and inverse limit of cohomologies.
result Second bounded cohomology of acylindrically hyperbolic groups embeds into cohomologies of virtually free subgroups.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Paper presents a novel hyperbolic neural network for efficient data representation.
problem Efficient representation of hierarchical data in hyperbolic space.
method Develops a fully hyperbolic neural network using projections and equivariant embeddings.
result Proves the proposed embedding is isometric and equivariant under Lorentz transformations.
The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.
We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.
problem Improving word embeddings quality and understanding their relationship with hyperbolic spaces.
method Analyzing squashed shifted PMI matrix and its relation to graph properties and hyperbolic geometry.
result Word embeddings can be connected to hyperbolic spaces through squashed shifted PMI matrix.
Algorithm morphs graphs on hyperbolic surfaces.
problem Morphing graphs on hyperbolic surfaces.
method Generalization of Tutte's spring embedding theorem.
result First algorithm for morphing graphs on hyperbolic surfaces.
Paper proves embedding theorem for conformally compact manifolds.
problem Embedding conformally compact manifolds into hyperbolic spaces.
method Proves analogous Nash Embedding Theorem for conformally compact manifolds.
result Conformally compact manifolds can be isometrically embedded into hyperbolic spaces.
We improve learning sub-tasks in hierarchical reinforcement learning using hyperbolic embeddings.
problem Learning meaningful sub-tasks in hierarchical reinforcement learning remains challenging.
method Combining routing in computer networks and graph-based skill discovery, we use hyperbolic embeddings to define sub-goals.
result Hyperbolic embeddings enforce a global topology on states, enabling the learning of meaningful sub-tasks.
For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
New findings show some hyperbolic 3-manifolds can't be geometrically bounded.
problem Understanding which cusped hyperbolic 3-manifolds can be geometrically bounded.
method Analyzing embeddings and geometric properties of hyperbolic 3-manifolds.
result Some cusped hyperbolic 3-manifolds cannot be geometrically bounded.
We embed directed acyclic graphs using hyperbolic spaces and geodesic cones.
problem Learning graph representations that preserve hierarchical structure.
method Use hyperbolic spaces and geodesic cones to define embeddings of directed acyclic graphs.
result Our method significantly outperforms existing approaches in graph representation learning.
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.