New criteria for relative hyperbolicity in hierarchically hyperbolic spaces.
problem Characterizing relative hyperbolicity in hierarchically hyperbolic spaces.
method New formulation of relative hyperbolicity in terms of hierarchy structures, applied to graphs associated to surfaces.
result The separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures.
Survey of tools for studying hierarchical hyperbolic spaces.
problem Understanding hierarchical hyperbolic spaces.
method Various tools developed for studying HHSs.
result Ease of use for non-experts in HHS machinery.
Introduces hierarchical hyperbolic spaces for non-experts.
problem Understanding hierarchical hyperbolic spaces for non-experts.
method No specific method mentioned; aimed at non-experts.
result Introduces hierarchical hyperbolic spaces for non-experts.
New combinatorial structure for hierarchically hyperbolic spaces.
problem Constructing new hierarchically hyperbolic spaces.
method Combinatorial hierarchical hyperbolicity criterion to construct and clarify HHS structures.
result HHSs admit a combinatorial structure, clarifying the application of the combinatorial HHS criterion.
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space X, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathc…
New insights into groups with uniform exponential growth.
problem Uniform exponential growth in hierarchically hyperbolic groups.
method Quasi-isometric characterization and new insights into group structure.
result Uniform exponential growth for hierarchically hyperbolic groups.
Hyperbolic space outperforms Euclidean in learning hierarchical data.
problem Learning hierarchical data in Euclidean space requires exponentially many samples.
method Established geometric obstruction in Euclidean space and showed hyperbolic space's advantage.
result Hyperbolic space enables learning with O(mRlogm) samples, matching information-theoretic optimum. Paper provides first theoretical guarantees for hyperbolic space learning.
problem Learning a classifier in hyperbolic space for hierarchical data.
method Efficient algorithm for large-margin hyperplane learning in hyperbolic space.
result The low embedding dimension in hyperbolic space leads to superior classifier learning guarantees.
Characterizes strongly quasiconvex subsets in hierarchically hyperbolic spaces.
problem Understanding the structure of strongly quasiconvex subsets in HHSs.
method Characterization through contracting properties, relative divergence, and hierarchical structure.
result Proves characterization of hyperbolically embedded subgroups in hierarchically hyperbolic groups.
Multicurve stabilizers' extensions are hierarchically hyperbolic.
problem Characterizing the structure of multicurve stabilizers' extensions.
method Proving the extensions of multicurve stabilizers are hierarchically hyperbolic groups.
result Extensions of multicurve stabilizers are hierarchically hyperbolic.
New largest acylindrical actions found for hierarchically hyperbolic groups.
problem Understanding non-positive curvature in hierarchically hyperbolic groups.
method Study of acylindrical actions and quasigeodesic stability in hierarchically hyperbolic groups.
result Hierarchically hyperbolic groups have a unique largest acylindrical action.
Graphs of multicurves are hierarchically hyperbolic spaces.
problem Understanding the geometric properties of graphs related to surfaces.
method Demonstrating hierarchical hyperbolicity and coarse median properties.
result Graphs of multicurves have a quadratic isoperimetric inequality and are Gromov hyperbolic under certain conditions.
Uniform undistortion in cyclic subgroups of certain groups.
problem Understanding undistorted subgroups in group actions.
method Using quasimorphisms and hierarchically hyperbolic groups.
result Sharp examples of undistorted subgroups in hierarchically hyperbolic groups.
The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.
problem Characterizing the hyperbolicity of curve graphs and their boundaries.
method Using hierarchical hyperbolicity and framed curves, the study examines the properties of curve graphs and their boundaries.
result The curve graphs and their boundaries are hierarchically hyperbolic but not Gromov hyperbolic.
We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to …
Study shows no boundary maps for certain groups.
problem Existence of boundary maps for hierarchically hyperbolic spaces.
method Analysis of right-angled Artin groups and mapping class groups.
result Negative results on boundary maps for some groups.
Poincaré VAEs improve hierarchical data representation.
problem Hierarchical data structures are difficult to represent in Euclidean latent spaces.
method Introducing Poincaré ball model of hyperbolic geometry as a latent space for VAEs.
result Better generalization and hierarchical structure recovery in hyperbolic space.
A new model embeds word and label hierarchies in hyperbolic space for HMLC.
problem Learning mappings from word hierarchies to label hierarchies in hierarchical multi-label classification.
method Proposes a Hyperbolic Interaction Model (HyperIM) to learn label-aware document representations in hyperbolic space.
result Demonstrates improved performance for HMLC compared to state-of-the-art methods.
New result classifies hierarchically hyperbolic groups based on their Morse boundaries.
problem Classifying hierarchically hyperbolic groups using Morse boundaries.
method Generalizing a result on Gromov boundaries to Morse boundaries, showing quasi-isometry if and only if there's a homeomorphism.
result Spaces are quasi-isometric if and only if there's a 2-stable, quasi-möbius homeomorphism between their Morse boundaries.
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.
Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.
problem Classifying 3-manifold groups with equivariant hierarchically hyperbolic structures.
method Construction of suitable quasimorphisms on Seifert pieces to construct actions on quasi-lines.
result 3-manifold groups admit equivariant hierarchically hyperbolic structures.
Explains what hierarchically hyperbolic spaces are.
problem None explicitly stated; focuses on definition and understanding.
method Heuristic discussion and detailed technical discussion.
result Provides a mental picture and technical definition of HHSs.
Paper proposes a novel method for aligning hierarchical data using optimal transport in hyperbolic spaces.
problem Aligning hierarchical data like ontologies without external supervision.
method Optimal transport over hyperbolic spaces.
result The method outperforms standard embedding alignment techniques.
Proposes HypCSE for enhanced hierarchical clustering.
problem Challenges in existing hierarchical clustering methods.
method Hyperbolic Continuous Structural Entropy (HypCSE) neural networks.
result Superior performance on seven datasets.
In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a numb…
Hyperbolic SVM improves classification in complex networks.
problem Accurately classifying points in hyperbolic space with hierarchical relationships.
method Introducing hyperbolic SVM, a hyperbolic formulation of SVM classifiers.
result Hyperbolic SVM outperforms Euclidean SVM in multi-class prediction tasks.
The study proves non-existence of Cannon-Thurston maps for certain groups.
problem Proving the non-existence of Cannon-Thurston maps for specific groups.
method Analyzing hierarchically hyperbolic groups and their boundaries.
result Non-existence of Cannon-Thurston maps for hyperbolic normal subgroups of hierarchically hyperbolic groups.
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
New group not biautomatic, geometrically constructed.
problem Constructing a non-biautomatic group.
method Using a CAT(0) hierarchically hyperbolic group and geodesic currents.
result First example of a non-biautomatic CAT(0) group of flat-rank 2.
Extensions of Veech groups using hierarchical hyperbolic spaces.
problem Generalizing Veech groups to non-lattice cases.
method Using a hyperbolic space E^ and a nice action of Γ on it. result Finitely generated Veech groups are hierarchically hyperbolic.
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.
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
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.
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.
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.
Study of Dehn filling quotients in hierarchically hyperbolic groups.
problem Understanding the structure of Dehn filling quotients in specific groups.
method Introduced a construction for cusped spaces of relatively hyperbolic groups and used it to study Dehn-filling-like quotients.
result Infinite hyperbolic quotients of mapping class groups of punctured spheres and braid groups are found.
We prove a geometric model for HHS hierarchies as CAT(0) cube complexes.
problem Modeling hierarchically hyperbolic spaces as cube complexes.
method Prove quasi-median quasi-isometry between hulls and cubical models.
result Hierarchical boundaries are locally modeled by CAT(0) cube complexes.
HypeGBMS clusters data in hyperbolic space, overcoming Euclidean limitations.
problem Clustering in hierarchical or tree-like datasets in curved spaces.
method Hyperbolic Gaussian Blurring Mean Shift with Möbius-weighted means.
result HypeGBMS effectively captures latent hierarchies in non-Euclidean data.
Paper introduces a hyperbolic Gaussian distribution for better learning in hierarchical data.
problem Learning hierarchical data in hyperbolic space.
method Developed a novel hyperbolic distribution for gradient-based learning.
result Demonstrated improved learning on various datasets.
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 study proves theorems about quasiflats in hierarchically hyperbolic spaces.
problem Understanding the structure of hierarchically hyperbolic groups.
method Proving theorems about quasiflats and hierarchically hyperbolic groups.
result Proves that a group is hyperbolic if it contains no Z2 subgroups and contains a uniform-quality quasiflat. Teichmüller geodesics can have flexible limit sets.
problem Teichmüller geodesics may not converge to a single point in the boundary.
method Analyzing the topology of the hierarchically hyperbolic space boundary.
result Limit sets of Teichmüller geodesics can be almost anything allowed by the topology.
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.
The study resolves conjectures about quasiflats in hierarchically hyperbolic spaces.
problem Understanding the structure and properties of quasiflats in hierarchically hyperbolic spaces.
method Proving that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants.
result The rank of a hierarchically hyperbolic space coincides with the maximal dimension of a quasiflat.
Poincaré embeddings learn hierarchical symbolic data representations.
problem Learning hierarchical representations for complex symbolic data like text and graphs.
method Embedding into hyperbolic space (Poincaré ball) for efficient Riemannian optimization.
result Poincaré embeddings outperform Euclidean embeddings on data with latent hierarchies.
New graphs show hierarchical hyperbolic properties, extending previous work.
problem Characterizing hierarchically hyperbolic properties of multiarc and curve graphs.
method Analyzing the geometric intersection number and using PMod(S) action.
result Multiarc and curve graphs are hierarchically hyperbolic.
Survey discusses new ideas in geometric group theory and their applications.
problem Understanding geodesic metric spaces and their equivariant wall structures.
method Introduces and highlights the impact of injective metric spaces and cubical approximation theorem.
result Rich equivariant wall structures in various geodesic metric spaces.
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.