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.
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.
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.
Boundary properties of hyperbolic groups are invariant under a maximization procedure.
problem Proving boundary properties of hierarchically hyperbolic groups are invariant.
method Proving boundary invariance under a maximization procedure.
result Boundary properties of hierarchically hyperbolic groups are invariant under maximization.
We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizati…
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.
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.
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.
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
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. 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…
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 provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our c…
Characterizes boundaries of HHGs and their properties.
problem Understanding boundaries and properties of HHGs.
method Simplicial structure on hierarchically hyperbolic boundaries, modifications of HHG structures.
result Characterizes relative hyperbolicity and thickness of HHGs.
New groups are hyperbolic and rigid in mapping class groups.
problem Understanding the structure of Veech groups in mapping class groups.
method Showed that Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
result Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
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.
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.
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.
Outer automorphism group of hyperbolic groups is HHG under certain conditions.
problem Characterizing the outer automorphism group of hyperbolic groups.
method Proving finite-index subgroups are central extensions of orbifold mapping class groups with bounded Euler class.
result Outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group.
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.
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 …
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Mas…
We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right…
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…
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.
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.
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.
Different from the traditional classification tasks which assume mutual exclusion of labels, hierarchical multi-label classification (HMLC) aims to assign multiple labels to every instance with the labels organized under hierarchical relations. Besides the labels, since linguistic ontologies are intrinsic hierarchies, …
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…
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.
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.
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 provide negative answers to questions posed by Durham, Hagen, and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from right-angled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.
Logarithmic separation profile in hyperbolic groups shows hierarchical structure.
problem Understanding hierarchical structure in hyperbolic groups with logarithmic separation.
method Proving groups with logarithmic separation split over cyclic groups and providing counterexamples.
result Not all groups with hierarchical structure have logarithmic separation profile.
This paper generalizes property (QT) to a broader class of groups.
problem Proving property (QT) for a wider range of groups.
method Using projection complex machinery and hierarchical hyperbolic groups.
result Established sufficient conditions for groups to have property (QT).
Proposes a method to improve hierarchical clustering using set-level structural priors.
problem Lack of supervision for non-leaf structure in hierarchical clustering.
method Introduces set-level structural priors for semi-supervised hyperbolic hierarchical clustering.
result Improves label consistency and similarity-based tree quality over baselines.
Stable subgroups identified in genus two handlebody group.
problem Characterizing stable subgroups in genus two handlebody group.
method Proving genus two handlebody group is hierarchically hyperbolic, using quasi-isometric embedding properties and Hamenstädt-Hensel construction.
result Stable subgroups identified and characterized.
Hyperbolic GANs improve image generation metrics.
problem Improving image generation quality in neural networks.
method Integrating hyperbolic layers into GAN architectures.
result Hyperbolic GANs achieve better metrics than Euclidean counterparts.
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.
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.
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. …
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.
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.
The variational auto-encoder (VAE) is a popular method for learning a generative model and embeddings of the data. Many real datasets are hierarchically structured. However, traditional VAEs map data in a Euclidean latent space which cannot efficiently embed tree-like structures. Hyperbolic spaces with negative curvatu…
The choice of approximate posterior distributions plays a central role in stochastic variational inference (SVI). One effective solution is the use of normalizing flows \cut{defined on Euclidean spaces} to construct flexible posterior distributions. However, one key limitation of existing normalizing flows is that they…
A new drug embedding method using hierarchical drug relations and chemical structures.
problem Learning accurate drug representations from chemical structures and hierarchies.
method Semi-supervised drug embedding using VAE in hyperbolic space.
result The method accurately places drugs in a hierarchy and predicts side-effects.
The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.
problem Conditions for amalgamation of subgroups in hierarchically hyperbolic groups.
method Study of amalgamation conditions and preservation of convexity properties.
result Conditions under which amalgamation preserves hierarchical quasiconvexity and strong quasiconvexity.