Hyperbolic Geometry Research Articles

Rational surfaces with a non‐arithmetic automorphism group

AbstractIn [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, Cambridge, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur in Section 7 of [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14–50]. We give examples of rational surfaces with the same property. Our examples are Looijenga pairs, that is, there is a connected singular nodal curve such that .

Graphlet-based hyperbolic embeddings capture evolutionary dynamics in genetic networks.

Spatial Analysis of Functional Enrichment (SAFE) is a popular tool for biologists to investigate the functional organisation of biological networks via highly intuitive 2D functional maps. To create these maps, SAFE uses Spring embedding to project a given network into a 2D space in which nodes connected in the network are near each other in space. However, many biological networks are scale-free, containing highly connected hub nodes. Because Spring embedding fails to separate hub nodes, it provides uninformative embeddings that resemble a "hairball". In addition, Spring embedding only captures direct node connectivity in the network and does not consider higher-order node wiring patterns, which are best captured by graphlets, small, connected, non-isomorphic, induced subgraphs. The scale-free structure of biological networks is hypothesised to stem from an underlying low-dimensional hyperbolic geometry, which novel hyperbolic embedding methods try to uncover. These include coalescent embedding, which projects a network onto a 2D disk. To better capture the functional organisation of scale-free biological networks, whilst also going beyond simple direct connectivity patterns, we introduce Graphlet Coalescent (GraCoal) embedding, which embeds nodes nearby on a disk if they frequently co-occur on a given graphlet together. We use GraCoal to extend SAFE-based network analysis. Through SAFE-enabled enrichment analysis, we show that GraCoal outperforms graphlet-based Spring embedding in capturing the functional organisation of the genetic interaction networks of fruit fly, budding yeast, fission yeast and E. coli. We show that depending on the underlying graphlet, GraCoal embeddings capture different topology-function relationships. We show that triangle-based GraCoal embedding captures functional redundancies between paralogs. https://gitlab.bsc.es/swindels/gracoal_embedding. Supplementary data are available at Bioinformatics online.

Boundary-dominated photonic Chern insulators with hyperbolic lattice geometry

Boundary-dominated photonic Chern insulators with hyperbolic lattice geometry

Euclidean and Poincaré space ensemble Xgboost

The Hyperbolic space has garnered attention for its unique properties and efficient representation of hierarchical structures. Recent studies have explored hyperbolic alternatives to hyperplane-based classifiers, such as logistic regression and support vector machines. Hyperbolic methods have even been fused into random forests by constructing data splits with horosphere, which proved effective for hyperbolic datasets. However, the existing incorporation of the horosphere leads to substantial computation time, diverting attention from its application on most datasets. Against this backdrop, we introduce an extension of Xgboost, a renowned machine learning (ML) algorithm to hyperbolic space, denoted as PXgboost. This extension involves a redefinition of the node split concept using the Riemannian gradient and Riemannian Hessian. Our findings unveil the promising performance of PXgboost compared to the algorithms in the literature through comprehensive experiments conducted on 64 datasets from the UCI ML repository and 8 datasets from WordNet by fusing both their Euclidean and hyperbolic-transformed (hyperbolic UCI) representations. Furthermore, our findings suggest that the Euclidean metric-based classifier performs well even on hyperbolic data. Building upon the above finding, we propose a space fusion classifier called, EPboost. It harmonizes data processing across various spaces and integrates probability outcomes for predictive analysis. In our comparative analysis involving 19 algorithms on the UCI dataset, our EPboost outperforms others in most cases, underscoring its efficacy and potential significance in diverse ML applications. This research marks a step forward in harnessing hyperbolic geometry for ML tasks and showcases its potential to enhance algorithmic efficacy.

Navigating Memorability Landscapes: Hyperbolic Geometry Reveals Hierarchical Structures in Object Concept Memory.

Why are some object concepts (e.g., birds, cars, vegetables, etc.) more memorable than others? Prior studies have suggested that features (e.g., color, animacy, etc.) and typicality (e.g., robin vs. penguin) of object images influences the likelihood of being remembered. However, a complete understanding of object memorability remains elusive. In this study, we examine whether the geometric relationship between object concepts explains differences in their memorability. Specifically, we hypothesize that image concepts will be geometrically arranged in hierarchical structures and that memorability will be explained by a concept's depth in these hierarchical trees. To test this hypothesis, we construct a Hyperbolic representation space of object concepts (N=1,854) from the THINGS database (Hebart et al., 2019), which consists of naturalistic images of concrete objects, and a space of 49 feature dimensions derived from data-driven models. Using ALBATROSS (Stier, A. J., Giusti, C., & Berman, M. G., In prep), a stochastic topological data analysis technique that detects underlying structures of data, we demonstrate that Hyperbolic geometry efficiently captures the hierarchical organization of object concepts above and beyond a traditional Euclidean geometry and that hierarchical organization is related to memorability. We find that concepts closer to the center of the representational space are more prototypical and also more memorable. Importantly, Hyperbolic distances are more predictive of memorability and prototypicality than Euclidean distances, suggesting that concept memorability and typicality are organized hierarchically. Taken together, our work presents a novel hierarchical representational structure of object concepts that explains memorability and typicality.

A Steward inequality in the Hyperbolic Geometry

The main aim of this study is to show that the steward inequality holds for any hyperbolic domain and distinct points , where denote the hyperbolic metric in and be a hyperbolic geodesic triangle with vertices and sides and be a point on .

Geometric model of quantum navigation during (anti-)search on a plane

A model of joint random walk of two agents in an infinite plane is considered. The agents possess no means of mutual classical communication, but have access to quantum-entanglement resources which are used according to a pre-arranged protocol. The trajectories of the agents diverge on average, but the rate of divergence might be slowed down or speeded up (compared to ordinary random walk) depending on the details of the protocol. This effect has purely quantum nature and can be interpreted in terms of spherical or hyperbolic geometries for slowdown or speedup, respectively.

Dynamic community detection algorithm based on hyperbolic graph convolution

PurposeCommunity detection is a key factor in analyzing the structural features of complex networks. However, traditional dynamic community detection methods often fail to effectively solve the problems of deep network information loss and computational complexity in hyperbolic space. To address this challenge, a hyperbolic space-based dynamic graph neural network community detection model (HSDCDM) is proposed.Design/methodology/approachHSDCDM first projects the node features into the hyperbolic space and then utilizes the hyperbolic graph convolution module on the Poincaré and Lorentz models to realize feature fusion and information transfer. In addition, the parallel optimized temporal memory module ensures fast and accurate capture of time domain information over extended periods. Finally, the community clustering module divides the community structure by combining the node characteristics of the space domain and the time domain. To evaluate the performance of HSDCDM, experiments are conducted on both artificial and real datasets.FindingsExperimental results on complex networks demonstrate that HSDCDM significantly enhances the quality of community detection in hierarchical networks. It shows an average improvement of 7.29% in NMI and a 9.07% increase in ARI across datasets compared to traditional methods. For complex networks with non-Euclidean geometric structures, the HSDCDM model incorporating hyperbolic geometry can better handle the discontinuity of the metric space, provides a more compact embedding that preserves the data structure, and offers advantages over methods based on Euclidean geometry methods.Originality/valueThis model aggregates the potential information of nodes in space through manifold-preserving distribution mapping and hyperbolic graph topology modules. Moreover, it optimizes the Simple Recurrent Unit (SRU) on the hyperbolic space Lorentz model to effectively extract time series data in hyperbolic space, thereby enhancing computing efficiency by eliminating the reliance on tangent space.

When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry

A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as the Klein model)—motivates the study of four cyclic antipodal pairs of a circle, ordered arbitrarily counterclockwise. The translation of results from Euclidean geometry into hyperbolic geometry is obtained by means of hyperbolic trigonometry, called gyrotrigonometry, to which Einstein addition gives rise. Identities that extend the Pythagorean identity in both Euclidean and hyperbolic geometry are obtained.

Deep learning approach to identifying cancer subtypes using convolutional hyperbolic k nearest neighbours method

ABSTRACT To properly treat a cancer patient, radiologists must first review a large volume of cancer images. Based on this information, the appropriate treatment is prescribed. However, the manual cancer image classification is susceptible to errors due to the intricate nature of different tumours. This study introduces a novel convolutional hyperbolic k-nearest neighbours (ChkNN) method to address this challenge. Using deep learning algorithms, convolutional neural networks (CNN), hyperbolic mapping, and hyperbolic geometry, the proposed method effectively extracts and embeds features in hyperbolic space. This embedding not only enhances classification accuracy rates up to 99.92% but also facilitates informed decision-making in cancer diagnosis. The ChkNN method holds immense potential for radiologists worldwide, empowering them to make more accurate and timely treatment decisions, ultimately improving patient outcomes.

Detecting local perturbations of networks in a latent hyperbolic embedding space.

This paper introduces two novel scores for detecting local perturbations in networks. For this, we consider a non-Euclidean representation of networks, namely, their embedding onto the Poincaré disk model of hyperbolic geometry. We numerically evaluate the performances of these scores for the detection and localization of perturbations on homogeneous and heterogeneous network models. To illustrate our approach, we study latent geometric representations of real brain networks to identify and quantify the impact of epilepsy surgery on brain regions. Results suggest that our approach can provide a powerful tool for representing and analyzing changes in brain networks following surgical intervention, marking the first application of geometric network embedding in epilepsy research.

Valuations, completions, and hyperbolic actions of metabelian groups

AbstractActions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.

The worst approximable rational numbers

We classify and enumerate all rational numbers with approximation constant at least 13 using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.

MSO‐DETR: Metric space optimization for few‐shot object detection

AbstractIn the metric‐based meta‐learning detection model, the distribution of training samples in the metric space has great influence on the detection performance, and this influence is usually ignored by traditional meta‐detectors. In addition, the design of metric space might be interfered with by the background noise of training samples. To tackle these issues, we propose a metric space optimisation method based on hyperbolic geometry attention and class‐agnostic activation maps. First, the geometric properties of hyperbolic spaces to establish a structured metric space are used. A variety of feature samples of different classes are embedded into the hyperbolic space with extremely low distortion. This metric space is more suitable for representing tree‐like structures between categories for image scene analysis. Meanwhile, a novel similarity measure function based on Poincaré distance is proposed to evaluate the distance of various types of objects in the feature space. In addition, the class‐agnostic activation maps (CCAMs) are employed to re‐calibrate the weight of foreground feature information and suppress background information. Finally, the decoder processes the high‐level feature information as the decoding of the query object and detects objects by predicting their locations and corresponding task encodings. Experimental evaluation is conducted on Pascal VOC and MS COCO datasets. The experiment results show that the effectiveness of the authors’ method surpasses the performance baseline of the excellent few‐shot detection models.

Convex co‐compact representations of 3‐manifold groups

AbstractA representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Modeling Tree-like Heterophily on Symmetric Matrix Manifolds.

Tree-like structures, characterized by hierarchical relationships and power-law distributions, are prevalent in a multitude of real-world networks, ranging from social networks to citation networks and protein-protein interaction networks. Recently, there has been significant interest in utilizing hyperbolic space to model these structures, owing to its capability to represent them with diminished distortions compared to flat Euclidean space. However, real-world networks often display a blend of flat, tree-like, and circular substructures, resulting in heterophily. To address this diversity of substructures, this study aims to investigate the reconstruction of graph neural networks on the symmetric manifold, which offers a comprehensive geometric space for more effective modeling of tree-like heterophily. To achieve this objective, we propose a graph convolutional neural network operating on the symmetric positive-definite matrix manifold, leveraging Riemannian metrics to facilitate the scheme of information propagation. Extensive experiments conducted on semi-supervised node classification tasks validate the superiority of the proposed approach, demonstrating that it outperforms comparative models based on Euclidean and hyperbolic geometries.

Tunable non-specular effects on hyperbolic crystal coated with a single layer of graphene

In this work, we have theoretically investigated the tunable non-specular effects, specifically Imbert-Fedorov and Goos-Hänchen shifts, on a graphene-based uniaxial hyperbolic crystal geometry. The hyperbolic crystal is made of hexagonal boron nitride and the source of excitation is a circularly polarized light beam. The influence of chemical potential and absolute temperature of the graphene on these effects is examined. Notably, the research reveals that the shifts exhibit complex and significantly varying behavior within and outside the infrared reststrahlen frequency-bands of hyperbolic crystal.

Finding Fibonacci in the Hyperbolic Plane

Summary We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a surprising link to Fibonacci numbers.

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

Hyperbolic Deep Learning in Computer Vision: A Survey

Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning in computer vision. Specifically, hyperbolic learning has shown a strong potential to embed hierarchical structures, learn from limited samples, quantify uncertainty, add robustness, limit error severity, and more. In this paper, we provide a categorization and in-depth overview of current literature on hyperbolic learning for computer vision. We research both supervised and unsupervised literature and identify three main research themes in each direction. We outline how hyperbolic learning is performed in all themes and discuss the main research problems that benefit from current advances in hyperbolic learning for computer vision. Moreover, we provide a high-level intuition behind hyperbolic geometry and outline open research questions to further advance research in this direction.