Hyperbolic Geometry Research Articles

Residual Hyperbolic Graph Convolution Networks

Hyperbolic graph convolutional networks (HGCNs) have demonstrated representational capabilities of modeling hierarchical-structured graphs. However, as in general GCNs, over-smoothing may occur as the number of model layers increases, limiting the representation capabilities of most current HGCN models. In this paper, we propose residual hyperbolic graph convolutional networks (R-HGCNs) to address the over-smoothing problem. We introduce a hyperbolic residual connection function to overcome the over-smoothing problem, and also theoretically prove the effectiveness of the hyperbolic residual function. Moreover, we use product manifolds and HyperDrop to facilitate the R-HGCNs. The distinctive features of the R-HGCNs are as follows: (1) The hyperbolic residual connection preserves the initial node information in each layer and adds a hyperbolic identity mapping to prevent node features from being indistinguishable. (2) Product manifolds in R-HGCNs have been set up with different origin points in different components to facilitate the extraction of feature information from a wider range of perspectives, which enhances the representing capability of R-HGCNs. (3) HyperDrop adds multiplicative Gaussian noise into hyperbolic representations, such that perturbations can be added to alleviate the over-fitting problem without deconstructing the hyperbolic geometry. Experiment results demonstrate the effectiveness of R-HGCNs under various graph convolution layers and different structures of product manifolds.

Multi-geometry embedded transformer for facial expression recognition in videos

Dynamic facial expressions in videos express more realistic emotional states, and recognizing emotions from in-the-wild facial expression videos is a challenging task due to the changeable posture, partial occlusion and various light conditions. Although current methods have designed transformer-based models to learn spatial–temporal features, they cannot explore useful local geometry structures from both spatial and temporal views to capture subtle emotional features for the videos with varied poses and facial occlusion. To this end, we propose a novel multi-geometry embedded transformer (MGET), which adapts multi-geometry knowledge into transformers and excavates spatial–temporal geometry information as complementary to learn effective emotional features. Specifically, from a new perspective, we first design a multi-geometry distance learning (MGDL) to capture emotion-related geometry structure knowledge under Euclidean and Hyperbolic spaces. Especially based on the advantages of hyperbolic geometry, it finds the more subtle emotional changes among local spatial and temporal features. Secondly, we combine MGDL with transformer to design spatial–temporal MGETs, which capture important spatial and temporal multi-geometry features to embed them into their corresponding original features, and then perform cross-regions and cross-frame interaction on these multi-level features. Finally, MGET gains superior performance on DFEW, FERV39k and AFEW datasets, where the unweighted average recall (UAR) and weighted average recall (WAR) are 58.65%/69.91%, 41.91%/50.76% and 53.23%/55.40%, respectively, and the gained improvements are 2.55%/0.66%, 3.69%/2.63% and 3.66%/1.14% compared to M3DFEL, Logo-Forme and EST methods.

3d super hyperbolic geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSpC(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The invariant quadratic form is x1x2−xx¯+ϕψ+ϕ¯ψ¯ in Wigner's coordinates x1,x2∈R and x∈C, where ϕ,ψ are Dirac fermions. The extended action is linear in the super space variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie super group with its Lie algebra. This extension of Wigner's representation opens the door to studying geometry in super hyperbolic three-space with this metric, initiated here with foundations on super geodesics, triangles and tetrahedra, and culminating in a proof of divergence of the volume of a typical ideal tetrahedron and a discussion of the non-zero fermionic correction to the Schläfli formula.

Flame self-interactions in MILD combustion of homogeneous and inhomogeneous mixtures

Flame self-interaction (FSI) events in Moderate or Intense Low-Oxygen Dilution (MILD) combustion of homogeneous and inhomogeneous mixtures of methane and oxidiser have been analysed using three-dimensional Direct Numerical Simulations (DNS). The simulations have been conducted at the same global equivalence ratio (⟨ϕ⟩=0.8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle = 0.8$$\\end{document}) for different levels of O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {O_2}$$\\end{document} concentration (dilution) and initial turbulence intensities. It has been reported that both homogeneous and inhomogeneous mixture MILD combustion cases exhibit significant occurrences of FSI events, with the peak frequency of FSI events occurring towards the burned gas side in all cases. Moreover, the frequency of FSI events increases with increasing dilution level and turbulence intensity, but the presence of mixture inhomogeneity leads to a reduction in total FSI events. In all cases, the cylindrical FSI topologies (i.e. tunnel formation and tunnel closure) were found to have a higher likelihood of occurrence compared to spherical FSI topologies (i.e. unburned and burned gas pockets). The geometries of FSI topologies were also analysed using the mean and Gaussian curvatures. It has been shown that the inward propagating spherical FSI topologies (i.e. unburned gas pockets) are associated with negative mean curvature, while outward propagating spherical FSI topologies (i.e. burned gas pockets) are associated with positive mean curvature. Moreover, tunnel formation (tunnel closure) FSI topologies predominantly exhibit either elliptic geometries with positive (negative) mean curvature or hyperbolic saddle geometries with negative (positive) mean curvature. It has been shown for the first time in MILD combustion that the mean values of kinematic restoration and dissipation terms in the transport equation of the magnitude of the reaction progress variable conditional upon the reaction progress variable tend to cancel each other in the vicinity of the critical points associated with cylindrical topologies. Thus, the singular contributions in these terms, which are obtained from analytical descriptions in the vicinity of tunnel formation and tunnel closure topologies, do not affect the balance equation of the magnitude of the gradient of the reaction progress variable. Consequently, there is no need for a separate model treatment for singularities in modelling approaches based on the magnitude of the gradient of the reaction progress variable. The FSI events in the reaction dominated and propagating flame regions of MILD combustion have also been analysed for the first time. It has been found that more FSI events occur in the reaction dominated region, particularly towards the burned gas side. However, the majority of spherical FSI topologies are found in the propagating flame region. The findings from this study indicate that turbulence intensity, dilution level and mixture inhomogeneity effects need to be considered in any attempt to extend flame surface-based modelling approaches to MILD combustion.

A Comprehensive Overview of Photon-Proton Scattering and QCD Methodology

The AdS/CFT correspondence is used as a helpful reference in the light front hyperbolic geometry technique, which solves this issue by mapping a confining gauge theory parameterized on the light front to a greater anti-de Sitter space. Three different interaction processes exist Direct or angular the target photon quark and the photon pair directly. When a lepton-antilepton pair is produced, only quantum electrodynamics (QED) is used; however, when a quark-antiquark pair is produced, both QED and perturbative quantum chromodynamics (QCD) are used. A deep-inelastic electron-photon scattering experiment was used to study the photon structure function, which describes the photon inherent quark composition: Single resolved: the desired spectroscopy quark combination creates the vector meson, one of the constituents of relationships to the investigating photon. The main focus of the current study was on photon-proton and QCD methodology. The main theoretical conclusion resulting from the work carried out can be used in the development of the conceptual concept of further researchers and this work also will be a guideline for future researchers.

Long‐range quantum energy teleportation and distribution on a hyperbolic quantum network

AbstractTeleporting energy to remote locations is new challenge for quantum information science and technology. Developing a method for transferring local energy in laboratory systems to remote locations will enable non‐trivial energy flows in quantum networks. From the perspective of quantum information engineering, we propose a method for distributing local energy to a large number of remote nodes using hyperbolic geometry. Hyperbolic networks are suitable for energy allocation in large quantum networks since the number of nodes grows exponentially. To realise long‐range quantum energy teleportation (QET), we propose a hybrid method of quantum state telepotation and QET. By transmitting local quantum information through quantum teleportation and performing conditional operations on that information, QET can theoretically be realized independent of geographical distance. The method we present will provide new insights into new applications of future large‐scale quantum networks and potential applications of quantum physics to information engineering.

Formalization, Automatization and Visualization of Hyperbolic Geometry

Formalization, Automatization and Visualization of Hyperbolic Geometry

The angular derivative problem for petals of one-parameter semigroups in the unit disk

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals, we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals, we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy–Wolff point.

A combinatorial curvature flow in spherical background geometry

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.

HyperED: A hierarchy‐aware network based on hyperbolic geometry for event detection

AbstractEvent detection plays an essential role in the task of event extraction. It aims at identifying event trigger words in a sentence and classifying event types. Generally, multiple event types are usually well‐organized with a hierarchical structure in real‐world scenarios, and hierarchical correlations between event types can be used to enhance event detection performance. However, such kind of hierarchical information has received insufficient attention which can lead to misclassification between multiple event types. In addition, the most existing methods perform event detection in Euclidean space, which cannot adequately represent hierarchical relationships. To address these issues, we propose a novel event detection network HyperED which embeds the event context and types in Poincaré ball of hyperbolic geometry to help learn hierarchical features between events. Specifically, for the event detection context, we first leverage the pre‐trained BERT or BiLSTM in Euclidean space to learn the semantic features of ED sentences. Meanwhile, to make full use of the dependency knowledge, a GNN‐based model is applied when encoding event types to learn the correlations between events. Then we use a simple neural‐based transformation to project the embeddings into the Poincaré ball to capture hierarchical features, and a distance score in hyperbolic space is computed for prediction. The experiments on MAVEN and ACE 2005 datasets indicate the effectiveness of the HyperED model and prove the natural advantages of hyperbolic spaces in expressing hierarchies in an intuitive way.

Earthquakes on the Once-Punctured Torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behavior of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Scalable and Effective Temporal Graph Representation Learning With Hyperbolic Geometry.

Real-life graphs often exhibit intricate dynamics that evolve continuously over time. To effectively represent continuous-time dynamic graphs (CTDGs), various temporal graph neural networks (TGNNs) have been developed to model their dynamics and topological structures in Euclidean space. Despite their notable achievements, the performance of Euclidean-based TGNNs is limited and bounded by the representation capabilities of Euclidean geometry, particularly for complex graphs with hierarchical and power-law structures. This is because Euclidean space does not have enough room (its volume grows polynomially with respect to radius) to learn hierarchical structures that expand exponentially. As a result, this leads to high-distortion embeddings and suboptimal temporal graph representations. To break the limitations and enhance the representation capabilities of TGNNs, in this article, we propose a scalable and effective TGNN with hyperbolic geometries for CTDG representation (called STGNh ). It captures evolving behaviors and stores hierarchical structures simultaneously by integrating a memory-based module and a structure-based module into a unified framework, which can scale to billion-scale graphs. Concretely, a simple hyperbolic update gate (HuG) is designed as the memory-based module to store temporal dynamics efficiently; for the structure-based module, we propose an effective hyperbolic temporal Transformer (HyT) model to capture complex graph structures and generate up-to-date node embeddings. Extensive experimental results on a variety of medium-scale and billion-scale graphs demonstrate the superiority of the proposed STGNh for CTDG representation, as it significantly outperforms baselines in various downstream tasks.

Rethinking Few-Shot Class-Incremental Learning With Open-Set Hypothesis in Hyperbolic Geometry

Rethinking Few-Shot Class-Incremental Learning With Open-Set Hypothesis in Hyperbolic Geometry

Curved Geometric Networks for Visual Anomaly Recognition.

Learning a latent embedding to understand the underlying nature of data distribution is often formulated in Euclidean spaces with zero curvature. However, the success of the geometry constraints, posed in the embedding space, indicates that curved spaces might encode more structural information, leading to better discriminative power and hence richer representations. In this work, we investigate the benefits of the curved space for analyzing anomalous, open-set, or out-of-distribution (OOD) objects in data. This is achieved by considering embeddings via three geometry constraints, namely, spherical geometry (with positive curvature), hyperbolic geometry (with negative curvature), or mixed geometry (with both positive and negative curvatures). Three geometric constraints can be chosen interchangeably in a unified design, given the task at hand. Tailored for the embeddings in the curved space, we also formulate functions to compute the anomaly score. Two types of geometric modules (i.e., geometric-in-one (GiO) and geometric-in-two (GiT) models) are proposed to plug in the original Euclidean classifier, and anomaly scores are computed from the curved embeddings. We evaluate the resulting designs under a diverse set of visual recognition scenarios, including image detection (multiclass OOD detection and one-class anomaly detection) and segmentation (multiclass anomaly segmentation and one-class anomaly segmentation). The empirical results show the effectiveness of our proposal through consistent improvement over various scenarios. The code is made available at https://github.com/JHome1/GiO-GiT.

Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds.

Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements Möbius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.

ON THE GEOMETRIC PICTURE OF THE WORLD IN THE LIGHT OF HEIDEGGER’S ONTOLOGY

Until recently, there was one unsolved riddle regarding the logic that Lobachevsky followed when creating his non-Euclidean (hyperbolic) geometry. This article shows that such a logic is implicitly present in him, and it coincides with the logic already formed now, called complementary-dialectical logic. This logic stems from Heidegger’s fundamental ontology. In the field of the geometric discipline of thought, complementary-dialectical logic makes it possible to combine the historical and logical aspects of the genesis of Lobachevsky’s geometry. Allows us to understand how and why imaginary points appear on a hyperbolic line, how they are related to points at infinity, etc. constructing a geometric picture of the world as part of the overall scientific picture.

Hyperbolic string tadpole

Hyperbolic geometry on the one-bordered torus is numerically uniformized using Liouville theory. This geometry is relevant for the hyperbolic string tadpole vertex describing the one-loop quantum corrections of closed string field theory. We argue that the Lamé equation, upon fixing its accessory parameter via Polyakov conjecture, provides the input for the characterization. The explicit expressions for the Weil-Petersson metric as well as the local coordinates and the associated vertex region for the tadpole vertex are given in terms of classical torus conformal blocks. The relevance of this vertex for vacuum shift computations in string theory is highlighted.

A Visual, In-Expensive, and Wireless Capillary Rheometer for Characterizing Wholly-Cellular Bioinks.

Rheological measurements with in situ visualization can elucidate the microstructural origin of complex flow behaviors of an ink. However, existing commercial rheometers suffer from high costs, the need for dedicated facilities for microfabrication, a lack of design flexibility, and cabling that complicates operation in sterile or enclosed environments. To address these limitations, a low-cost ($300) visual, in-expensive and wireless rheometer (VIEWR) using 3D-printed and off-the-shelf components is presented. VIEWR measurements are validated by steady-state and transient flow responses for different complex fluids, and microstructural flow profiles and evolution of yield-planes are revealed via particle image velocimetry. Using the VIEWR, a wholly-cellular bioink system comprised of compacted cell aggregates is characterized, and complex yield-stress and viscoelastic responses are captured via concomitantly visualizing the spatiotemporal evolution of aggregate morphology. A symmetric hyperbolic extensional-flow geometry is further constructed inside a capillary tube using digital light processing. Such geometries allow for measuring the extensional viscosity at varying deformation rates and further visualizing the alignment and stretching of aggregates under external flow. Synchronized but asymmetric evolution of aggregate orientation and strain through the neck is visualized. Using varying geometries, the jamming and viscoelastic deformation of aggregates are shown to contribute to the extensional viscosity of the wholly-cellular bioinks.

Empirical Calibration of a Cylindrical Ion Trap for Mass-Selected Gas-Phase Fluorescence Spectroscopy.

The ion motion in a quadrupole ion trap of hyperbolic geometry is well described by the Mathieu equations. A simpler cylindrical ion trap has also gained significance and has been used by us for fluorescence-spectroscopy experiments. This design allows for the easy replacement of the end-cap with a mesh, enhancing the photon collection. It is crucial to obtain a firm understanding of the ion motion in cylindrical ion traps and their capability as mass spectrometers. We present here an empirical method of calibrating a cylindrical ion trap based on fluorescence detection. This can be done nearly background-free in a pulsed experiment. The ions are located at the center of the trap, where the field is primarily quadrupolar, and here an effective Mathieu description is found through an effective geometry parameter. In spectroscopy experiments, high buffer-gas pressures are needed to efficiently cool the ions, which complicates the ions' motion and hence their stability. Still, simulations show that the stability diagram closely aligns with the Mathieu diagram, albeit shifted due to collisions. We map the stability diagram for six molecular ions by fluorescence collection from four cations and two anions spanning m/z from 212 to 647. The stability diagram is parametrized through the Mathieu functions with an m/z-dependent effective geometry parameter and a q-dependent shrinkage of the diagram. Based on the calibration, we estimate the mass resolution to be +7/-3 Da for ions with masses in the hundreds of Da.

Extension of Pick’s Theorem to Spherical Geometry using Girard’s Theorem

In this article, Pick’s theorem is extended to three-dimensional bodies with two-dimensional surfaces, namely spherical geometry. The equation for the area of a polygon consisting of equilateral spherical triangles is obtained by combining Girard’s theorem used to find area of any spherical triangle and Pick’s theorem used to find area of a simple polygon with lattice point vertices in Euclidian geometry. Vertices of the polygon are represented by integer points. In this way, an equation to find area of a spherical polygon is presented. This equation could give an idea to be applied on cylindrical surfaces, hyperbolic geometry and more general surfaces. The theorem proposed in this article which is the extension of Pick’s theorem using Girard’s theorem seems to be a special case of a more general theorem.