Hyperbolic Space Research Articles

Random walk on the Poincaré disk

We consider three interpretations of the standard random walk of [Formula: see text] on a hyperbolic space. We will consider the hyperbolic disk model, and we study properties such as transient behavior, synchronicity and properties of the stationary measure.

Almost η-Ricci solitons on two classes of almost Kenmotsu manifolds

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting almost η-Ricci solitons. In this context, we have shown that in a (k, µ) and (k, µ)' -almost Kenmotsu manifold admitting an almost η-Ricci soliton the curvature conditions (i) the manifold is Einstein, (ii) the manifold is Ricci symmetric (∇S = 0), (iii) the manifold is Ricci semisymmetric (R · S = 0) and (iv) the manifold is projective Ricci semisymmetric (P · S = 0) are equivalent. Also, we have shown that the curvature condition Q · P = 0 in a (k, µ)-almost Kenmotsu manifold admitting an almost η-Ricci soliton holds if and only if the manifold is locally isometric to the hyperbolic space H2n+1(−1) and if a (k, µ)' -almost Kenmotsu manifold admitting an almost η-Ricci soliton satisfies the curvature condition Q · R = 0, then it is locally isometric to the Riemannian product H n+1(−4) × ℝn. n.

Towards Improved Privacy in Digital Marketing: A Unified Approach to User Modeling with Deep Learning on a Data Monetization Platform

This paper introduces an innovative method for safeguarding user privacy in digital marketing campaigns through the application of deep learning techniques on a data monetization platform. This framework empowers users to maintain authority over their personal data while enabling marketers to pinpoint suitable target audiences. The system consists of several key stages Data representation learning in hyperbolic space captures latent user interests across various data sources with hierarchical structures. Subsequently, Generative Adversarial Networks generate synthetic user interests from these embedding. To preserve user privacy, Federated Learning is utilized for decentralized user monetization, Data privacy, modeling training, ensuring data remains undisclosed to marketers. Lastly, a hyperbolic embedding, Federated learning targeting strategy, rooted in recommendation systems, utilizes learned user interests to identify optimal target audiences for digital marketing campaigns. In sum, this approach offers a comprehensive solution for privacy-preserving user modeling in digital marketing.

Infinitely many commensurability classes of compact Coxeter polyhedra in [formula omitted] and [formula omitted]

We prove that certain families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups in these dimensions.

A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study

Abstract This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ \Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w 2 {w}^{2} -stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.

Fermions, quantum gravity, and holography in two dimensions

We study a model comprising N flavors of Kähler Dirac fermion propagating on a triangulated two-dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit N→∞ that the partition function is dominated by a regular triangulation of two-dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However, we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a nonholographic phase. In contrast, for massless fermions, we find evidence that the boundary is conformal even for finite N. This is consistent with theoretical results in quantum Liouville theory. Published by the American Physical Society 2024

Exploiting relation of video segments for temporal action detection

With the development of urban intelligence, intelligent identification of human activities has become imminent. Temporal Action Detection (TAD) is designed to identify real human social activities, which is a challenging task in video understanding. The current methods mainly use global features for boundary matching or predefine all possible proposals, while ignoring the interference of background and the causal relevance of the temporal action, resulting in the generation of more redundant proposals and the decline of detection accuracy. To fill this gap, we propose a novel Dilated Convolution Locate and Action Relevant Score model called DCAR. Specifically, DCAR includes a Dilated Location Network (DL-Net) and Action Relevance Calculation (ARC) block. For the DL-Net, we design a Boundary Feature Enhancement (BFE) block, which enhances the boundary feature of actions and fuses the similar features of different channels by pooling and channel squeeze to reduce the interference of the background. Meanwhile, we also design multiple dilated convolutional structures to aggregate long contextual information in time point/interval after boundary enhancement. For the ARC block, we use the hyperbolic space distance and cross attention to calculate the causal correlation of action proposals classification, which removes the misclassification of action proposals. We conduct extensive experiments on Thumos14 and ActivityNet-1.3, which shows our method significantly improves the performance and achieves state-of-the-art results. On Thumos14, at tIoU 0.3, 0.4 and 0.5, it achieves mAP of 63.2%, 57.0% and 48.5%, respectively; On ActivityNet-1.3 it reaches 9.42% at tIoU 0.95.

One-loop partition functions in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed AdS3

We study the geometry of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed BTZ black hole and find it can be regarded as a quotient of hyperbolic space. We then consider the massive scalar field propagating in the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed BTZ black hole background. The one-loop partition function of scalar field is calculated using the heat kernel method and the Wilson spool proposal. These two methods give consistent result which implies the Wilson spool proposal still holds under \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document} deformation. Moreover, we also calculate the one-loop partition function of graviton in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed BTZ black hole. We find the deformed one-loop partition functions are modified in a simple way, which corresponds to a replacement of the modular parameter. The result precisely matches the large c expansion of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed CFT partition function. These results provide a further check about the correspondence between \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document}-deformed CFT2 and AdS3 with mixed boundary condition.

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.

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.

Intra-community link formation and modularity in ultracold growing hyperbolic networks

Hyperbolic network models offer a straightforward yet powerful method for understanding the small-world, scale-free, highly clustered, and modular characteristics typical of complex systems that are often called as real-world networks. These models involve randomly positioning nodes in a hyperbolic space and connecting them based on a probability that diminishes with distance. In this study, we examine the community structure within networks created by the Popularity Similarity Optimization model, a fundamental hyperbolic model, when the temperature parameter (responsible for tuning the clustering coefficient) is set to the limiting value of zero. We focus on link formation within communities and show their close relation with non-linear preferential attachment processes. Based on this, we derive analytical expressions that significantly improve previous estimates of the expected modularity for partitions formed by equally sized angular sectors in the 2d hyperbolic space. Our formulas can now predict average modularity, confirmed by numerical simulations, with high accuracy over a broader range of model parameters and community sizes relative to the whole network. These results advance our understanding of module formation in hyperbolic networks, highlighting the surprising emergence of communities despite the lack of explicit community formation steps in the model definition.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Inverse mean curvature flow with a free boundary in hyperbolic space

Inverse mean curvature flow with a free boundary in hyperbolic space

Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.

Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continuous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.

Barycenters and a law of large numbers in Gromov hyperbolic spaces

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.