Hyperbolic Space Research Articles

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.

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.

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

Barycenters and a law of large numbers in Gromov hyperbolic spaces

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.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic manifolds

AbstractIn this paper, we build a compactification by a strictly pseudoconvex CR structure for a complete and non-compact Kähler manifold whose curvature tensor is asymptotic to that of the complex hyperbolic space. To do so, we study in depth the evolution of various geometric objects that are defined on the leaves of some foliation of the complement of a suitable convex subset, called an essential subset, whose leaves are the equidistant hypersurfaces above this latter subset. With a suitable renormalization which is closely related to the anisotropic nature of the ambient geometry, the above mentioned geometric objects converge near infinity, inducing the claimed structure on the boundary at infinity.

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.

Michael-Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows

Abstract In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 ( Ω ) ) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k (r) = ω n (λ′) k−1, W 1 ( Ω ) = 1 n | M | ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q 1 ( r ) = W 1 S r n + 1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q 1 − 1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).

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.

Asymptotic behavior of the first Dirichlet eigenvalue of AHE manifolds

AbstractIn this article, we investigate the rate at which the first Dirichlet eigenvalue of geodesic balls decreases as the radius approaches infinity. We prove that if the conformal infinity of an asymptotically hyperbolic Einstein manifold is of nonnegative Yamabe type, then the two‐term asymptotic of the eigenvalues is the same as that in hyperbolic space.

The boundary correspondence of some hyperbolic space

Suppose that (X,d) is a complete metric space and M is a nonempty closed proper subset of X. Let μM be the Gromov hyperbolic metric on X∖M defined by Z. Ibragimov. In this paper, we show that there exists a natural quasisymmetric correspondence between the Gromov boundary of (X∖M,μM) and its metric boundary ∂M.

Complex hyperbolic representations of $$\textrm{PSL}_2(\textbf{R})$$

AbstractWe construct a two-parameter family of irreducible representations of $$\textrm{PSL}_2(\textbf{R})$$ PSL 2 ( R ) in the isometry group of the infinite-dimensional complex hyperbolic space. To this end, we introduce the notion of horospherical combination of two representations. Our family then appears as horospherical combinations of two known one-parameter families.

Hyperbolic Deep Learning in Computer Vision: A Survey

AbstractDeep 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.

Hyperbolic Graph Diffusion Model

Diffusion generative models (DMs) have achieved promising results in image and graph generation. However, real-world graphs, such as social networks, molecular graphs, and traffic graphs, generally share non-Euclidean topologies and hidden hierarchies. For example, the degree distributions of graphs are mostly power-law distributions. The current latent diffusion model embeds the hierarchical data in a Euclidean space, which leads to distortions and interferes with modeling the distribution. Instead, hyperbolic space has been found to be more suitable for capturing complex hierarchical structures due to its exponential growth property. In order to simultaneously utilize the data generation capabilities of diffusion models and the ability of hyperbolic embeddings to extract latent hierarchical distributions, we propose a novel graph generation method called, Hyperbolic Graph Diffusion Model (HGDM), which consists of an auto-encoder to encode nodes into successive hyperbolic embeddings, and a DM that operates in the hyperbolic latent space. HGDM captures the crucial graph structure distributions by constructing a hyperbolic potential node space that incorporates edge information. Extensive experiments show that HGDM achieves better performance in generic graph and molecule generation benchmarks, with a 48% improvement in the quality of graph generation with highly hierarchical structures.

Poincaré Differential Privacy for Hierarchy-Aware Graph Embedding

Hierarchy is an important and commonly observed topological property in real-world graphs that indicate the relationships between supervisors and subordinates or the organizational behavior of human groups. As hierarchy is introduced as a new inductive bias into the Graph Neural Networks (GNNs) in various tasks, it implies latent topological relations for attackers to improve their inference attack performance, leading to serious privacy leakage issues. In addition, existing privacy-preserving frameworks suffer from reduced protection ability in hierarchical propagation due to the deficiency of adaptive upper-bound estimation of the hierarchical perturbation boundary. It is of great urgency to effectively leverage the hierarchical property of data while satisfying privacy guarantees. To solve the problem, we propose the Poincar\'e Differential Privacy framework, named PoinDP, to protect the hierarchy-aware graph embedding based on hyperbolic geometry. Specifically, PoinDP first learns the hierarchy weights for each entity based on the Poincar\'e model in hyperbolic space. Then, the Personalized Hierarchy-aware Sensitivity is designed to measure the sensitivity of the hierarchical structure and adaptively allocate the privacy protection strength. Besides, Hyperbolic Gaussian Mechanism (HGM) is proposed to extend the Gaussian mechanism in Euclidean space to hyperbolic space to realize random perturbations that satisfy differential privacy under the hyperbolic space metric. Extensive experiment results on five real-world datasets demonstrate the proposed PoinDP’s advantages of effective privacy protection while maintaining good performance on the node classification task.