Job recommendation systems face two critical challenges: treating users as a homogeneous group despite diverse job-seeking behaviors, and inadequately modeling the hierarchical nature of job markets. This study proposes Segment-Aware LorentzFM, a novel framework combining behavioral user segmentation with Lorentz-model hyperbolic embeddings. We analyze 1.6 million job applications from 103,896 users on Kariyer.net, revealing three distinct behavioral segments through unsupervised clustering: Ideal Candidates (38.1%) with focused patterns, Career Explorers (51.4%) with diverse behaviors, and Balanced Seekers (10.5%) with moderate exploration. We establish a theoretical foundation connecting segmentation with hyperbolic geometry, proving optimality through information-theoretic and geometric perspectives. Our triangle scoring mechanism in hyperbolic space models interactions among user characteristics, job attributes, and reference points, naturally encoding job market hierarchies. Experimental results demonstrate that Segment-Aware LorentzFM achieves AUC-ROC of 0.9718, outperforming the best baseline by 5.5% and original LorentzFM by 9.4%. The model shows exceptional cold-start performance with F1-score of 0.6068 for new users and jobs. Key contributions include: (1) first framework connecting behavioral segmentation with hyperbolic geometry, (2) novel triangle scoring function in Lorentz space, (3) significant improvements in both warm-start and cold-start scenarios, and (4) practical deployment guidelines. This work advances job recommendation systems by combining user behavioral understanding with geometric representation learning.

Medical hierarchical image classification via dual-geometry image-text learning.

Anomaly detection plays a crucial role in various applications such as industrial defect inspection and safety perception. Existing mainstream approaches typically rely on modeling the distribution of normal samples to build anomaly discrimination models. However, these methods often face challenges in practical scenarios due to ambiguous decision boundaries and limited capability to capture complex semantic and structural variations in defects. To overcome these limitations, we propose a novel industrial defect detection framework based on hyperbolic space. This framework exploits the negative curvature property of hyperbolic geometry to dynamically extract semantic prototypes and embed them into the hyperbolic space, enhancing the model's ability to represent intricate semantic and structural changes. Furthermore, a semantic prototype-guided attention mechanism is integrated to assist the reconstruction of images, enabling accurate localization of anomalies through reconstruction error. Extensive experiments demonstrate that our method achieves state-of-the-art results on multiple industrial anomaly detection datasets. Notably, on the MVTec AD benchmark, our approach attains 99.8% and 99.1% AUROC scores for image-level and pixel-level tasks, respectively, significantly surpassing current leading methods.

Weakly supervised video anomaly detection (WSVAD) is fundamentally constrained by the absence of frame-level annotations, which leads to noisy instance selection in Multiple Instance Learning (MIL) and weak correspondence between temporal video segments and semantic descriptions. Vision-language models address this by enabling cross-modal alignment between visual features and textual labels, but when these representations are learned in Euclidean space, they struggle to capture subtle semantic variations and often produce ambiguous instance ranking under weak supervision. To address this limitation, we propose PoinCLIP-VAD, a vision-language framework that performs cross-modal fusion in hyperbolic space. The model embeds visual and textual features into a shared Poincaré ball geometry, where non-linear distance scaling provides a more expressive representation of latent semantic relationships induced by cross-modal interactions, without relying on predefined hierarchical structures. This geometry-consistent formulation enables more reliable similarity estimation and better preserves distinctions between normal and anomalous patterns. The framework adopts a dual-block architecture consisting of a classification block for coarse anomaly scoring and a video-text alignment block for fine-grained correspondence using negative Poincaré distance. Extensive experiments on benchmark datasets demonstrate that PoinCLIP-VAD achieves an AUC of 90.62% on UCF-Crime and an AP of 86.93% on XD-Violence, confirming improved anomaly discrimination and more consistent cross-modal alignment under weak supervision.

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.

Graph neural networks (GNNs) have excelled in predicting graph properties in various applications ranging from identifying trends in social networks to drug discovery and malware detection. With the abundance of new architectures and increased complexity, GNNs are becoming highly specialized when tested on a few well-known datasets. However, how the performance of GNNs depends on the topological and features properties of graphs is still an open question. In this work, we introduce a comprehensive benchmarking framework for graph machine learning, called HypBench, focusing on the performance of GNNs across varied network structures. Utilizing the geometric soft configuration model in hyperbolic space, we generate synthetic networks with realistic topological properties and node feature vectors. This approach enables us to assess the impact of network properties, such as topology-feature correlation, degree distributions, local density of triangles, and homophily, on the effectiveness of different GNN architectures. Our results highlight the dependency of model performance on the interplay between network structure and node features, providing insights for model selection in various scenarios. This study contributes to the field by offering a versatile tool for evaluating GNNs, thereby assisting in developing and selecting suitable models based on specific data characteristics.

In this paper, we introduce a new four-step iterative scheme, referred to as the Jungck–HR iteration, for approximating the unique common fixed point of a pair of contractive mappings in hyperbolic spaces. We establish strong convergence, stability, and Δ-convergence results for the proposed method. A comparative analysis shows that the Jungck–HR iteration converges faster than the Jungck–AI and Jungck–DK iterative schemes and remains convergent for certain contractive mappings where the Jungck–AI iteration fails. Numerical experiments are presented to demonstrate the convergence behaviour of the generated sequences. In addition, we conduct numerical simulations and present graphical illustrations showing the convergence of the orbits under the Jungck-HR iteration, thereby extending its theoretical applicability. As an application, the effectiveness of the proposed scheme is demonstrated by solving a two-dimensional nonlinear Volterra integral equation.

In laparoscopic liver resection, precise registration between preoperative 3D models and intraoperative laparoscopic point clouds remains challenging due to liver deformation, respiratory motion, and limited visibility. This study aims to develop a robust registration method achieving stable alignment under low-overlap and non-rigid conditions. We propose a novel framework centered on a Hyperbolic-Topology Interaction Module. The module maps rotation-invariant features extracted by the backbone into hyperbolic space, leveraging its negative curvature to amplify subtle geometric differences, while simultaneously constructing a topological graph that propagates spatial relationships to enhance feature consistency. Finally, based on the refined features, a coarse-to-fine matching strategy combined with a hypothesis generation mechanism establishes robust correspondence estimation. Evaluation of our method with comparative methods on both simulated and real datasets shows that our method achieves state-of-the-art results. On the public DePOLL dataset, our method achieved the lowest surface target registration error (TRE) of 6.2mm and the lowest internal TRE of 7.0mm. Additional non-rigid experiments further validate the strong generalization capability of the proposed features under varying deformation conditions. The proposed method effectively combines local geometric discrimination with global topological reasoning, achieving notable gains in accuracy, robustness, and efficiency. It delivers reliable rigid initialization for augmented reality (AR)-guided resection navigation and establishes a solid foundation for subsequent non-rigid estimation, demonstrating strong potential for clinical use.

Antiferromagnetic Ising models on frustrated lattices can realize classical spin liquids, with highly degenerate ground states and, possibly, fractionalized excitations and emergent gauge fields. Motivated by the recent interest in many-body systems in negatively curved space, we study hyperbolic frustrated Ising models. Specifically, we consider nearest-neighbor Ising models on tesselations with odd-length loops in two-dimensional hyperbolic space. For finite systems with open boundaries we determine the ground-state degeneracy exactly, and we perform extensive finite-temperature Monte Carlo simulations to obtain thermodynamic data as well as correlation functions. We show that the shape of the boundary, constituting an extensive part of the system, can be used to control low-energy states: Depending on the boundary, we find ordered or disordered ground states. Our results demonstrate how geometric frustration acts in curved space to produce classical spin liquids.

TIH: Transformer in hyperbolic space for 3D point cloud semantic segmentation

Abstract The superintegrability of five Hamiltonians defined on the 3-dimensional spaces with constant curvature κ , sphere S κ 3 ( κ > 0 ) and hyperbolic space H κ 3 ( κ < 0 ), was recently studied in a previous work. Three of the Hamiltonians were oscillator related, while the other two were the Kepler related. In all the cases the systems had additional nonlinear terms. Now we present a similar study of two new Hamiltonians, neither related to the oscillator nor to the Kepler system, that were previously studied by Evans on the 3-dimensional Euclidean space E 3 . The formalism consider the curvature κ as a parameter. All the mathematical expressions are presented by using κ as a parameter, in such a way that particularizing for κ > 0 , κ = 0, or κ < 0 , the corresponding properties are obtained for the system on the sphere S κ 3 , the Euclidean space E 3 , or the hyperbolic space H κ 3 , respectively.

ABSTRACT We investigate the asymptotic behavior of solutions to the defocusing energy‐critical complex Ginzburg‐Landau equation on exterior domains and hyperbolic spaces. Employing the energy method, we establish a rigorous convergence theory for the zero‐dispersion limit from the energy‐critical complex Ginzburg‐Landau equation to the energy‐critical nonlinear heat equation. Moreover, we derive the inviscid limit connecting the energy‐critical complex Ginzburg‐Landau equation to the energy‐critical nonlinear Schrödinger equation.

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by ClHV. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra Cl(HV) simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space HV are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with ClHV are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived.

Abstract This note serves as an addendum to our previous work (Flores et al 2025 Class. Quantum Grav. 42 215020), where the Bartnik Splitting Conjecture (BSC) was first established for globally hyperbolic Lorentzian length spaces. Here, we strengthen and generalize that result by removing the assumption of a global topological product structure, which is not intrinsic in the setting of Lorentzian length spaces. Instead, we only require the existence of a compact Cauchy set. Consequently, there is no hypothesis on the asymptotic behaviour of vertical curves—whose existence is now obtained a posteriori —but rather the more natural timelike geodesic completeness condition is assumed. This refinement (theorem 1.2) yields a stronger and more flexible version of the BSC, extending its applicability and bringing it closer to its smooth counterpart.

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 2 , separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on ℂ P n for all n and on ℂ H n for n = 2 , 3 .

The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees

Recent advances in action segmentation have greatly enhanced our understanding of complex and dynamic scenes in video content. Despite these improvements, the field continues to face persistent challenges, particularly in terms of model efficiency and the substantial cost associated with manual annotation. In this work, we introduce a novel framework that integrates active learning within hyperbolic space to effectively address these issues. By leveraging the hierarchical representational capacity of hyperbolic space, which is naturally suited for modeling structured data, and combining it with the selective efficiency of active learning, our method introduces hyperbolic uncertainty metrics to guide the targeted selection of the most informative video frames and sequences for annotation. This enables the model to prioritize annotation efforts where they are most impactful. Furthermore, the model iteratively refines pseudo labels using all available annotations, significantly reducing the need for exhaustive labeling while preserving high segmentation accuracy. To further mitigate reliance on precise annotations, we enhance the MS-TCN model by incorporating soft pseudo labels and a weighting mechanism that dynamically adjusts learning based on label confidence, allowing for more robust training in the presence of noisy or weakly labeled data. Extensive experiments conducted on two widely used action segmentation benchmark datasets validate the effectiveness of our approach, demonstrating that it can substantially reduce annotation effort while maintaining overall segmentation performance.

Abstract In this paper, we investigate the extension of uniformisation results for Gromov hyperbolic spaces beyond the standard geodesic setting. By establishing a Gehring-Hayman type theorem for conformal deformations of any intrinsic Gromov hyperbolic space, we provide a framework for analysing spaces that do not necessarily admit geodesics. As a primary application, we prove that any complete intrinsic hyperbolic space with at least two points in the Gromov boundary can be uniformised by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification between the Gromov boundary of the original space and the metric boundary of the deformed space.

ABSTRACT In this paper, we investigate the existence of global nontrivial weak solutions to certain partial differential systems defined in the exterior domain of hyperbolic space. We establish sufficient conditions for the nonexistence of such global solutions, providing insights into the behavior of these solutions in the context of hyperbolic geometry. Our results highlight the challenges in the existence of global solutions in non‐Euclidean settings.

Abstract We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by 50 reflections of , and the other by a rotation of order 21 and a reflection of . For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.