Hyperbolic Representation Learning

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Overview

Hyperbolic representation learning studies how negatively curved spaces can encode tree-like hierarchies and scale-free relationships more compactly than flat Euclidean spaces. Common approaches learn embeddings in Poincaré balls or Lorentz hyperboloids, using graph neural networks, entailment cones, contrastive objectives, and curvature-adaptive or manifold-aware layers. Applications span graph prediction, vision-language alignment, time-series forecasting, and biomedical data, with evaluations often testing retrieval, classification, and transfer across levels of a hierarchy. A central open question is when the geometry itself improves results: outcomes depend on objectives, curvature, and optimization stability, and some controlled studies find that hierarchy-aware supervision matters more than the choice of geometry.

Latest papers

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92 papers18 papers in the last four weeks, up 6% on the four weeks before. 0.1% of all new papers.

Jun 4, 2026

HyperVis: Continuous Latent Visual Relational Graphs on the Lorentz Hyperboloid for Compositional Reasoning

Vision-Language Models (VLMs) struggle with compositional reasoning that requires understanding inter-object relationships. A natural remedy is to inject explicit scene graph triplets ⟨s,p,o⟩\langle s, p, o \rangle from an off-the-shelf scene graph generator (SGG), but we show this backfires: discrete text labels collide with the continuous visual modality, degrading GQA accuracy from 60.38% to 58.86%. We propose HyperVis, which bypasses the SGG semantic bottleneck entirely. From NN class-agnostic region proposals, we compute a dense O(N2)O(N^2) visual relation tensor via spatially-biased cross-attention, project it onto a Lorentz hyperboloid, and enforce hierarchy through spatial physics, namely IoA-driven entailment cones and exterior-angle repulsion. We discover that HyperVis contributes in two complementary ways: (1) as a training-time regularizer, the hyperbolic relational losses shape LoRA representations that improve generative VQA (GQA 61.03% vs.\ 57.21% for LoRA fine-tuning without relational losses, recovering and surpassing the baseline); and (2) as an inference-time relational encoder, hyperbolic prefix tokens boost discriminative compositional scoring (SugarCrepe 79.94%, ++6.25pp over baseline). The learned curvature stabilises at κ=4.0κ{=}4.0, an order of magnitude above prior hyperbolic VLMs where κκ typically collapses toward zero, indicating that continuous visual features genuinely require the exponential volume of strongly curved space. A controlled Euclidean ablation confirms this decomposition: the relational pipeline regularises LoRA comparably in flat space (GQA 60.81%), but the compositionality gain is specifically hyperbolic (SugarCrepe ++4.58pp over Euclidean), with entailment loss ∼6×{\sim}6{\times} higher in Euclidean training. Codes are available at TBA.

Moshiur Farazi, Sameera Ramasinghe, Mahbub Ahmed Turza +1

Data Science and AI, University of Doha for Science and Technology, Qatar · Pluralis Research, Australia · Department of Electrical and Computer Engineering, North South University, Bangladesh

Hyperbolic Representation LearningVisual ReasoningRelational Deep LearningCompositional Representation LearningVLM ReasoningVisually Grounded Reasoning+2
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Jun 2, 2026

SkelHCC: A Hyperbolic CLIP-Driven Cache Adaptation Framework for Skeleton-based One-Shot Action Recognition

Skeleton-based action recognition aims to understand human behaviors from body joint sequences and is especially challenging in the one-shot setting, where only a single labeled exemplar is available for each novel action. A key challenge is learning representations that capture the hierarchical and compositional structure of human motion while aligning effectively with high-level action semantics under extreme data scarcity. Existing approaches, largely based on Euclidean embeddings and low-level motion cues, struggle to model the tree-like organization of skeleton data, limiting cross-modal alignment and generalization to unseen action categories. We propose SkelHCC, a unified skeleton hyperbolic CLIP-driven cache adaptation framework for one-shot skeleton-based action recognition. SkelHCC introduces an Explicitly Hierarchical Hyperbolic CLIP (EH-HCLIP) module that embeds skeleton sequences and action language into a shared hyperbolic space. By leveraging the negative curvature and exponential volume growth of hyperbolic geometry, EH-HCLIP naturally encodes the joint-part-body hierarchy of human anatomy and yields structurally consistent cross-modal representations. To support efficient one-shot adaptation, SkelHCC further integrates a training-free LLM-guided Multi-granularity Voting Cache (LMV-Cache) for context-aware inference. Experiments on NTU RGB+D 60, NTU RGB+D 120, and PKU-MMD demonstrate that SkelHCC consistently outperforms state-of-the-art methods.

Yanan Liu, Anqi Zhu, Jingmin Zhu +6

Yunnan University, Kunming, China · Monash University · Lancaster University +1

Skeleton-Based Action RecognitionCross-Modal Representation LearningHierarchical Representation LearningFew-Shot LearningHyperbolic Representation Learning
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May 28, 2026

SAHG: Sector-Anisotropic Hyperbolic Graph Model for Social Bot Detection

LLM-driven social bots can generate fluent, human-like text, reducing the discriminative advantage of content-based detection alone. However, coordinated campaigns still leave relational patterns -- interactions, behavioral similarity, shared neighborhoods, community positions, and coordinated activity -- that graph-based methods can exploit. Existing graph detectors face two challenges when exploiting such evidence. First, Euclidean GNNs distort hierarchical and scale-free social graphs; while hyperbolic geometry addresses this volume-growth mismatch, fixed-curvature models still assign uniform geometric resolution to structural directions with different densities and separation needs. Second, relational evidence is not always reliable: sophisticated bots forge heterophilic connections with genuine users, causing neighborhood aggregation to mix bot and human signals and dilute account-level evidence. We propose SAHG (Sector-Anisotropic Hyperbolic Graph), addressing both challenges. SAHG learns a direction-dependent curvature field γ(u)γ(u) that adapts geometric resolution across structural directions, and uses sector prototypes to convert angular concentration and alignment into classifier-readable features. To prevent contaminated aggregation from overwhelming account-level evidence, SAHG encodes per-account features and graph-neighborhood representations in two independent SAH channels, fusing them only at the classifier. Experiments on Fox8-23, BotSim-24, and MGTAB show that SAHG achieves the highest accuracy and F1 on all three benchmarks, outperforming feature-based, graph-based, LLM-based, and isotropic hyperbolic baselines. Ablation and geometric analyses confirm the effectiveness of the anisotropic geometry and dual-channel design.

Hanning Lu, Yingguang Yang, Jinwei Su +8

University of Leeds · University of Science and Technology of China · South China Normal University +6

Social Media AnalysisGraph Neural NetworksHyperbolic Representation LearningSocial Bot Detection
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May 27, 2026

Temporal Hyperbolic Graph Representation Learning for Scale-Free Internet Routing and Delay Prediction

Predicting Internet round-trip time (RTT) is critical for routing optimization, quality-of-service (QoS) provisioning, and traffic engineering, yet remains challenging due to long-term temporal dependencies, evolving routing dynamics, and heavy-tailed latency distributions. While Temporal Graph Neural Networks (TGNNs) can model evolving network topologies, most existing approaches operate in Euclidean space, which poorly captures the hierarchical and scale-free structure of Internet routing graphs. Hyperbolic geometry provides a more suitable representation space. We propose HERMIT (Hyperbolic Edge-aware RTT Modeling via Integrated Topology), a hybrid framework combining a hyperbolic manifold-preserving temporal GNN with a Random Forest regressor for joint link prediction and RTT prediction. Built on HMPTGN, HERMIT introduces RTT-aware edge features and a learnable edge encoder to improve modeling of evolving link states and routing behavior. The resulting hyperbolic node representations are combined with historical RTT statistics for robust latency prediction. We evaluate HERMIT on a large-scale real Internet dataset spanning 2015-2024. HERMIT consistently outperforms a strong Random Forest baseline using only historical RTT statistics, achieving a 6% RMSE improvement while reducing large errors on heavy-tailed samples. It also surpasses prior hyperbolic TGNN models, including HMPTGN and HTGN, in link prediction performance. These results demonstrate that combining hyperbolic temporal graph learning with tree-based regression provides a scalable solution for RTT prediction in real-world Internet topologies.

Yi-Ling Kuo, Hao-Yu Tien, Shih-Yu Tsai

Department of Information Management and Finance, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan

Temporal GNNsTemporal Link PredictionHyperbolic Representation Learning
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May 12, 2026

H2G: Hierarchy-Aware Hyperbolic Grouping for 3D Scenes

Hierarchical 3D grouping aims to recover scene groups across multiple granularities, from fine object parts to complete objects, without relying on semantic labels or a fixed vocabulary. The main challenge is to transform 2D foundation-model cues into coherent hierarchy supervision and embed that hierarchy in a 3D representation. We propose H2G, a hyperbolic affinity field for hierarchical 3D grouping. Our method derives semantically organized tree supervision by interpreting foundation-model affinities through Dasgupta's objective for similarity-based hierarchical clustering. This supervision is distilled into a single Lorentz hyperbolic feature field, whose geometry is well suited for tree-like branching structures. A hierarchy-aware objective aligns the field with fine-level assignments, coarse object structure, compact feature clusters, and LCA (Lowest Common Ancestor) ordering. This formulation represents multiple grouping levels in one feature space, enabling semantic hierarchical grouping grounded in 2D foundation-model knowledge.

ByungHa Ko, Youngmin Lee, Dong Hwan Kim

Department of Computer Science and Engineering, Korea University · Intelligence and Interaction Research Center, Korea Institute of Science and Technology

ClusteringHierarchical Representation Learning3D Representation LearningHyperbolic Representation Learning
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May 11, 2026

HYPERPOSE: Hyperbolic Kinematic Phase-Space Attention for 3D Human Pose Estimation

We introduce HYPERPOSE, a novel 3D human pose estimation framework that performs spatio-temporal reasoning entirely within the Lorentz model of hyperbolic space Hd\mathbb{H}^d to natively preserve the hierarchical tree topology of the human skeleton. Current state-of-the-art pose estimators aim to capture complex joint dynamics by relying on transformers and graph convolutional networks. Since these architectures operate exclusively in Euclidean space which fundamentally mismatches the inherent tree structure of the human body, these methods inevitably suffer from exponential volume distortion and struggle to maintain structural coherence. To this end, we depart from flat spaces and aim to improve geometric fidelity with Hyperbolic Kinematic Phase-Space Attention (HKPSA), natively embedding complex joint relationships without distortion, alongside a multi-scale windowed hyperbolic attention mechanism that efficiently models temporal dynamics in O(TW)O(TW) complexity. Furthermore, to overcome the well-known instability of training non-Euclidean manifolds, HYPERPOSE introduces a novel Riemannian loss suite and an uncertainty-weighted curriculum, enforcing physical geodesic constraints like bone length and velocity consistency. Extensive evaluations on the Human3.6M and MPI-INF-3DHP datasets demonstrate that HYPERPOSE achieves state-of-the-art structural and temporal coherence, significantly reducing both volume distortion and velocity error, while establishing new state-of-the-art benchmarks in overall positional accuracy.

Vinduja Thekkath, Ashish Musale, Ajay Waghumbare +1

3D Human Pose EstimationAttention MechanismsHyperbolic Representation Learning
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May 10, 2026

HyNeuralMap: Hyperbolic Mapping of Visual Semantics to Neural Hierarchies

Understanding the intricate mappings between visual stimuli and neural responses is a fundamental challenge in cognitive neuroscience. While current approaches predominantly align images and functional magnetic resonance imaging (fMRI) responses in Euclidean space, this geometry often struggles to preserve fine-grained semantic relationships and latent hierarchical structures across visual and neural modalities. To overcome this, we propose HyNeuralMap, a framework that employ hyperbolic Lorentz model to map visual semantics into a shared, cross-subject neural hierarchy. By leveraging the negative curvature of hyperbolic space as an inductive bias, the proposed framework better captures hierarchical semantic organization and cross-subject neural similarities. Specifically, visual and neural embeddings are jointly optimized through hyperbolic geometric alignment, where geodesic distances preserve semantic proximity and hierarchical relationships more effectively than Euclidean embeddings. Experiments demonstrate that HyNeuralMap consistently outperforms state-of-the-art Euclidean baselines in both multi-label semantic prediction and cross-modal retrieval tasks. This confirms hyperbolic geometry's superiority for cross-modal semantic alignment and hierarchical modeling, providing a new avenue for vision-neural representation learning.

Zihan Ma, Tian Xia, Kexin Wang +3

School of Electronic Information (School of Artificial Intelligence), the Xi’an Key Laboratory of Radiomics and Intelligent Perception, Northwest University, Xi’an 710127, China

Neural Representation GeometryCross-Modal AlignmentHyperbolic Representation Learning
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May 9, 2026

Semantic Alignment in Hyperbolic Space for Open-Vocabulary Semantic Segmentation

Open-vocabulary semantic segmentation requires adapting image-level vision-language models such as CLIP to dense pixel-level prediction, which is challenging due to the mismatch between hierarchical structure and semantic alignment in the embedding space. While recent works leverage hyperbolic geometry to model hierarchical relationships, they align embeddings across hierarchical levels but overlook semantic misalignment among embeddings within the same level. In this work, we propose HyRo, a hyperbolic fine-tuning framework that decouples hierarchical and semantic alignment in the Poincaré ball model. HyRo aligns hierarchical levels by adjusting the hyperbolic radius and refines semantic relationships through angular alignment using an orthogonal transformation that theoretically preserves the hyperbolic radius. Experiments on standard open-vocabulary semantic segmentation benchmarks demonstrate that HyRo achieves state-of-the-art performance over prior methods.

Hoang M. Truong, Hai Nguyen-Truong, Dang Huynh

Fulbright University Vietnam

VLM AdaptationHierarchical Representation LearningOpen-Vocabulary Semantic SegmentationVision-Language AlignmentHyperbolic Representation Learning
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May 9, 2026

Curvature-Aware Captioning:Leveraging Geodesic Attention for 3D Scene Understanding

Accurate 3D scene description is fundamental to robotic navigation and augmented reality, yet current dense captioning methods face significant limitations in processing sparse point cloud data. % Existing approaches that apply Euclidean embedding spaces struggle to simultaneously preserve fine-grained local geometric details and model exponentially growing global semantic hierarchies, leading to either inaccurate localization or disjointed, shallow scene descriptions. % In this work, we propose a novel Curvature-Aware Captioning framework, integrating novel non-Euclidean geodesic attention mechanisms, to resolve the localization-contextualization conflict. % Specifically, self-attention within Oblique space enforces dimensional homogeneity while establishing long-range dependencies. Bidirectional geodesic cross-attention within Lorentz space models hierarchical semantic relationships across scene instances, enabling simultaneous precision in object localization and coherence in scene descriptions. % Theoretical analysis confirms that the curvature complementarity between the Oblique manifold and Lorentz hyperboloid resolves the Euclidean-hyperbolic conflict, ensuring feature stability via isotropic optimization while preserving inherent hierarchical relationships. Extensive experiments on ScanRefer and Nr3D benchmarks demonstrate state-of-the-art performance, with significant gains in both localization accuracy and descriptive richness.

Ziyao He, Yingjie Liu, ZhangYangRui +3

East China Normal University

Attention MechanismsHyperbolic Representation Learning3D Scene Understanding3D VLMs3D Visual Grounding
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May 7, 2026

Hyperbolic Concept Bottleneck Models

Concept Bottleneck Models (CBMs) have become a popular approach to enable interpretability in neural networks by constraining classifier inputs to a set of human-understandable concepts. While effective, current models embed concepts in flat Euclidean space, treating them as independent, orthogonal dimensions. Concepts, however, are highly structured and organized in semantic hierarchies. To resolve this mismatch, we propose Hyperbolic Concept Bottleneck Models (HypCBM), a post-hoc framework that grounds the bottleneck in this structure by reformulating concept activation as asymmetric geometric containment in hyperbolic space. Rather than treating entailment cones as a pre-training penalty, we show they encode a natural test-time activation signal: the margin of inclusion within a concept's entailment cone yields sparse, hierarchy-aware activations without any additional supervision or learned modules. We further introduce an adaptive scaling law for hierarchically faithful interventions, propagating user corrections coherently through the concept tree. Empirically, HypCBM rivals post-hoc Euclidean models trained on 20×\times more data in sparse regimes required for human interpretability, with stronger hierarchical consistency and improved robustness to input corruptions.

Daniel Uyterlinde, Swasti Shreya Mishra, Pascal Mettes

Informatics Institute, University of Amsterdam, The Netherlands

Concept Bottleneck ModelsHierarchical Representation LearningNeural Network InterpretabilityHyperbolic Representation Learning
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May 6, 2026

BatMIL: Geometry-Embedding State Space Model for Whole-Slide Image Representation

Accurate analysis of histopathological images is critical for disease diagnosis and treatment planning. Whole-slide images (WSIs), which digitize tissue specimens at gigapixel resolution, are fundamental to this process but require aggregating thousands of patches for slide-level predictions. Multiple Instance Learning (MIL) tackles this challenge with a two-stage paradigm, decoupling tile-level embedding and slide-level prediction. However, most existing methods implicitly embed patch representations in homogeneous Euclidean spaces, overlooking the hierarchical organization and regional heterogeneity of pathological tissues. This limits current models' ability to capture global tissue architecture and fine-grained cellular morphology. To address this limitation, we introduce a hybrid hyperbolic-Euclidean representation that embeds WSI features in dual geometric spaces, enabling complementary modeling of hierarchical tissue structures and local morphological details. Building on this formulation, we develop BatMIL, an MIL method that leverages both geometric spaces. To model long-range dependencies among thousands of patches, we employ a structured state space sequence model backbone that encodes patch sequences with linear computational complexity. Furthermore, to account for regional heterogeneity, we introduce a mixture-of-experts module that dynamically routes each patch to specialized expert sub-networks based on its features, improving representational capacity while reducing redundant computation. Experiments on seven datasets spanning six cancer types demonstrate that BatMIL achieves outstanding performance against fifteen state-of-the-art MIL approaches, demonstrating that geometry-embedding representation learning offers a promising direction for computational pathology.

Enhui Chai, Sicheng Chen, Tianyi Zhang +4

PuzzleLogic Pte Ltd, Singapore 229594, Singapore · Nhu Department of Electrical Engineering and Computer Science, University of California, Irvine · Department of Electrical & Computer Engineering, National University of Singapore

Histopathology Image ClassificationMultiple Instance LearningWhole-Slide Image AnalysisHyperbolic Representation Learning
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May 4, 2026

Hyp2Former: Hierarchy-Aware Hyperbolic Embeddings for Open-Set Panoptic Segmentation

Recognizing unknown objects is crucial for safety-critical applications such as autonomous driving and robotics. Open-Set Panoptic Segmentation (OPS) aims to segment known thing and stuff classes while identifying valid unknown objects as separate instances. Prior OPS approaches largely treat known categories as a flat label set, ignoring the semantic hierarchy that provides valuable structural priors for distinguishing unknown objects from in-distribution classes. In this work, we propose Hyp2Former, an end-to-end framework for OPS that does not require explicit modeling of unknowns during training, and instead learns hierarchical semantic similarities continuously in hyperbolic space. By explicitly encoding hierarchical relationships among known categories, the model learns a structured embedding space that captures multiple levels of semantic abstraction. As a result, unknown objects that cannot be confidently classified as known categories still remain in close proximity to higher-level concepts (e.g., an unknown animal remains closer to "animal" or "object" than to unrelated concepts such as "electronics" or "stuff") and can therefore be reliably detected, even if their fine-grained category was not represented during training. Empirical evaluations across multiple public datasets such as MS COCO, Cityscapes, and Lost&Found demonstrate that Hyp2Former outperforms existing methods on OPS, achieving the best balance between unknown object discovery and in-distribution robustness.

Yao Lu, Rohit Mohan, Florian Drews +2

Department of Computer Science, University of Freiburg, Germany · Bosch Research, Robert Bosch GmbH, Germany

Image SegmentationPanoptic SegmentationHierarchical Representation LearningOpen-Set RecognitionHyperbolic Representation Learning
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Apr 30, 2026

A Unified Framework of Hyperbolic Graph Representation Learning Methods

Hyperbolic geometry has emerged as an effective latent space for representing complex networks, owing to its ability to capture hierarchical organization and heterogeneous connectivity patterns using low-dimensional embeddings. As a result, numerous hyperbolic graph representation learning methods have been proposed in recent years. However, their practical adoption and systematic comparison remain challenging, as implementations are fragmented and shared tools for reproducible and fair evaluation are lacking. In this work, we introduce a unified open-source framework for hyperbolic graph representation learning that integrates several widely used embedding methods under a common optimization interface. The novel framework enables consistent training, visualization, and evaluation of hyperbolic embeddings, and interfaces seamlessly with standard network analysis tools. Leveraging this unified setup, we conduct an experimental study of hyperbolic embedding methods on real-world networks, focusing on two canonical downstream tasks: link prediction and node classification. Beyond predictive accuracy, the study offers practical insights into the strengths and limitations of existing approaches, thereby facilitating informed method selection and fostering reproducible research in hyperbolic graph representation learning.

Sofía Pérez Casulo, Marcelo Fiori, Bernardo Marenco +1

Facultad de Ingenier´ıa, Universidad de la Rep´ublica, Uruguay · Centro Interdisciplinario en Ciencia de Datos y Aprendizaje Autom´atico (CICADA), Universidad de la Rep´ublica, Uruguay

Graph Representation LearningGraph EmbeddingHyperbolic Representation Learning
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Apr 26, 2026

HAC: Parameter-Efficient Hyperbolic Adaptation of CLIP for Zero-Shot VQA

Recent advances in representation learning have shown that hyperbolic geometry can offer a more expressive alternative to the Euclidean embeddings used in CLIP models, capturing hierarchical structures and leading to better-organized representations. However, current hyperbolic CLIP variants are trained entirely from scratch, which is computationally expensive and resource-intensive. In this work, we propose HAC (Hyperbolic Adaptation of CLIP), a parameter-efficient framework that enables pretrained CLIP models to transition into hyperbolic space via lightweight fine-tuning. We apply HAC to Visual Question Answering (VQA), where models must interpret visual elements and align them with textual queries. Notably, HAC's training is performed on a dataset with no overlap with any VQA benchmark, resulting in a strict zero-shot evaluation paradigm that underscores HAC's task-agnostic adaptability. We evaluate HAC across a diverse suite of VQA benchmarks spanning General, Reasoning, and OCR categories. Both HAC-S (small) and HAC-B (medium) consistently surpass Euclidean baselines and prior hyperbolic approaches, with HAC-B delivering up to a +1.9 point average improvement over CLIP-B on reasoning-intensive tasks. Our code is available at https://github.com/fdibiton/HAC

Francesco Dibitonto, Cigdem Beyan, Vittorio Murino

Department of Computer Science, University of Verona, Verona, Italy · EVS - Embedded Vision Systems Srl, Verona, Italy · AI for Good (AIGO), Istituto Italiano di Tecnologia, Genova, Italy

Visual Question AnsweringZero-Shot LearningVLM AdaptationHyperbolic Representation LearningParameter-Efficient Fine-Tuning
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Apr 22, 2026

HypEHR: Hyperbolic Modeling of Electronic Health Records for Efficient Question Answering

Electronic health record (EHR) question answering is often handled by LLM-based pipelines that are costly to deploy and do not explicitly leverage the hierarchical structure of clinical data. Motivated by evidence that medical ontologies and patient trajectories exhibit hyperbolic geometry, we propose HypEHR, a compact Lorentzian model that embeds codes, visits, and questions in hyperbolic space and answers queries via geometry-consistent cross-attention with type-specific pointer heads. HypEHR is pretrained with next-visit diagnosis prediction and hierarchy-aware regularization to align representations with the ICD ontology. On two MIMIC-IV-based EHR-QA benchmarks, HypEHR approaches LLM-based methods while using far fewer parameters. Our code is publicly available at https://github.com/yuyuliu11037/HypEHR.

Yuyu Liu, Sarang Rajendra Patil, Mengjia Xu +1

Department of Computer Science, Stony Brook University · Department of Data Science, New Jersey Institute of Technology · Department of Biomedical Informatics, Stony Brook University

HealthcareHierarchical Representation LearningClinical QAMedical QAElectronic Health RecordsEHR QA+1
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Apr 19, 2026

HSG: Hyperbolic Scene Graph

Scene graph representations enable structured visual understanding by modeling objects and their relationships, and have been widely used for multiview and 3D scene reasoning. Existing methods such as MSG learn scene graph embeddings in Euclidean space using contrastive learning and attention based association. However, Euclidean geometry does not explicitly capture hierarchical entailment relationships between places and objects, limiting the structural consistency of learned representations. To address this, we propose Hyperbolic Scene Graph (HSG), which learns scene graph embeddings in hyperbolic space where hierarchical relationships are naturally encoded through geometric distance. Our results show that HSG improves hierarchical structure quality while maintaining strong retrieval performance. The largest gains are observed in graph level metrics: HSG achieves a PP IoU of 33.17 and the highest Graph IoU of 33.51, outperforming the best AoMSG variant (25.37) by 8.14, highlighting the effectiveness of hyperbolic representation learning for scene graph modeling. Code: https://github.com/AIGeeksGroup/HSG.

Liyang Wang, Zeyu Zhang, Hao Tang

School of Computer Science, Peking University

Graph Representation LearningHierarchical Representation LearningGraph-Based Retrieval3D Scene RepresentationHyperbolic Representation Learning
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Apr 19, 2026

Modeling Multi-Dimensional Cognitive States in Large Language Models under Cognitive Crowding

Modeling human cognitive states is essential for advanced artificial intelligence. Existing Large Language Models (LLMs) mainly address isolated tasks such as emotion analysis or stance detection, and fail to capture interactions among cognitive dimensions defined in psychology, including emotion, thinking style, stance, and intention. To bridge this gap, we construct CognitiveBench, the first benchmark with unified annotations across the above four dimensions. Experiments on CognitiveBench show that although LLMs perform well on single dimension tasks, their performance drops sharply in joint multi-dimensional modeling. Using Gromov δδ-hyperbolicity analysis, we find that CognitiveBench exhibits a strong hierarchical structure. We attribute the performance bottleneck to ``Cognitive Crowding'', where hierarchical cognitive states require exponential representational space, while the Euclidean space of LLMs grows only polynomially, causing representation overlap and degraded performance. To address this mismatch, we propose HyCoLLM, which models cognitive states in hyperbolic space and aligns LLM representations via Hyperbolic Guided Alignment Tuning. Results show that HyCoLLM substantially improves multi-dimensional cognitive understanding, allowing 8B parameter model to outperform strong baselines, including GPT-4o.

Lin Zhong, Siyu Zhu, Zizhen Yuan +5

Harbin Institute of Technology, Shenzhen, China · City University of Macau, Macao SAR, China · Peng Cheng Laboratory, Shenzhen, China

LLM AlignmentCognitive ModelingHyperbolic Representation Learning
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Apr 18, 2026

Hyperbolic Enhanced Representation Learning for Incomplete Multi-view Clustering

Incomplete Multi-View Clustering (IMVC) faces the challenge of learning discriminative representations from fragmentary observations while maintaining robustness against missing views. However, prevalent Euclidean-based methods suffer from a geometric mismatch when modeling real-world data with intrinsic hierarchies, leading to semantic blurring where representations drift towards spatially proximal but semantically distinct neighbors. To bridge this gap, we propose HERL, a Hyperbolic Enhanced Representation Learning framework for IMVC. Operating within the Poincaré ball, HERL constructs a structure-aware latent space to enhance representation learning. Specifically, we design a dual-constraint hyperbolic contrastive mechanism optimizing: an angular-based loss to preserve semantic identity via directional alignment, and a distance-based loss to enforce hierarchical compactness. Furthermore, a hyperbolic prototype head is introduced to rectify global structural drift by aligning cross-view hierarchy-aware prototype distributions. Consequently, HERL disentangles fine-grained semantic correlations to sharpen cluster boundaries and imposes geometric constraints to rectify the data recovery process. Extensive experimental results demonstrate that HERL consistently outperforms state-of-the-art approaches.

Tianyi Chen, Haobo Wang, Kai Tang +5

School of Software Technology, Zhejiang University, Zhejiang 310027, China · College of Computer Science and Technology, Zhejiang University, Zhejiang 310027, China · Beijing Key Laboratory of Traffic Data Analysis and Mining, Beijing Jiaotong University, Beijing 100044, China +2

Contrastive LearningHierarchical Representation LearningMulti-View LearningMulti-View ClusteringHyperbolic Representation Learning
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Apr 18, 2026

Lorentz Framework for Semantic Segmentation

Semantic segmentation in hyperbolic space enables compact modeling of hierarchical structure while providing inherent uncertainty quantification. Prior approaches predominantly rely on the Poincaré ball model, which suffers from numerical instability, optimization, and computational challenges. We propose a novel, tractable, architecture-agnostic semantic segmentation framework (pixel-wise and mask classification) in the hyperbolic Lorentz model. We employ text embeddings with semantic and visual cues to guide hierarchical pixel-level representations in Lorentz space. This enables stable and efficient optimization without requiring a Riemannian optimizer, and easily integrates with existing Euclidean architectures. Beyond segmentation, our approach yields free uncertainty estimation, confidence map, boundary delineation, hierarchical and text-based retrieval, and zero-shot performance, reaching generalized flatter minima. We introduce a novel uncertainty and confidence indicator in Lorentz cone embeddings. Further, we provide analytical and empirical insights into Lorentz optimization via gradient analysis. Extensive experiments on ADE20K, COCO-Stuff-164k, Pascal-VOC, and Cityscapes, utilizing state-of-the-art per-pixel classification models (DeepLabV3 and SegFormer) and mask classification models (mask2former and maskformer), validate the effectiveness and generality of our approach. Our results demonstrate the potential of hyperbolic Lorentz embeddings for robust and uncertainty-aware semantic segmentation. Code is available at https://github.com/mxahan/Lorentz_semantic_segmentation.

Zahid Hasan, Masud Ahmed, Nirmalya Roy

Department of Information Systems, University of Maryland, Baltimore County

Segmentation Uncertainty EstimationSemantic SegmentationHyperbolic Representation Learning
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Apr 14, 2026

EEG-Based Multimodal Learning via Hyperbolic Mixture-of-Curvature Experts

Electroencephalography (EEG)-based multimodal learning integrates brain signals with complementary modalities to improve mental state assessment, providing great clinical potential. The effectiveness of such paradigms largely depends on the representation learning on heterogeneous modalities. For EEG-based paradigms, one promising approach is to leverage their hierarchical structures, as recent studies have shown that both EEG and associated modalities (e.g., facial expressions) exhibit hierarchical structures reflecting complex cognitive processes. However, Euclidean embeddings struggle to represent these hierarchical structures due to their flat geometry, while hyperbolic spaces, with their exponential growth property, are naturally suited for them. In this work, we propose EEG-MoCE, a novel hyperbolic mixture-of-curvature experts framework designed for multimodal neurotechnology. EEG-MoCE assigns each modality to an expert in a learnable-curvature hyperbolic space, enabling adaptive modeling of its intrinsic geometry. A curvature-aware fusion strategy then dynamically weights experts, emphasizing modalities with richer hierarchical information. Extensive experiments on benchmark datasets demonstrate that EEG-MoCE achieves state-of-the-art performance, including emotion recognition, sleep staging, and cognitive assessment. Code is available at https://github.com/zhourunhe/EEG-MoCE.

Runhe Zhou, Shanglin Li, Guanxiang Huang +5

Nanyang Technological University, Singapore · BIFOLD, Berlin Institute for the Foundations of Learning and Data, Berlin, Germany · ATR, Kyoto, Japan +2

Multimodal LearningMultimodal FusionMixture-of-Experts ModelsHyperbolic Representation Learning
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Apr 8, 2026

Busemann energy-based attention for emotion analysis in Poincaré discs

We present EmBolic - a novel fully hyperbolic deep learning architecture for fine-grained emotion analysis from textual messages. The underlying idea is that hyperbolic geometry efficiently captures hierarchies between both words and emotions. In our context, these hierarchical relationships arise from semantic ambiguities. EmBolic aims to infer the curvature on the continuous space of emotions, rather than treating them as a categorical set without any metric structure. In the heart of our architecture is the attention mechanism in the hyperbolic disc. The model is trained to generate queries (points in the hyperbolic disc) from textual messages, while keys (points at the boundary) emerge automatically from the generated queries. Predictions are based on the Busemann energy between queries and keys, evaluating how well a certain textual message aligns with the class directions representing emotions. Our experiments demonstrate strong generalization properties and reasonably good prediction accuracy even for small dimensions of the representation space. Overall, this study supports our claim that affective computing is one of the application domains where hyperbolic representations are particularly advantageous.

Zinaid Kapić, Vladimir Jaćimović

Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, bb., 81000 Podgorica, Montenegro

Emotion RecognitionAffective ComputingAttention MechanismsHyperbolic Representation Learning
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Apr 5, 2026

Hierarchical Co-Embedding of Font Shapes and Impression Tags

Font shapes can evoke a wide range of impressions, but the correspondence between fonts and impression descriptions is not one-to-one: some impressions are broadly compatible with diverse styles, whereas others strongly constrain the set of plausible fonts. We refer to this graded constraint strength as style specificity. In this paper, we propose a hyperbolic co-embedding framework that models font--impression correspondence through entailment rather than simple paired alignment. Font images and impression descriptions, represented as single tags or tag sets, are embedded in a shared hyperbolic space with two complementary entailment constraints: impression-to-font entailment and low-to-high style-specificity entailment among impressions. This formulation induces a radial structure in which low style-specificity impressions lie near the origin and high style-specificity impressions lie farther away, yielding an interpretable geometric measure of how strongly an impression constrains font style. Experiments on the MyFonts dataset demonstrate improved bidirectional retrieval over strong one-to-one baselines. In addition, traversal and tag-level analyses show that the learned space captures a coherent progression from ambiguous to more style-specific impressions and provides a meaningful, data-driven quantification of style specificity.

Yugo Kubota, Kaito Shiku, Seiichi Uchida

Kyushu University, Fukuoka, Japan

Multimodal EmbeddingVision-Language RetrievalHierarchical Representation LearningHyperbolic Representation Learning
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Mar 30, 2026

LA-Sign: Looped Transformers with Geometry-aware Alignment for Skeleton-based Sign Language Recognition

Skeleton-based isolated sign language recognition (ISLR) demands fine-grained understanding of articulated motion across multiple spatial scales, from subtle finger movements to global body dynamics. Existing approaches typically rely on deep feed-forward architectures, which increase model capacity but lack mechanisms for recurrent refinement and structured representation. We propose LA-Sign, a looped transformer framework with geometry-aware alignment for ISLR. Instead of stacking deeper layers, LA-Sign derives its depth from recurrence, repeatedly revisiting latent representations to progressively refine motion understanding under shared parameters. To further regularise this refinement process, we present a geometry-aware contrastive objective that projects skeletal and textual features into an adaptive hyperbolic space, encouraging multi-scale semantic organisation. We study three looping designs and multiple geometric manifolds, demonstrating that encoder-decoder looping combined with adaptive Poincare alignment yields the strongest performance. Extensive experiments on WLASL and MSASL benchmarks show that LA-Sign achieves state-of-the-art results while using fewer unique layers, highlighting the effectiveness of recurrent latent refinement and geometry-aware representation learning for sign language recognition.

Muxin Pu, Mei Kuan Lim, Chun Yong Chong +1

School of Information Technology, Monash University · S-Lab, Nanyang Technological University

Sign Language RecognitionRecurrent TransformersGeometric Representation LearningHyperbolic Representation Learning
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Mar 24, 2026

ARGENT: Adaptive Hierarchical Image-Text Representations

Large-scale Vision-Language Models (VLMs) such as CLIP learn powerful semantic representations but operate in Euclidean space, which fails to capture the inherent hierarchical structure of visual and linguistic concepts. Hyperbolic geometry, with its exponential volume growth, offers a principled alternative for embedding such hierarchies with low distortion. However, existing hyperbolic VLMs use entailment losses that are unstable: as parent embeddings contract toward the origin, their entailment cones widen toward a half-space, causing catastrophic cone collapse that destroys the intended hierarchy. Additionally, hierarchical evaluation of these models remains unreliable, being largely retrieval-based and correlation-based metrics and prone to taxonomy dependence and ambiguous negatives. To address these limitations, we propose an adaptive entailment loss paired with a norm regularizer that prevents cone collapse without heuristic aperture clipping. We further introduce an angle-based probabilistic entailment protocol (PEP) for evaluating hierarchical understanding, scored with AUC-ROC and Average Precision. This paper introduces a stronger hyperbolic VLM baseline ARGENT, Adaptive hieRarchical imaGe-tExt represeNTation. ARGENT improves the SOTA hyperbolic VLM by 0.7, 1.1, and 0.8 absolute points on image classification, text-to-image retrieval, and proposed hierarchical metrics, respectively.

Chuong Huynh, Hossein Souri, Abhinav Kumar +3

University of Maryland, College Park · Samsung Research America, AI Center – Mountain View

Vision-Language ModelsHierarchical Representation LearningHyperbolic Representation LearningVLM Evaluation
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Mar 24, 2026

HyFI: Hyperbolic Feature Interpolation for Brain-Vision Alignment

Recent progress in artificial intelligence has encouraged numerous attempts to understand and decode human visual system from brain signals. These prior works typically align neural activity independently with semantic and perceptual features extracted from images using pre-trained vision models. However, they fail to account for two key challenges: (1) the modality gap arising from the natural difference in the information level of representation between brain signals and images, and (2) the fact that semantic and perceptual features are highly entangled within neural activity. To address these issues, we utilize hyperbolic space, which is well-suited for considering differences in the amount of information and has the geometric property that geodesics between two points naturally bend toward the origin, where the representational capacity is lower. Leveraging these properties, we propose a novel framework, Hyperbolic Feature Interpolation (HyFI), which interpolates between semantic and perceptual visual features along hyperbolic geodesics. This enables both the fusion and compression of perceptual and semantic information, effectively reflecting the limited expressiveness of brain signals and the entangled nature of these features. As a result, it facilitates better alignment between brain and visual features. We demonstrate that HyFI achieves state-of-the-art performance in zero-shot brain-to-image retrieval, outperforming prior methods with Top-1 accuracy improvements of up to +17.3% on THINGS-EEG and +9.1% on THINGS-MEG.

Sangmin Jo, Wootaek Jeong, Da-Woon Heo +2

Department of Artificial Intelligence, Korea University, Seoul, South Korea

Cross-Modal AlignmentEEG-Based Visual DecodingCross-Modal RetrievalHyperbolic Representation Learning
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Mar 23, 2026

Uncertainty-guided Compositional Alignment with Part-to-Whole Semantic Representativeness in Hyperbolic Vision-Language Models

While Vision-Language Models (VLMs) have achieved remarkable performance, their Euclidean embeddings remain limited in capturing hierarchical relationships such as part-to-whole or parent-child structures, and often face challenges in multi-object compositional scenarios. Hyperbolic VLMs mitigate this issue by better preserving hierarchical structures and modeling part-whole relations (i.e., whole scene and its part images) through entailment. However, existing approaches do not model that each part has a different level of semantic representativeness to the whole. We propose UNcertainty-guided Compositional Hyperbolic Alignment (UNCHA) for enhancing hyperbolic VLMs. UNCHA models part-to-whole semantic representativeness with hyperbolic uncertainty, by assigning lower uncertainty to more representative parts and higher uncertainty to less representative ones for the whole scene. This representativeness is then incorporated into the contrastive objective with uncertainty-guided weights. Finally, the uncertainty is further calibrated with an entailment loss regularized by entropy-based term. With the proposed losses, UNCHA learns hyperbolic embeddings with more accurate part-whole ordering, capturing the underlying compositional structure in an image and improving its understanding of complex multi-object scenes. UNCHA achieves state-of-the-art performance on zero-shot classification, retrieval, and multi-label classification benchmarks. Our code and models are available at: https://github.com/jeeit17/UNCHA.git.

Hayeon Kim, Ji Ha Jang, Junghun James Kim +1

Dept. of Electrical and Computer Engineering, Seoul National University, Republic of Korea · INMC & IPAI, Seoul National University, Republic of Korea

Vision-Language ModelsHierarchical Representation LearningVision-Language AlignmentHyperbolic Representation Learning
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Mar 17, 2026

A Depth-Aware Comparative Study of Euclidean and Hyperbolic Graph Neural Networks on Bitcoin Transaction Systems

Bitcoin transaction networks are large scale socio- technical systems in which activities are represented through multi-hop interaction patterns. Graph Neural Networks(GNNs) have become a widely adopted tool for analyzing such systems, supporting tasks such as entity detection and transaction classification. Large-scale datasets like Elliptic have allowed for a rise in the analysis of these systems and in tasks such as fraud detection. In these settings, the amount of transactional context available to each node is determined by the neighborhood aggregation and sampling strategies, yet the interaction between these receptive fields and embedding geometry has received limited attention. In this work, we conduct a controlled comparison of Euclidean and tangent-space hyperbolic GNNs for node classification on a large Bitcoin transaction graph. By explicitly varying the neighborhood while keeping the model architecture and dimensionality fixed, we analyze the differences in two embedding spaces. We further examine optimization behavior and observe that joint selection of learning rate and curvature plays a critical role in stabilizing high-dimensional hyperbolic embeddings. Overall, our findings provide practical insights into the role of embedding geometry and neighborhood depth when modeling large-scale transaction networks, informing the deployment of hyperbolic GNNs for computational social systems.

Ankit Ghimire, Saydul Akbar Murad, Nick Rahimi

School of Computing Sciences and Computer Engineering University of Southern Mississippi Hattiesburg, MS, 39406, USA

Graph Neural NetworksHyperbolic Representation LearningNode Classification
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Mar 14, 2026

A Hyperbolic Perspective on Hierarchical Structure in Object-Centric Scene Representations

Slot attention has emerged as a powerful framework for unsupervised object-centric learning, decomposing visual scenes into a small set of compact vector representations called slots, each capturing a distinct region or object. However, these slots are learned in Euclidean space, which provides no geometric inductive bias for the hierarchical relationships that naturally structure visual scenes. In this work, we propose a simple post-hoc pipeline to project Euclidean slot embeddings onto the Lorentz hyperboloid of hyperbolic space, without modifying the underlying training pipeline. We construct five-level visual hierarchies directly from slot attention masks and analyse whether hyperbolic geometry reveals latent hierarchical structure that remains invisible in Euclidean space. Integrating our pipeline with SPOT (images), VideoSAUR (video), and SlotContrast (video), We find that hyperbolic projection exposes a consistent scene-level to object-level organisation, where coarse slots occupy greater manifold depth than fine slots, which is absent in Euclidean space. We further identify a "curvature--task tradeoff": low curvature (c=0.2c{=}0.2) matches or outperforms Euclidean on parent slot retrieval, while moderate curvature (c=0.5c{=}0.5) achieves better inter-level separation. Together, these findings suggest that slot representations already encode latent hierarchy that hyperbolic geometry reveals, motivating end-to-end hyperbolic training as a natural next step. Code and models are available at github.com/NeeluMadan/HHS.

Neelu Madan, Àlex Pujol, Andreas Møgelmose +4

Aalborg University & Pioneer Centre for AI, Denmark · Universitat de Barcelona & Computer Vision Center, Spain · Milestone Systems, Denmark +1

Hierarchical Representation LearningObject-Centric Representation LearningHyperbolic Representation LearningSlot Attention
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Mar 5, 2026

Hyperbolic Multiview Pretraining for Robotic Manipulation

3D-aware visual pretraining has proven effective in improving the performance of downstream robotic manipulation tasks. However, existing methods are constrained to Euclidean embedding spaces, whose flat geometry limits their ability to model structural relations among embeddings. As a result, they struggle to learn structured embeddings that are essential for robust spatial perception in robotic applications. To this end, we propose HyperMVP, a self-supervised framework for Hyperbolic MultiView Pretraining. Hyperbolic space offers geometric properties well suited for capturing structural relations. Methodologically, we extend the masked autoencoder paradigm and design a GeoLink encoder to learn multiview hyperbolic representations. The pretrained encoder is then finetuned with visuomotor policies on manipulation tasks. In addition, we introduce 3D-MOV, a large-scale dataset comprising multiple types of 3D point clouds to support pretraining. We evaluate HyperMVP on COLOSSEUM, RLBench, and real-world scenarios, where it consistently outperforms strong baselines across diverse tasks and perturbation settings. Our results highlight the potential of 3D-aware pretraining in a non-Euclidean space for learning robust and generalizable robotic manipulation policies.

Jin Yang, Ping Wei, Yixin Chen +1

National Key Laboratory of Human-Machine Hybrid Augmented Intelligence, Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University

Robotic Manipulation3D Representation LearningHyperbolic Representation LearningMulti-View Learning
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Feb 27, 2026

Intrinsic Lorentz Neural Network

Real-world data frequently exhibit latent hierarchical structures, which can be naturally represented by hyperbolic geometry. Although recent hyperbolic neural networks have demonstrated promising results, many existing architectures remain partially intrinsic, mixing Euclidean operations with hyperbolic ones or relying on extrinsic parameterizations. To address it, we propose the Intrinsic Lorentz Neural Network (ILNN), a fully intrinsic hyperbolic architecture that conducts all computations within the Lorentz model. At its core, the network introduces a novel point-to-hyperplane fully connected layer (FC), replacing traditional Euclidean affine logits with closed-form hyperbolic distances from features to learned Lorentz hyperplanes, thereby ensuring that the resulting geometric decision functions respect the inherent curvature. Around this fundamental layer, we design intrinsic modules: GyroLBN, a Lorentz batch normalization that couples gyro-centering with gyro-scaling, consistently outperforming both LBN and GyroBN while reducing training time. We additionally proposed a gyro-additive bias for the FC output, a Lorentz patch-concatenation operator that aligns the expected log-radius across feature blocks via a digamma-based scale, and a Lorentz dropout layer. Extensive experiments conducted on CIFAR-10/100 and two genomic benchmarks (TEB and GUE) illustrate that ILNN achieves state-of-the-art performance and computational cost among hyperbolic models and consistently surpasses strong Euclidean baselines. The code is available at magenta{this url}.

Xianglong Shi, Ziheng Chen, Yunhan Jiang +1

University of Science and Technology of China · University of Trento · Peking University

Hyperbolic Representation LearningImage Classification
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