Hyperbolic Representation Learning

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17 papers in the last four weeks, up 467% on the four weeks before. 0.2% of all new papers.

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Latest papers 60

Oct 5, 2026cs.LG

The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective

We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space

\mathbb{H}^n

. Our framework accommodates general permutation invariant relative margin (PERM) losses, a class that includes cross entropy and other standard multiclass losses. Our analysis is based on a decomposition: at large radius, the distance to each prototype splits into a radial term and a direction-dependent term described by the Busemann function. This yields two main results. First, we prove a radial dichotomy: the sign of a drift coefficient

μ

determines whether the radius is pushed toward the ideal boundary or back toward the interior; if the positive drift persists, then

r(t)=\frac{1}{2}\log t+O(1)

, while persistent negative drift returns the trajectory to the large-radius threshold in finite time. Second, we show that the boundary direction converges to a critical point of the Busemann risk on

\partial\mathbb{H}^n

. These results provide a rigorous asymptotic perspective on two phenomena we refer to as boundary saturation and near-boundary clustering in hyperbolic representation learning.

Xingrun Li, Sho Kuno, Yusuke Mukuta +2

The University of Tokyo · RIKEN

Riemannian Optimization Implicit Bias Multiclass Classification Hyperbolic Representation Learning

Oct 5, 2026cs.LG

Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another

Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from

-1

to

0

, while the embedding remains at curvature

-1

. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.

Federico Larroca, Paola Bermolen, Marcelo Fiori +1

Facultad de Ingenier´ıa Universidad de la Rep´ublica Uruguay

Riemannian Optimization Hierarchical Representation Learning Graph Embedding Hyperbolic Representation Learning

Oct 5, 2026cs.LG

On the Geometry of Multimodal Saturation: Riemannian VICReg

In self-supervised learning, a third modality should improve, or at least preserve, performance. Across nine image-text-tabular datasets, we show that it instead harms performance: the trimodal model underperforms its own best bimodal subset in 55.6% of paired runs under VICReg. The same failure occurs in 51.1% of paired runs under SimSiam. We call this failure multimodal saturation. We propose that the failure lies in the alignment geometry. Riemannian VICReg (R-VICReg) generalizes classical VICReg: it aligns views by squared geodesic distance on learnable negative-curvature product factors and recovers VICReg exactly as curvature vanishes. Over the same 45 paired runs, R-VICReg raises the probability that the third modality helps from 44.4% to 64.4%, with gains concentrated where VICReg saturates.

Nessim Ben Abbes, Duc Han Le, Sabri Mtibaa +1

Protectline · Télécom Paris

Self-Supervised Learning Geometric Representation Learning Hyperbolic Representation Learning Multimodal Learning

Sep 30, 2026cs.LG

Curvature Under Attack in hZACH-ViT: Gauge Symmetry, Boundary Saturation, and Adversarial Failure

Curvature is often treated as an intrinsic property of a representation, although its empirical effect also depends on coordinate scale, learned logit temperature, and numerical safeguards. We study this interaction in hZACH-ViT, a compact Vision Transformer with Euclidean, Poincare, and spherical prototype heads. The backbone architecture, seed-specific initialization, 50-per-class training subset, and optimization protocol are matched across three MedMNIST datasets and five seeds. At the fixed comparison curvature

c=1

, Poincare has the lowest class-macro PGD attack-success rate in all 12 dataset-budget cells and under a stronger CE+DLR multi-restart attack on all three datasets, but it also has the lowest clean MacroF1. An end-to-end curvature intervention changes the interpretation. Reducing Poincare curvature to

c=0.1

improves clean MacroF1 in every one of the 15 paired seed-dataset comparisons and removes hard boundary clipping, yet on OrganAMNIST it increases strong attack success from

89.7\%

to

99.3\%

(paired difference

+9.57

points; 95% hierarchical bootstrap CI

[+5.52,+14.03]

). At

c=1

,

40

-

47\%

of clean Poincare features are hard-clipped, the radial Jacobian of the inherited map is nearly zero, and dimensionless attack trajectories are unusually long and inefficient. The spherical head provides a control: its curvature change is an exact scale gauge to floating-point precision and produces much smaller attack differences. These results do not establish intrinsic hyperbolic robustness. They identify an implementation-sensitive regime in which curvature, scale, and proximity to the Poincare boundary jointly organize clean recognition and adversarial representation motion.

Athanasios Angelakis, Marta Gomez-Barrero

BioML, Research Institute CODE, University of the Bundeswehr Munich, Munich, Germany · Amsterdam UMC, University of Amsterdam, Amsterdam, Netherlands

Vision Transformer Adversarial Robustness Hyperbolic Representation Learning

Sep 29, 2026cs.CV

Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (

\mathbb{T}^K

). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces (

\mathbb{H}^m

) circumvent this via constant negative curvature (

K<0

) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce \textbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (

\mathbb{R}^{1,m}

). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish \textbf{MAN-2D} (

\mathbb{R}^{1,1} \to \mathbb{H}^1

) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into

D/2

independent two-dimensional Minkowski blocks. We further formulate \textbf{MAN-4D} (

\mathbb{R}^{1,3} \to \mathbb{H}^3

) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of

\mathrm{SO}^+(1,3)

to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.

Zhongping Ji

Visual Representation Learning Hierarchical Representation Learning Geometric Representation Learning Hyperbolic Representation Learning

Sep 29, 2026cs.CV

CurvSpec: Adaptive Multi-Curvature Learning for Partial Relevant Video Retrieval

Partially Relevant Video Retrieval (PRVR) seeks to retrieve untrim-med videos containing a moment that matches a text query, without temporal annotations. The relevant moment may last only seconds within a video spanning several minutes, creating an extremely low signal-to-noise ratio that makes PRVR more challenging than standard full-video retrieval. This task presents two intertwined challenges: (1) signal dilution, where coarse global representations blur the brief relevant signal into the dominant irrelevant surroundings;(2) curvature rigidity, where embedding all videos in the same fixed-geometry space distorts representations for videos that range from flat atomic events to deep compositional hierarchies. Existing PRVR methods have improved moment selection and cross-modal matching, but they still typically encode all videos in a single fixed-curvature retrieval space, limiting their ability to model diverse video structures. To address both challenges, we propose CurvSpec, a framework that learns content-adaptive curvature for video retrieval representations rather than imposing a fixed geometric prior. CurvSpec processes features through parallel Euclidean and hyperbolic attention layers, with independently learned curvatures assigned to the hyperbolic layers, and a content-aware fusion mechanism routes each input to its most suitable geometric regime. To further suppress signal dilution, CurvSpec represents each video with semantic centroids whose number is determined by the video's content complexity, projects them onto the learned manifold, and matches each query against its nearest centroid by geodesic distance. Experiments on ActivityNet Captions, TVR, and Charades-STA demonstrate state-of-the-art retrieval performance.

Zhen Liu, Letian Li, Jinpeng Wang +4

SIGS, Tsinghua University Shenzhen, China · Harbin Institute of Technology, Shenzhen Shenzhen, China

Cross-Modal Retrieval Video Representation Learning Information Retrieval Hyperbolic Representation Learning

Sep 28, 2026cs.LG

Explaining Hyperbolic Neural Networks via Geometry-Aware Relevance Propagation

Hyperbolic neural networks introduce geometric operations that require explicit treatment in relevance propagation. Equivalent geometric realizations can produce different feature attributions, even when local relevance is conserved. We study this problem through Geometric Representation Invariance (GRI), a specialization of Implementation Invariance, and zero-curvature consistency, which requires identity relevance propagation when a geometric module approaches the identity. We propose LRP-radial-all for origin-centered radial modules, treating geometric scaling as modulation and assigning relevance entirely to the signal branch. The rule conserves relevance, is invariant to equivalent radial factorizations, and satisfies zero-curvature consistency, yielding GRI for a specified Poincaré-Lorentz logarithmic-map construction. In contrast, a conservative LRP-half baseline can violate both consistency criteria. Experiments on hyperbolic MNIST, sEEG, and CIFAR-10 classifiers assess attribution fidelity, qualitative explanations, and runtime. LRP-radial-all achieves competitive attribution fidelity across datasets with runtime comparable to Gradient

\times

Input and substantially lower than Integrated Gradients. These findings motivate geometry-aware propagation rules that distinguish relevance conservation from consistency across equivalent computations.

Ping Xiong, Shanglin Li, Yi Ding +2

BIFOLD, Germany · Technische Universität Berlin, Germany · Nanyang Technological University, Singapore +3

Neural Network Interpretability Layer-Wise Relevance Propagation Hyperbolic Representation Learning

Sep 24, 2026cs.CV

Hyperbolic Multimodal Continual Learning: A Closest-Admissible Solution

Existing continual-learning methods protect parameters, replayed examples, or Euclidean feature subspaces. When applied to hyperbolic multimodal models, they do not explicitly preserve the Lorentz geometry that jointly encodes within-modality similarity, cross-modal correspondence, and semantic hierarchy; sequential updates can therefore retain task scores while still distorting previously learned relations. We address this gap with Hyperbolic Multimodal Continual Learning (HMCL). We show that preserving the old multimodal geometry amounts to restricting all modalities to one shared hyperbolic isometry, which induces a family of admissible first-order parameter changes. We formulate a joint closest-admissible (CA) correction that retains the shared rotation best matching the candidate modal updates; its minimal-rotation (MR) special case fixes this rotation to zero. Both variants correct the displacement realized by AdamW, and task anchoring bounds within-task accumulation while preserving learning freedom. Across a unified 16-task classification-retrieval stream with three hyperbolic backbones, HMCL improves final performance and backward transfer over sequential fine-tuning and four continual-learning baselines; HMCL-CA gives the highest Overall score on every backbone. A modality-extended stream confirms the retrieval gains. Representation analyses find 81.2 to 95.5 percent less radial, angular, cross-modal, and paired-distance drift; ImageNet-WordNet results show better semantic ancestry and radial hierarchy.

Jiahong Liu, Ming Shen, Xiaohao Liu +4

The Chinese University of Hong Kong · National University of Singapore · Yale University +1

Cross-Modal Alignment Hyperbolic Representation Learning Multimodal Learning

Sep 23, 2026cs.LG

hyperbolix: Hyperbolic Deep Learning in JAX

We present hyperbolix, an open-source library for hyperbolic deep learning in JAX, built on Flax NNX. To our knowledge, it is the first comprehensive, general-purpose hyperbolic deep learning library in JAX. It includes six manifolds with a common interface: Euclidean space, the Poincaré ball, the hyperboloid, the

κ

-stereographic model, mixed-curvature product spaces, and the proper velocity space. We implement layer families that cover linear layers, convolutions, attention, normalization, positional encoding, regression, and vector quantization. These building blocks span methods ranging from Ganea's original hyperbolic neural networks to recent fully hyperbolic architectures such as Hypformer and Lorentzian ResNet. Additionally, hyperbolix contains Riemannian optimizers implemented as optax transformations, wrapped distributions, and hyperbolic dimensionality-reduction techniques. Its API uses idiomatic JAX: Manifolds are stateless, with curvature being passed at call time, while manifold operations act on single points, with jax.vmap enabling batch operations. The precision of every checked operation is tested against a closed-form NumPy/SciPy transcription from the source paper or a finite difference, for both float32 and float64. On the hyperboloid, standard formulas for two-point operations, such as the distance, lose precision far from the origin, because they subtract two large, nearly equal terms. hyperbolix replaces these subtractions with cancellation-free formulas that stay accurate in float32 at distances where prior implementations return NaN. hyperbolix is available under the MIT license at https://github.com/timoklein/hyperbolix .

Timo Klein, Thomas Lang, Yllka Velaj +1

Faculty of Computer Science, University of Vienna, Vienna, Austria · UniVie Doctoral School Computer Science, University of Vienna, Vienna, Austria · ds:UniVie, University of Vienna, Vienna, Austria

Hyperbolic Representation Learning

Sep 22, 2026cs.CV

A Hierarchy-Aware Video-Language Model Evaluation and Hyperbolic Baseline for Surgery

Surgical procedures follow a phase-to-step hierarchy, yet the video-language models used to recognize them are evaluated with flat per-level metrics that ignore cross-level coherence and error structure. In this paper we make two contributions to address this problem, (i) we introduce SurgHiBench, the first hierarchy-aware evaluation suite for surgical video understanding, with three tasks measuring recognition, consistency, and severity across granularity levels. We evaluate a general-purpose CLIP model, a Euclidean surgical model, and, as second contribution: (ii) HyperSurg, a new hyperbolic model that enforces phase-step containment via entailment cones, across four (existing) datasets spanning three procedure types. The suite reveals that two models with the same accuracy can produce predictions of very different error severity, ranging from sibling confusions within the correct phase to unrelated cross-phase predictions. Hyperbolic geometry shifts predictions toward the correct procedural neighborhood, and these gains scale with the tree-likeness of each dataset's annotation hierarchy, providing a principled indicator when hierarchy-aware geometry helps.

Ana Manzano Rodríguez, Pascal Mettes, Marlies P. Schijven +1

Data Science Center HAVA-Lab, University of Amsterdam · Video & Image Sense Lab, Informatics Institute, University of Amsterdam · Amsterdam UMC Location University of Amsterdam, Surgery +2

Video-Language Model Evaluation Hierarchical Representation Learning Surgical Video Understanding Hyperbolic Representation Learning Temporal Video Understanding

Sep 22, 2026cs.LG

Geometry-Aware Hyperbolic Residual Quantization

Residual Vector Quantization turns continuous representations into discrete, multi-level token sequences. Yet most methods operate in Euclidean space, despite the coarse-to-fine structure of the resulting codes and the latent hierarchies present in many data domains. Hyperbolic geometry offers a natural alternative for hierarchical representations, but naive hyperbolic extensions introduce geometric inconsistencies: non-associative hyperbolic addition prevents consistent residual aggregation, while standard straight-through gradient estimation ignores the geometry of the latent space. We propose a geometry-aware hyperbolic residual quantization that addresses these issues in both the forward and backward passes. In the forward pass, Hyperbolic Residual Aggregation restores the telescoping behavior of residual quantization on the Poincare ball. In the backward pass, a discounted Hyperbolic Straight-Through Estimator routes the reconstruction gradient through the quantizer as a single geometric block, avoiding unstable recursive gradient transport across residual stages. Evaluations on hierarchical prediction, recommendation, image tokenization, and neural audio coding tasks show that our method improves the stability and structural organization of hyperbolic residual codes over naive hyperbolic baselines. At the same time, we observe a clear structure-compression trade-off: Euclidean residual quantization remains preferable for pure compression, while geometry-aware hyperbolic quantization is most useful for hierarchically organized discrete latent spaces.

Alessio Colombo, Melika Ayoughi

Universiteit van Amsterdam, Amsterdam, Netherlands

Residual VQ Hierarchical Representation Learning Hyperbolic Representation Learning

Sep 22, 2026quant-ph

Hyperbolic Restricted Boltzmann Machine Neural Quantum State

We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from

N=14

to

N=24

, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2

S_2

and von Neumann

S_{vN}

absolute entanglement entropy reconstruction errors. More importantly, for all tested QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatze, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatze in the QSK volume-law system as the Hilbert space dimension increases exponentially.

H. L. Dao

Neural Quantum States Quantum Many-Body Physics Restricted Boltzmann Machines Hyperbolic Representation Learning

Sep 21, 2026cs.CV

HyperCLIP++: Fine-tuning CLIP forOpen-vocabulary Semantic Segmentation in Hyperbolic Space

CLIP, a foundational vision-language model, has emerged as a powerful tool for open-vocabulary semantic segmentation. While freezing CLIP's text encoder is known to preserve its generalization capability, recent studies show that fine-tuning both CLIP's text and image encoders jointly significantly enhances segmentation performance, especially for classes from open sets. In this work, we explain this phenomenon from the perspective of hierarchy alignment, since during fine-tuning, the hierarchical level of image embeddings shifts from image-level to pixel-level. We achieve this by leveraging hyperbolic space, which naturally encodes hierarchical structures. Our key observation is that, during fine-tuning, the hyperbolic radius of CLIP's text embeddings decreases, facilitating better alignment with the pixel-level granularity of visual data. Building on this, we propose HyperCLIP++, a novel and parameter-efficient adaptation strategy. HyperCLIP++ directly adjusts the hyperbolic radius of CLIP's embeddings via scaling transformations to achieve a hierarchy alignment to the target task, i.e., segmentation. To ensure this hierarchy alignment is effected consistently across both modalities and preserves their cross-modal alignment during training, HyperCLIP++ integrates a Dual Cross-Relation Communication (DCRC) module that synchronizes these adjustments between the vision and text pathways. Our experiments show that HyperCLIP++ achieves state-of-the-art performance across three benchmarks while fine-tuning only approximately 5% of CLIP's total parameters. More importantly, we observe that after adjustment, CLIP's text embeddings exhibit a relatively fixed hyperbolic radius across datasets, suggesting that the hierarchical level required for this segmentation task might be quantified using the hyperbolic radius.

Zelin Peng, Zhengqin Xu, Changsong Wen +4

MoE Key Lab of Artificial Intelligence, AI Institute, School of Computer Science, Shanghai Jiao Tong University, Shanghai 200240, China · State Key Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Science, Shanghai, 200083, China · Meituan, Shanghai, 200051, China

VLM Adaptation Open-Vocabulary Semantic Segmentation Vision-Language Alignment Hyperbolic Representation Learning Parameter-Efficient Fine-Tuning

Sep 16, 2026cs.AI

Hyperbolic Graph Representation Learning for Differential Diagnosis on Biomedical Knowledge Graphs

Biomedical knowledge graphs combine ontology-derived hierarchies with transversal associations among heterogeneous entities such as phenotypes, diseases, genes, proteins, and patients. This hybrid structure raises the question of whether hyperbolic embeddings, which naturally capture tree-like organization, remain useful beyond purely hierarchical graphs. We present a preliminary study of hyperbolic graph representation learning for Mendelian-disease differential diagnosis on a patient-integrated biomedical graph. Experiments on isolated ontology subgraphs show that hyperbolic models achieve strong performance in substantially lower dimensions than Euclidean baselines. We then evaluate the models on a link-prediction task that ranks candidate diseases for each patient. Results suggest that hyperbolic embeddings can exploit biomedical hierarchical structure while supporting diagnostic reasoning over heterogeneous patient-level graphs.

Pietro Miotto, Lucia Mellini, Tommaso Marzi +5

Department of Computer Science, Università degli Studi di Milano, Milan, Italy · Università della Svizzera italiana (USI), Lugano, Switzerland · Politecnico di Milano, Milan, Italy +2

KG Embedding Knowledge Graphs Differential Diagnosis Hyperbolic Representation Learning Biomedical KGs

Sep 15, 2026cs.LG

HyCoSeq: Contextual Hyperbolic Representation Learning for Genomic Sequences

Hyperbolic geometry provides a natural inductive bias for genomic representation learning, but existing hyperbolic genomic models primarily use Lorentz convolutions to learn local sequence representations, while their residual pathways do not directly aggregate full Lorentz representations. We propose HyCoSeq, a contextual hyperbolic representation learning framework for genomic sequences. HyCoSeq incorporates weighted Lorentzian residual aggregation into multi-curvature Lorentz encoding, allowing full Lorentz representations to participate directly in geometry-consistent local aggregation. It further introduces a bidirectional long short-term memory network that integrates information from both sequence directions to learn contextual relationships among local representations at different positions within a genomic sequence, thereby extending local hyperbolic convolutional encoding to sequence-level contextualized representations. Extensive experiments across diverse genomic tasks show that HyCoSeq outperforms existing hyperbolic baselines and, without large-scale genomic pretraining, achieves competitive performance against substantially larger pretrained DNA language models.

Chenhao Zeng, Zhibin Pu, Shufei Ge

Institute of Mathematical Sciences, ShanghaiTech University, Shanghai, China

Sequence Modeling Hyperbolic Representation Learning Molecular Representation Learning

Sep 14, 2026cs.CV

Hyperbolic Contrastive Learning with Entailment for Spatial Transcriptomics

Spatial Transcriptomics (ST) has transformed biomedical research by enabling the spatial mapping of gene expression across tissue sections. However, high operational costs, specialized equipment requirements, and sensitivity to experimental noise limit the accessibility and scalability of ST. Recent computer vision approaches aim to overcome these limitations by predicting spatial gene expression directly from histopathology images. While effective, current approaches often suffer from gene expression over-smoothing and overly uniform predictions across tissue regions, suggesting that further progress depends on learning representations that reflect the hierarchical and asymmetric structure of gene regulation and tissue morphology. To address these issues, we propose Hyperbolic Contrastive Learning with Entailment for Spatial Transcriptomics (HyCLoST), a hyperbolic contrastive learning model that captures the intrinsic hierarchical relationships within ST data. By leveraging hyperbolic geometry and a gene-to-image entailment loss, HyCLoST learns structured, biologically grounded representations that improve gene expression prediction accuracy, achieving a 6% reduction in MSE and an 8% increase in PCC across 26 ST datasets, over previous methods. Our source code is publicly available at https://github.com/BCV-Uniandes/HyCLoST

Daniela Vega, Paula Cárdenas, Hannah Ceballos +2

Center for Research and Formation in Artificial Intelligence Universidad de los Andes, Colombia

Contrastive Learning Spatial Transcriptomics Spatial Gene Expression Prediction Gene Expression Prediction from Histopathology Images Whole-Slide Image Analysis Hyperbolic Representation Learning

Sep 10, 2026cs.CV

Hyperbolic Geometry for Open-World Object Detection in Remote Sensing Imagery

Open-world object detection (OWOD) extends closed-set detection by requiring models to identify unknown objects and incrementally learn them once annotations become available. In remote sensing imagery, object categories often exhibit latent hierarchical relationships that may be inadequately represented in the Euclidean spaces commonly adopted by existing methods, limiting unknown-object recall and incremental-learning performance. To address this issue, we investigate hyperbolic geometry for OWOD in remote sensing imagery and propose HyRS-OWOD. To improve unknown object recall, we design a two-step unknown-object discovery mechanism: a Decoupled Objectness Learning (DOL) module that disentangles foreground perception from semantic information to separate foreground proposals from background regions, followed by a Hyperbolic Uncertainty Learning (HUL) component that leverages the radius of hyperbolic embeddings as an uncertainty-aware cue for known-unknown discrimination. For incremental learning, we develop a Hyperbolic Metric Learning (HML) strategy that enhances inter-class separability, facilitating the incorporation of novel categories while mitigating catastrophic forgetting. Experiments on three remote sensing benchmarks demonstrate consistent improvements in unknown recall and incremental learning over state-of-the-art OWOD methods.

Wuzhou Li, Jiawei Zhou, Shenghang Wang +1

School of Computer Science and Artificial Intelligence, Wuhan Textile University, Wuhan 430072, China · Electronic Information School, Wuhan University, Wuhan 430072, China · Electrical and Computer Engineering, Ohio State University, Columbus, OH, USA +1

Open-Vocabulary Object Detection Continual Learning Incremental Learning Aerial Object Detection Hyperbolic Representation Learning

Sep 9, 2026cs.CL

RiLM: Parameter-Efficient Language Modeling via Geodesic Decoding

Language models under one million parameters matter for edge deployment, domain adaptation, and reproducible research, yet a two-layer LSTM or Transformer at embedding width d = 128 still spends roughly one third of its capacity on the output matrix W_out in R^(d x |V|). We propose Riemannian Language Models (RiLM), which remove that layer entirely: context unfolds as a trajectory on a Riemannian manifold, and next-token probabilities arise from squared geodesic distance between the current state and vocabulary embeddings. The same embedding map serves input and output -- decoding is geometry. We instantiate the framework on flat R^d (Flat RiLM) and the Poincare ball H^d (HypRiLM) with a shared MLP composition map phi (~290k parameters, d = 128, |V| = 2000). Across five seeds on WikiText-2, HypRiLM reaches 54.2 +/- 0.2 validation perplexity versus 87.6 +/- 0.6 for Flat RiLM; tied and matched LSTM, Transformer, and SSM controls remain at 113-147 PPL on WT-2 -- HypRiLM leads by roughly 2x over the strongest tied recurrent baseline (SSM, 113.0 +/- 3.8). Penn Treebank and a 10k-vocabulary stress test confirm that geodesic decoding transfers across corpora and larger |V|, while hyperbolic curvature helps selectively. We also characterize boundary collapse in naive hyperbolic recurrence and show how Mobius stabilization restores trainability. Claims are scoped to controlled small-model comparisons, not full-vocabulary state of the art.

Fang Li

Department of Computer Science Oklahoma Christian University Edmond, USA

Autoregressive Language Modeling Language Modeling Hyperbolic Representation Learning

Sep 9, 2026cs.CE

Geometric organization of olfactory descriptor data in the Poincaré disk

Odor quality is commonly represented using high dimensional descriptor profiles, yet their low dimensional organization remains unclear. We investigated whether a two-dimensional hyperbolic embedding can provide an interpretable representation of this structure. We applied hyperbolic metric multidimensional scaling to two complementary datasets: 480 Sagar rating profiles from three participants rating 160 odorants on 15 continuous descriptors, and 4983 GoodScents--Leffingwell molecules annotated with 138 binary descriptors. The embeddings substantially preserved pairwise descriptor distances, supporting subsequent analyses of radial and angular organization. In Sagar, rating profile entropy was strongly and negatively associated with hyperbolic radius, with diffuse profiles closer to the center and concentrated profiles closer to the boundary. This radial organization emerged primarily at the level of the full descriptor profile, rather than any individual descriptor, and remained robust across alternative descriptor representations, participant specific analyses, and averaged ratings. Sweet, musky, fruity, pleasantness showed the strongest directional trends. In GoodScents--Leffingwell, active label entropy, reflecting descriptor multiplicity, increased with radius, whereas orthogonalized descriptor entropy, reflecting spread across orthogonal modes, decreased with radius. Related binary descriptors occupied coherent localized high-density regions. These findings reveal complementary radial and angular organization in the hyperbolic representation of olfactory descriptor data. They support hyperbolic mapping as an interpretable descriptive framework in which radius summarizes global profile properties, while the angular component captures continuous descriptor gradients and categorical organization.

Aniss Aiman Medbouhi, Farzaneh Taleb, Giovanni Luca Marchetti +1

Department of Robotics, Perception and Learning School of Electrical Engineering and Computer Science KTH Royal Institute of Technology, Sweden · Department of Mathematics School of Engineering Sciences KTH Royal Institute of Technology, Sweden

Dimensionality Reduction Hyperbolic Representation Learning

Sep 8, 2026cs.LG

HypLTSF: A Hyperbolic Geometric View of Multi-Scale Hierarchies for Long-Term Time Series Forecasting

Multi-scale modeling has become an effective approach for long-term time series forecasting, capturing temporal patterns that range from fine-grained local dynamics to coarse global trends. Representations across these temporal scales are inherently hierarchical, with coarser scales abstracting and aggregating information from finer ones. While existing approaches readily exchange information across these scales, the hierarchy itself is typically left as an emergent byproduct of such interactions rather than captured as a geometric structure in its own right. In this paper, we introduce HypLTSF, a framework that endows the multi-scale hierarchy with a concrete geometric form by embedding scale-wise representations into the Poincaré ball, whose exponentially expanding volume naturally accommodates hierarchical structures. To align this geometry with the temporal hierarchy, HypLTSF imposes two constraints: (1) a radial constraint that orders embeddings by their level of abstraction, and (2) an angular constraint that groups fine-scale patterns sharing a common coarser-scale ancestor. Extensive experiments on long-term time series forecasting benchmarks show that HypLTSF achieves state-of-the-art performance, suggesting that explicitly modeling the multi-scale hierarchy as a geometric structure is effective for forecasting.

Namwoo Kim, Hyungryul Baik, Yoonjin Yoon

Urban AI Institute, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Department of Mathematical Science, KAIST, Daejeon, Republic of Korea · Department of Civil and Environmental Engineering, KAIST, Daejeon, Republic of Korea

Long-Term Time Series Forecasting Hierarchical Representation Learning Multiscale Time-Series Modeling Hyperbolic Representation Learning

Aug 12, 2026cs.LG

HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce \textbf{HY}perbolic \textbf{D}ynamic \textbf{R}epresentation \textbf{A}rchitecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincaré ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.

Zhao Su, Yuxin Xia, Haoran Li +4

School of Information Science and Engineering, Lanzhou University · Department of Data Science and AI, Monash University · School of Computing and Information Technology, University of Wollongong +1

Kolmogorov-Arnold Networks Neural Network Approximation Theory Neural Network Compression Hyperbolic Representation Learning

Aug 12, 2026cs.AI

HyperANFIS: Enhancing Rule Representation and Interpretability in Adaptive Neuro-Fuzzy Systems via Hyperbolic Geometry

The adaptive neuro-fuzzy inference system (ANFIS) is an interpretable reasoning framework capable of generating explicit IF-THEN fuzzy rules, making it suitable for tasks requiring transparent reasoning. However, existing ANFIS models generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. To address this issue, we propose Hyperbolic ANFIS (HyperANFIS), a hyperbolic extension of ANFIS. HyperANFIS preserves the fuzzy semantics and core architecture of conventional ANFIS while performing rule-prototype learning, rule activation, and consequent aggregation in hyperbolic space. It also retains the ability to generate interpretable IF-THEN rules. By exploiting the representational properties of hyperbolic geometry, HyperANFIS strengthens the fuzzy inference process, thereby improving predictive accuracy, inter-rule collaboration, and the credibility of its interpretable rules. Experimental results show that HyperANFIS consistently outperforms the standard ANFIS baseline and various ANFIS variants across all datasets, while also generating higher-quality fuzzy rules.

Haoran Pei, Zhao Su, Zetao Lin +6

school of Information Science and Engineering, Lanzhou University · Department of Data Science and AI, Monash University · School of Computing and Information Technology, University of Wollongong +1

Interpretable ML Fuzzy Logic Hyperbolic Representation Learning

Aug 10, 2026cs.LG

Hyperbolic Multimodal Continual Learning

Hyperbolic geometry has recently emerged as a powerful representation space for multimodal learning, as it naturally captures hierarchical semantic structure across modalities. Despite this progress, how such representations behave under continual learning poses fundamentally different challenges that remain underexplored. This work provides a geometric perspective on this problem and establishes a theoretical foundation for representation preservation in hyperbolic space, showing that preventing forgetting requires cross-modal invariance under a shared hyperbolic isometry. We further show that forgetting in hyperbolic continual learning involves both semantic relation drift and hierarchy-related distortion, motivating preservation of both cross-modal relational structure and hierarchical geometry. Guided by these insights, a principled continual learning framework is derived that preserves essential geometric structure while allowing effective adaptation to new tasks. Experiments on continual multimodal benchmarks corroborate the effectiveness of the proposed approach.

Jiahong Liu, Ming Shen, Xiaohao Liu +4

Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, China · School of Computing, National University of Singapore, Singapore · Department of Computer Science, Yale University, New Haven, CT, United States +1

Cross-Modal Representation Learning Invariant Representation Learning Hyperbolic Representation Learning Multimodal Learning Catastrophic Forgetting

Aug 7, 2026cs.CV

H2AL: Hyperbolic Hierarchy-aware Aggregative Learning for Registration-based Few-shot Medical Image Segmentation

Registration-based Few-shot medical image segmentation (RFMIS) aims to generate pseudo-labels for unlabeled images by warping a labeled image through registration. However, existing methods primarily perform pixel-level optimization and inference in Euclidean space, treating anatomical structures as flat and disjoint. This neglect of inherent hierarchies degrades pseudo-label quality and weakens the discrimination of ambiguous regions, limiting the segmentation performance. To overcome this challenge, we propose a Hyperbolic Hierarchy-aware Aggregative Learning framework for RFMIS, termed H2AL, that enhances both deformation plausibility and anatomical discrimination for dual-task learning. Specifically, we introduce a Hyperbolic Hierarchy-aware Infusion (H2I) module, which leverages the hierarchical modeling capability of hyperbolic space to learn precise hierarchy-aware representations via transformation-guided supervised hyperbolic contrastive learning, and injects such hierarchical priors into Euclidean space through a gated infusion block while preserving semantic richness. Furthermore, we propose an end-to-end joint optimization algorithm by gradient aggregation, where the gradients from the registration and segmentation decoders, embedding semantic and hierarchical cues, are aggregated to update the shared encoder to promote collaborative learning across tasks. Extensive experiments on two anatomical regions, with five experimental settings, demonstrate the effectiveness and efficiency of our method in both registration and segmentation. The code is publicly available at https://github.com/JiamingCai469/H2AL.

Jia Wang, Jiaming Cai, Zunying Hu +4

Hierarchical Representation Learning Medical Image Registration Few-Shot Medical Image Segmentation Hyperbolic Representation Learning

Aug 7, 2026cs.LG

Hyperbolic Graph Embedders for Link Prediction and Topology Reconstruction

Hyperbolic embeddings provide compact geometric representations of complex networks in hyperbolic spaces, but systematic comparisons of methods developed in machine learning, network science, and algorithmics remain rare. We benchmark 13 unsupervised hyperbolic graph embedders under a unified protocol for link prediction and topology reconstruction on synthetic and empirical networks. The protocol captures both missing-link recovery and the preservation of local and global network structure. Maximum-likelihood and representation-learning-based approaches, including hybrid variants, achieve the strongest overall performance, although no method dominates across all tasks and structural regimes. Performance is more strongly associated with embedding paradigm than with disciplinary origin. We identify the network regimes in which different paradigms succeed or fail and provide practical guidance for method selection in downstream applications.

Robert Jankowski, Maksim Kitsak, Dorota Celińska-Kopczyńska

TU Delft · Indiana University · Institute of Informatics, University of Warsaw

Link Prediction Unsupervised Representation Learning Hyperbolic Representation Learning

Aug 5, 2026cs.SD

HyPASE: Hyperbolic Geometry for Parameter-Efficient Speech Emotion Fine-Tuning Framework for Large Audio-Language Models

Large Audio-Language Models (LALMs) excel at general speech understanding; however, adapting them to fine-grained tasks like Speech Emotion Recognition (SER) remains a significant bottleneck. Current Parameter-Efficient Fine-Tuning (PEFT) methods typically operate in flat Euclidean space, and this geometry fails to capture the multi-granularity nature of emotion cues, which range from low-level prosody to high-level semantics. To address this, we propose HyPASE, a hyperbolic PEFT framework for LALM-based SER. HyPASE leverages the Poincare ball model, using the hyperbolic radius as an explicit proxy for representational granularity. The framework consists of two core components: a Hyperbolic Geometric Adapter (HGA) for layer-adaptive weight modulation, and an Emotion-aware Multi-capacity Cross-modal Aggregator (EMCA) that compresses multi-scale features into compact audio prefixes. Empirical results on standard benchmarks show that HyPASE outperforms Euclidean PEFT baselines across all metrics on MELD and achieves a notable Unweighted Accuracy gain on IEMOCAP, particularly in class-imbalanced emotion recognition, with the accompanying slight Weighted Accuracy trade-off reflecting hyperbolic space's geometric prioritization of minority-class representations; furthermore, HyPASE achieves robust zero-shot cross-dataset generalization within a constrained parameter budget. By grounding the adaptation process in hyperbolic geometry, HyPASE offers a highly efficient path for LALM fine-tuning.

Tian Jin, Ruikang Zhang, Zefeng Zhao +2

Tongji University Shanghai, China · Peking University Beijing, China · The Chinese University of Hong Kong, Shenzhen Shenzhen, China

Audio-Language Models Multimodal Emotion Recognition Speech Emotion Recognition Hyperbolic Representation Learning Parameter-Efficient Fine-Tuning

Aug 3, 2026cs.AI

ProWorld: Progress-Aware Hyperbolic World Models for Long-Horizon Visual Goal Reaching

JEPA-style visual world models offer an effective paradigm for visual goal planning by predicting future latent representations. Existing methods typically learn local transition consistency through next-step representation prediction. However, in long-horizon tasks, accurate local prediction alone need not ensure sustained progress toward the goal. First, multi-step rollouts can remain locally plausible while drifting away from goal-relevant trajectories. Second, locally similar future states can correspond to substantially different long-term progress, making them difficult to distinguish in a latent space optimized mainly for local consistency. To address these challenges, we introduce goal-conditioned progress order, a relative ordering of states according to how they advance toward a given goal. This order exhibits an asymmetric, coarse-to-fine structure: early states retain broader future possibilities, while later states concentrate on more specific goal-relevant regions. Such a structure is well suited to hyperbolic geometry. Motivated by this observation, we propose ProWorld, a progress-aware hyperbolic visual world model. ProWorld leverages goal-conditioned progress order to organize visual latent-space dynamics, maintains directional progress within trajectories via hyperbolic entailment learning, and mitigates progress ambiguity among locally similar future states via hyperbolic future discrimination. Furthermore, we design a progress-aware planning objective that scores candidate rollouts by jointly considering proximity to the goal and sustained progress across intermediate states. Experiments on four visual goal-reaching tasks demonstrate that ProWorld achieves an average absolute success-rate gain of 9.67 over LeWM. The code will be released after the paper is accepted.

Zihan Liu, Yuzhe Zhuang, Yuanzu Li +4

The Hong Kong University of Science and Technology (Guangzhou) · Independent Researcher · The Chinese University of Hong Kong +1

World Model-Based Planning Long-Horizon Planning Model-Based Planning Latent World Models Hyperbolic Representation Learning

Jul 27, 2026cs.CV

PointCHR: Point Cloud Analysis via Curvature-Aware Hyperbolic Rectification

High-curvature regions in 3D point clouds encapsulate critical fine-grained geometric semantics yet exhibit a distinct long-tail sparsity in their spatial distribution. The inherent limitations of polynomial volume growth in Euclidean space frequently render these intricate geometric features challenging to adequately resolve within a uniform-scale feature space. Consequently, these regions are frequently overshadowed by smooth global features dominated by low-curvature regions, thereby limiting the discriminative capacity of the network. To address this issue, we propose PointCHR, a curvature-aware hyperbolic rectification (CHR) for point cloud analysis. Utilising the property of exponential volume expansion in the vicinity of hyperbolic manifolds, CHR presents a learnable curvature-guided radial rectification mechanism. By adaptively projecting high-curvature points towards boundary regions endowed with larger effective embedding capacities, PointCHR effectively mitigates the representation crowding problem inherent in Euclidean settings. Extensive experimentation has demonstrated that PointCHR significantly enhances the ability of backbone to capture fine-grained geometric details, achieving state-of-the-art performance across multiple benchmarks.

Xinxing Yu, Liying Yang, Hao Mo +4

School of Computer Science and Engineering, Macau University of Science and Technology, Macau, China · School of Computing and Information Technology, Great Bay University, Dongguan, China · Shenzhen International Graduate School, Tsinghua University, Shenzhen, China +1

Geometric Representation Learning Point Cloud Understanding Hyperbolic Representation Learning

Jul 21, 2026cs.LG

Riemannian Deep Learning: Modules, Networks, and Geometries

Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.

Ziheng Chen

Doctoral School in Information and Communication Technology, Università di Trento

Geometric Representation Learning Hyperbolic Representation Learning

Jul 20, 2026cs.LG

After the Euclidean Highway: Hyperbolic Expert AI as the Next Innovation

Expert domains are trees; the Euclidean transformer is not, diluting parent-child structure exponentially at depth. The hyperbolic turn left one question unasked: not how much of a network to curve, but where curvature may touch the gradient. Placement is a law, not a knob: the same geometry on a trainable adapter collapses training (seventeen training collapses, ~220 GPU-hours), yet at the loss layer alone it trains without one -- this is HySAT (Hyperbolic Structure-Aware Training), hyperbolic losses at the loss layer only. Across six expert SLMs we constructed and deployed (Llama 3.1 and EXAONE 3.5; four adapter strategies; 18.0M-sample corpus; zero NaN over ~317K optimizer steps), a matched four-arm ablation isolates the preserved manifold invariant, and three propositions and a lemma prove why loss-only placement is stable where adapter-on-manifold is not. Four models are operationally deployed (one live, consumer-facing), two open-weight, with per-step traces and a seventeen-incident failure ledger on Zenodo (CC-BY-4.0).

Kwan Soo Shin, In Seok Kang, Munho Lee

1*PolymathMinds Lab, Asan, Republic of Korea. · 2aSSIST University, Seoul, Republic of Korea. · 3Samsung Engineering, Seoul, Republic of Korea.

Language Modeling Language Model Post-Training Hyperbolic Representation Learning