Hyperbolic Space

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.

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.

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.

We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver). The Euclidean part establishes three core theorems: (1) circuit separation---IDA achieves exact retrieval with $\mathcal{O}(1)$ resources while softmax requires $Ω((\log n)^2)$ width; (2) a Polyak--Lojasiewicz inequality with $Ω(e^{Δ^2/\sqrt{d}}/Δ^2)$ stronger constant than softmax, implying linear convergence, $\mathcal{O}(\log n)$ Lipschitz scaling under a low-rank/clustering assumption, $Θ(1)$ Hessian spread, and absence of spurious local minima; (3) a width-independent effective rank bound that limits noise memorization---softmax memorizes arbitrary labels when $d_h\ge n$, while IDA limits test error to $\mathcal{O}(η^2)$. The non-Euclidean extension then builds upon this prototype, replacing Euclidean distance with hyperbolic geodesic distance for storage and spherical geodesic distance for routing. The Riemann GeoResolver framework comprises ten integrated modules: four HIDA operators spanning $Θ(n^2)$ to $Θ(1)$ per token; Hyperbolic Curvature Compression (HCC) with provable error bounds; HyperGate with gradient lower-bound theorem; Spherical Inverse Distance Attention (SIDA) with sphere-analog PL inequalities; Dynamic Memory Genesis (DMG) with $\mathcal{O}(\log T)$ regret bounds; and Geodesic Sparse Routing (GSR) with quality and communication bounds. The Euclidean theorems are proved in full; the non-Euclidean extension theorems are proved with analogous arguments. This work establishes a theoretical arc: from Euclidean attention as a special case, to hyperbolic memory, to spherical retrieval.

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.

Planar tiled diffusion denoises overlapping windows of one rectangular canvas. The hyperbolic plane has no such canvas, and its area grows exponentially with radius. We introduce HyperbolicDiffusion, a training-free method for generating finite visual fields directly on the hyperbolic plane H2. Our Hyperbolic Blooming Cover reduces window placement to a compact dynamic program that runs in seconds while providing strong theoretical guarantees. Permanent surface IDs form a shared latent canvas: a standard diffusion model denoises local windows, whose predictions are fused back onto H2. Because curvature causes residual disagreement and blur at multi-window junctions, a geometry-derived second stage re-noises and repairs precisely those regions. The resulting fields are sharp, reprojectable, and consistent across viewpoints, providing a prompt-driven generative counterpart to Escher's Circle Limit series.

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.

Surgical vision-language foundation models typically adopt educational materials, such as surgical lecture videos, to transfer surgical knowledge encoded in language into visual representations. These knowledge are multi-dimensional and hierarchical: fine-grained action cues appear in narration, mid-level key steps are summarized in subsection headings, and global procedural context, such as patient history and surgical strategy, is described in abstract texts. Prior work largely collapses these heterogeneous signals into a single flat embedding space, implicitly assuming independence across hierarchy levels. However, this is suboptimal because it ignores cross-level semantic containment, e.g., actions belong to steps, steps compose phases, weakens long-range dependency modeling. To this end, we propose a hyperbolic surgical video-language pre-training framework that explicitly preserves the hierarchical structure by mitigating structural false negatives induced by procedural context and enforcing semantic consistency between parent phases and their constituent child steps. Extensive experiments on multiple surgical benchmarks show consistent gains in zero- and few-shot phase recognition across procedures and institutions.

In the first part of this work, we construct the first type of two-dimensional (2D) hyperbolic neural quantum state (NQS) in the form of the Lorentz 2DRNN (Recurrent Neural Network) and benchmark its performance against the Euclidean 2DRNN in the paradigmatic $N\times N$ 2D Transverse Field Ising Model (2DTFIM) setting with different lattice sizes up to $N=12$ and at different transverse magnetic field strengths. We find that hyperbolic Lorentz 2DRNN NQS definitively outperform Euclidean 2DRNN NQS when the system is at the phase transition point when the physics can be described by a conformal field theory (CFT), which is known to be dual to an Anti-de-Sitter (AdS) space whose spatial geometry is hyperbolic. In the second part of this work, we benchmark the performances of the recently introduced one-dimensional (1D) hyperbolic NQS including Poincaré RNN/GRU and Lorentz RNN/GRU against their Euclidean NQS versions in $N\times N$ 2DTFIM, which has to be converted to a one-dimensional setting to allow for the use of 1D NQS. The findings in this case extend our previous results that 1D hyperbolic NQS definitively outperform 1D Euclidean NQS, thanks to the combined effects of the hierarchical structure comprising the first and $N^{th}$ neighbor interactions present in the 1D system arising from the 2D lattice and the CFT physics at the critical point. While more studies with larger system sizes are required, our work serves as a proof-of-concept for the utility, effectiveness as well as the superior performances of one- and two-dimensional hyperbolic NQS ansatzes compared to the existing Euclidean NQS in many-body quantum physics systems, especially when these systems exhibit structural hierarchy or when they are at criticality, or a combination of both.

Vision-language models (VLMs) achieve strong cross-modal alignment but remain brittle to negation, often relying on shallow word associations rather than compositional reasoning. Fine-tuning on negation-specific data can also compromise their general purpose capabilities through catastrophic forgetting. We introduce HANCLIP (Hyperbolic, Angular, and Negation), a geometry-aware framework that improves negation sensitivity while preserving the structure of the pretrained joint embedding space. HANCLIP combines a hyperbolic contrastive objective, which models hierarchical relations and semantic asymmetries, with an angular triplet loss that separates negated descriptions from their affirmative counterparts. Using only 20,000 image-text quadruplets, HANCLIP consistently improves performance across CLIP, LongCLIP, and SmartCLIP backbones on the NegBench benchmark, while maintaining or improving zero-shot classification and image-text retrieval performance. These results show that lightweight, geometry-guided objectives can enhance negation understanding without large-scale retraining.

Echocardiography (echo) is a widely used imaging modality for assessing cardiac function, with Left Ventricular Filling Pressure (LVFP) serving as a critical physiological marker for conditions such as heart failure. Standard LVFP classification into normal \emph{vs} elevated categories relies on the Doppler-derived $E/e'$ ratio, which is operator-dependent and often unavailable in resource-limited settings, motivating methods that infer LVFP directly from B-mode echo. Existing deep learning approaches achieve high performance but remain largely black-box, limiting clinical interpretability. We propose HypOProto, a hyperbolic, ordinal prototype-based framework for interpretable LVFP classification using a frozen, explainable foundation model backbone. HypOProto arranges prototypes along the physiological $E/e'$ scale, placing borderline cases near the hyperboloid root where small angular differences separate similar cases, while normal and elevated cases occupy outward positions reflecting increasing diagnostic certainty. This hyperbolic geometry encodes clinically meaningful ordinal relationships and improves interpretability. We also introduce a novel Hyperbolic Prototype Angular Separation (HyperPAS) loss, enforcing inter-class prototype separation in hyperbolic space. HypOProto achieves SOTA performance while maintaining transparency, and highlights clinically relevant regions in visualizations. This work represents the first prototype-based framework for LVFP classification in echo. Our code can be found at https://github.com/DeepRCL/HypOProto.

Infrared and visible image fusion aims to integrate complementary modalities, while existing Euclidean methods impose rigid distance metrics that distort multi-modal interactions and parent-to-child semantic hierarchies. To overcome these limitations, we introduce a text-driven fusion framework empowered by hyperbolic manifold learning. During training, BLIP-extracted text prompts serve as topological anchors within the hyperbolic space, guiding vision-attribute alignment through hyperbolic embeddings that naturally accommodate varying semantic granularities. By exploiting the exponential volume growth dictated by the Poincaré ball's negative curvature, this approach seamlessly embeds hierarchical trees to encode coarse-to-fine semantics without metric saturation, while the vast peripheral space prevents texture distortion during cross-modal fusion. At inference, the fusion process autonomously adapts to input content using the learned text-attribute priors, completely eliminating the need for textual input. Experimental results show our method outperforms state-of-the-art approaches on benchmark datasets, with code available at https://github.com/Shaoyun2023/TEDFusion.

The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.

Graph foundation models (GFMs) emerged as a dominant paradigm in graph representation learning by leveraging large-scale pre-training for cross-domain inference. However, the parameterized knowledge encoded within these models is insufficient to cope with distribution shifts, limiting their generalization ability. To mitigate this issue, retrieval-augmented generation (RAG) has been introduced to incorporate external knowledge at inference time. Nevertheless, existing RAG frameworks operating in Euclidean space suffer from a fundamental geometric limitation: the polynomial volume growth of Euclidean space is inherently mismatched with the tree-structured external knowledge bases. This mismatch leads to the loss of semantic granularity in retrieval and gives rise to the hubness phenomenon.To address this limitation, we propose a Hyperbolic Retrieval-Augmented Generation (HyRAG) framework designed to enhance the generalization capabilities of GFMs. Specifically, the introduced Hyperbolic Knowledge Indexing module retains the tree-like hierarchies of the external knowledge base by modeling them within hyperbolic space. The Multi-granularity Retrieval module then provides GFMs with the global semantic anchors and local semantic nuances through coarse-grained and fine-grained knowledge retrieval, respectively. Finally, the Dual-path Fusion module achieves effective knowledge integration for graph tasks at both the feature and structural levels. Experiments on multiple graph benchmarks demonstrate significant improvements in the zero-shot setting, highlighting the generalization of our method for robust GFMs inference.

Emerging 6G and edge-intelligent networks require effective and balanced routing algorithms among varied and spatially distributed devices. Existing federated routing systems often prioritize aggregate latency or throughput above fairness and the underlying geometric structure of network topologies. This paper describes Geo-FairFed, a geometric fairness-aware routing system that blends hyperbolic graph neural networks (HGNNs) and federated optimization to provide equal performance across edge nodes. Each node learns topology-aware representations on a negatively curved manifold, which include hierarchical relationships and connection asymmetries. A global aggregator next enforces fairness using a curvature-regularized aim that minimizes routing loss, geometric inconsistency, and an inequality penalty based on Jain's fairness index. A theoretical analysis develops convergence guarantees under limited curvature and shows that the proposed fairness term results in a Pareto-improving equilibrium in routing performance. Extensive simulations on dynamic 6G-edge and IoT topologies reveal that Geo-FairFed minimizes average latency by 20%, reduces energy consumption by 17%, and improves fairness by up to 21% when compared to state-of-the-art federated and geometric routing protocols. The study found that embedding topology in a hyperbolic manifold and including fairness into federated updates can significantly enhance the efficiency and equity of large-scale network routing.

Large Vision-Language Models (LVLMs) have demonstrated impressive performance on multimodal tasks through scaled architectures and extensive training. Recent studies introduce Mixture of Experts (MoE) into LVLMs for improved computational efficiency. However, existing MoE approaches treat visual and linguistic modalities with symmetric architectures, overlooking the inherent asymmetry in how these two modalities are processed. This asymmetry causes two critical issues. First, text and vision form hierarchical rather than parallel relationships, as text queries typically describe partial aspects of complete visual scenes. Euclidean expert space struggles to encode such containment structures. Second, language experts in deeper layers progressively shift from evidence-based processing to parametric memory dependence, losing grounding in the provided visual and linguistic information. To address these issues, we propose AsyMoE, a novel architecture that explicitly models this asymmetry through three specialized expert groups. Intra-modality experts handle modality-specific processing. Hyperbolic inter-modality experts capture hierarchical cross-modal relationships through negative curvature geometry. Evidence-priority language experts suppress parametric memory activation and maintain contextual grounding throughout network depth. Extensive experiments demonstrate that AsyMoE achieves consistent improvements over baseline methods, with average gains of 1.5% over MoE variants and up to 3.8% on hallucination-sensitive tasks. AsyMoE activates 25.45% fewer parameters compared to dense models.

Multi-step reasoning remains a central challenge for large language models: single-pass generation is efficient but lacks accuracy; tree-search methods explore multiple paths but are computation-heavy. We address this gap by distilling reasoning progress into a hyperbolic geometric signal that guides step-by-step generation. Our approach is motivated by a structural observation: in combinatorial reasoning trees, solution-bearing states are few while dead ends are exponentially numerous. The hyperbolic space matches this asymmetry, with compact volume near the origin and exponentially expanding capacity toward the boundary, so that distance-to-origin naturally encodes solution proximity while angular separation distinguishes branches requiring different next operations. We train a lightweight head to project LLM hidden states into this space, then fine-tune a low-rank adapter interactively on its own reasoning attempts to act on the injected signal. Across multiple benchmarks, the geometric signal yields consistent gains, with larger improvements on deeper reasoning chains. Our code is publicly available at https://github.com/yuyuliu11037/HyperGuide.

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.

We introduce HYPERPOSE, a novel 3D human pose estimation framework that performs spatio-temporal reasoning entirely within the Lorentz model of hyperbolic space $\mathbb{H}^d$ 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 $O(TW)$ 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.

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.

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.

Graph few-shot learning, which focuses on effectively learning from only a small number of labeled nodes to quickly adapt to new tasks, has garnered significant research attention. Despite recent advances in graph few-shot learning that have demonstrated promising performance, existing methods still suffer from several key limitations. First, during the meta-training phase, these methods typically perform node representation learning in Euclidean space, which often fails to capture the inherently hierarchical structure existing in real-world graph data. Second, during the meta-testing phase, they usually fit an empirical target distribution derived from only a few support samples, even when this distribution significantly deviates from the true underlying distribution. To address these issues, we propose IMPRESS, a novel framework that IMproves graPh few-shot learning with hypeRbolic spacE and denoiSing diffuSion. Specifically, our model learns node representations in a hyperbolic space and enriches the support distribution through denoising diffusion mechanisms. Theoretically, IMPRESS achieves a tighter generalization bound. Empirically, IMPRESS consistently outperforms competitive baselines across multiple benchmark datasets.

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincare hyperbolic GRU, to new variants including Poincare RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing the new non-Euclidean hyperbolic NQS ansatzes, we generalize the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincare GRU NQS when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the Heisenberg $J_1J_2$ and $J_1J_2J_3$ models. Here, using larger systems consisting of 100 spins, we find that all four hyperbolic RNN/GRU NQS variants always outperform their respective Euclidean counterpart with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different $J_2$ and ($J_2, J_3$) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatzes are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Poincare RNN/GRU and Lorentz RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.

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.

This is our fourth work in the series on machine learning (ML) moment closure models for the radiative transfer equation (RTE). In the first three papers of this series, we considered the RTE in slab geometry in 1D1V (i.e. one dimension in physical space and one dimension in angular space), and introduced a gradient-based ML moment closure [1], then enforced the hyperbolicity through a symmetrizer [2], or together with physical characteristic speeds by learning the eigenvalues of the Jacobian matrix [3]. Here, we extend our framework to the RTE in 2D2V (i.e. two dimensions in physical space and two dimensions in angular space). The main idea is to preserve the leading part of the classical $P_N$ model and modify only the highest-order block row. By analyzing the structural properties of the $P_N$ model, we show that its coefficient matrices are symmetric and admit a block-tridiagonal structure. Then we use this property to introduce a block-diagonal symmetrizer for the ML moment model and derive explicit algebraic conditions on the closure blocks which guarantee the symmetrizable hyperbolicity of the resulting ML system. These conditions lead to a natural parametrization of the closure in terms of a symmetric positive definite matrix together with symmetric closure blocks, which can be learned from data while automatically enforcing symmetrizable hyperbolicity by construction. The numerical results show that the proposed framework improves upon the classical $P_N$ model while maintaining hyperbolicity.

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.