Hyperbolic Learning

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.

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the $\mathbb{SL}(n)$ space, a representation geometry defined by the simple $\det(A)=1$ constraint and a left invariant Schatten-$p$ Finsler structure. Despite this minimal construction, $\mathbb{SL}(n)$ exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, $\mathbb{SL}(n)$ consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by $44.3\%$ on KEGG and $40.5\%$ on HumanCyc, and improves Hits@20 by $42.8\%$ on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation 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.

Protein structure modeling rests on a single computational primitive: the interaction between what a residue is (sequence content) and where it sits (three-dimensional geometry). What is the expressive limit of this layer class? We show that the complete bilinear operator over content-geometry outer products--the sufficient statistic of all second-order interactions--is the expressive ceiling, while the additive message passing of mainstream geometric GNNs is provably blind to content-geometry binding. We then introduce Hyper-Fold, a rank-K separable convolutional backbone approaching this ceiling at message-passing cost: each radius neighborhood is organized into a sequence hyperedge and a contact hyperedge, modulated by an edge-conditioned matrix-valued operator factorized into K learned basis operators with geometry-generated coefficients. Across enzyme function prediction, fold classification, and ligand binding site detection, Hyper-Fold and its hierarchical variant Hyper-Fold-Deep achieve the best results among protein-specific structure encoders; Hyper-Fold-Pocket, an anchored set-prediction head, surpasses UniSite-3D on UniSite-DS and two zero-shot benchmarks with no sequence language model features, 68x fewer parameters, and 4.8x lower latency--suggesting that a sufficiently expressive 3D backbone recovers information that fusion architectures previously borrowed from evolution-scale pretraining.

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.

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.

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.

Infrared small target detection (IRSTD) has achieved substantial progress under domain-consistent evaluation, yet detector performance often degrades markedly when generalizing to unseen infrared domains. Existing methods primarily improve detection by enhancing target responses and suppressing background interference. However, when trained on only a limited set of source domains, their learned decision rules are inevitably established from a restricted range of source-domain target-background relation patterns. We formulate this cross-domain failure as target-background relation shift: unseen domains may exhibit relation patterns that are not observed during training, thereby weakening the discriminative capability learned from the source domains. To address this problem, we propose HyTBE, a Hyperbolic Target-Background Expert model that expands source-domain relation patterns and adaptively adjusts visual representations using explicit relation cues. The Target-Background Relation Intervention selectively perturbs either targets or backgrounds, broadening the observable relation patterns during training while maintaining valid supervision. Subsequently, the Hyperbolic Relation Modeling maps multi-scale visual cues into a Poincaré ball and characterizes the target-background relation of each feature token according to its relative distances to the target and background anchors. The Hyperbolic-guided MoE Adapter further uses these hyperbolic relation representations to calibrate multi-scale visual features and aggregate expert-specific feature corrections for different relation patterns. Leave-one-domain-out experiments on NUAA-SIRST, NUDT-SIRST, and IRSTD-1K demonstrate that HyTBE achieves stronger cross-domain generalization than competitive baselines.

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.

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).

Aerial-Ground Person Re-IDentification (AG-ReID) aims to retrieve the same person across heterogeneous aerial and ground camera platforms. Although great progress has been made, existing methods remain suboptimal due to the direct feature alignment across views, overlooking view-specific cues. To address this issue, we propose a novel Hierarchical Hyperbolic Representation (HiHR) framework for AG-ReID. More specifically, we first extract multi-granularity features based on pre-trained visual-text encoders. Then, we propose a Text-guided Multi-granularity Fusion (TMF) to fuse multi-granularity features and enhance the representation ability of identity features. Furthermore, we introduce the Hierarchical Hyperbolic Learning (HHL) to construct a hierarchical feature structure in a hyperbolic space. This hierarchy includes a coarse level that ensures identity separability and cross-view consistency, and a fine level that preserves view-specific discriminative cues. As a result, our proposed framework can effectively aggregate view-invariant and view-specific discriminative features for AG-ReID. Extensive experiments on four AG-ReID benchmarks demonstrate the effectiveness of our framework. The source code is available at https://github.com/YangQiWei3/HiHR.

Functional brain networks exhibit a hierarchical organization across ROI, community, and whole-brain levels, supporting local processing, inter-community coordination, and global integration. Recent studies have demonstrated that brain community-aware modeling is beneficial for both diagnosis and biomarker identification of brain networks. However, existing brain graph modeling methods often struggle to model ROI-community interactions, thereby failing to fully exploit the hierarchy across ROI, community, and whole-brain network levels. To address this issue, inspired by deep hyperbolic learning in modeling hierarchical structures, we propose a novel framework, termed Hyperbolic Learning on Brain Graphs (HLBG), for brain network analysis. The core idea of HLBG is to exploit the inherent hierarchical geometry of hyperbolic space to model the hierarchical relationships among ROIs, functional communities, and the whole-brain network, thereby learning hierarchy-aware and highly discriminative representations for brain network data. Specifically, HLBG first projects representations from ROIs, communities, and the whole-brain network into Lorentzian hyperbolic space. Then, the multi-level hierarchy is imposed via two geometric entailment constraints. In addition, we introduce a new Graph-aware Mamba (GaMamba) model, which incorporates topology-derived structural prompts into Mamba to capture long-range dependencies while preserving graph topological information. Experiments on ABIDE-I and REST-MDD demonstrate that HLBG outperforms state-of-the-art methods and identifies disorder-relevant functional biomarkers.

Tabular classification is often governed by local, condition-triggered rules rather than smooth global patterns. However, tabular deep neural networks (DNNs) are typically built upon Euclidean representations that favor smooth variations and semantic locality. This potential geometric mismatch can make it challenging for tabular DNNs to efficiently represent the discrete, rule-partitioned structures often underlying tabular classification. To address this issue, we propose HDE-Net, a manifold-constrained DNN that enables hierarchical decision modeling in hyperbolic space. We first abstract heterogeneous features into unified Latent Decision Nodes (LDNs) and embed them in the Poincaré ball, forming a continuous representation that resembles tree-structured reasoning. For numerical features, we introduce a Soft Decision Routing mechanism that approximates range-based local rules in a differentiable manner, bringing their LDN semantics closer to those of categorical features. An entropy-aware capacity allocation algorithm further adapts the number of LDNs per numerical feature to balance expressiveness and complexity. On the TALENT-tiny-core classification benchmark (30 datasets), HDE-Net achieves the \textit{best average rank}, outperforming both industrial GBDTs and recent tabular DNNs while maintaining high efficiency.

Semi-supervised hierarchical clustering aims to learn a tree structure consistent with data patterns and user-provided supervision. Supervision is usually given as leaf-level relations, such as pairwise must-link/cannot-link constraints or triplet-wise must-link-before constraints. Although useful for regulating local sample relations, such supervision does not directly indicate which samples should form coherent subtrees. Consequently, the non-leaf structure of the learned tree may deviate from the hierarchical organization preferred by ground-truth labels. To address this limitation, we propose a semi-supervised hyperbolic hierarchical clustering method with set-level structural priors. The main contribution is to introduce sets as basic modeling units for hierarchy learning. Each set denotes samples expected to cohere within a subtree and is induced from leaf-level supervision together with a learned constraint-consistent similarity structure. These sets act as soft structural priors for subtree-level supervision, allowing supervision to guide non-leaf hierarchy formation beyond local leaf-level relations. Specifically, we first learn constraint-consistent embeddings to obtain a reliable set partition, then construct constraint-induced sets and estimate inter-set similarities to form set-level structural priors. Finally, these priors are incorporated into a hyperbolic hierarchy objective for continuous tree optimization. Experiments on eleven benchmark datasets and ablation studies show that the proposed method consistently improves label consistency over representative hierarchical clustering baselines while also enhancing similarity-based tree quality.

Implicit neural representations (INRs) model a signal as a continuous coordinate-to-value function. For terrain elevation data, this supports analytic derivatives, arbitrary-resolution decoding, and a smooth surface model of the underlying heightfield. However, fitting and storing a separate INR for every tile does not scale to large terrain datasets. Amortized neural representations reduce this cost with a shared network: a new tile is mapped to a compact per-tile payload, and a shared decoder reconstructs the heightfield from it. Most such methods are hypernetworks that predict the payload in a single forward pass, while others recover it through a short per-tile optimization. These methods were developed primarily for natural images, and their suitability for terrain heightfields remains unclear. We introduce a controlled benchmark on a 1 m/pixel terrain dataset and evaluate three representative methods under a unified protocol. Observing a clear cross-domain gap, we propose HUVR+SIREN, a hypernetwork that adapts the strongest benchmarked method (HUVR) by replacing its coordinate decoder with a smooth, analytically differentiable one. It attains the best height and derivative fidelity on the benchmark with no additional per-tile storage and lower decode cost, and tolerates aggressive post-training quantization with negligible quality loss, giving a compact terrain neural format. Ablations and diagnostics further identify which design choices transfer to terrain and show that the per-tile bottleneck is already near its useful limit, leaving the remaining gap in the shared hypernetwork's architectural design.

Multimodal emotion recognition aims to integrate text, audio, and video sources to understand human affective states. Although multimodal large language models excel at multimodal reasoning, they typically treat emotion categories as independent labels, ignoring the rich hierarchical taxonomy of human psychology. Moreover, lacking external contextual knowledge makes them highly susceptible to over-interpreting noisy cues, further complicating fine-grained emotion classification. To address these issues, we propose \textbf{HyperEmo-RAG}, a retrieval-augmented generation framework that leverages a structured emotional knowledge base. Our framework introduces two key innovations. 1) Hierarchical hyperbolic grounding. Recognizing the inherent hierarchical tree structure of emotion taxonomies, we jointly embed hierarchical emotion labels and multimodal samples into a continuous hyperbolic space (Poincaré ball) and design a hierarchical beam-search deliberation process that progressively retrieves samples from coarse to fine-grained levels. 2) Structured evidence injection. Based on the retrieved evidence, we construct an evidence graph and inject the structured knowledge as explicit cognitive context into the LLM through a Tree-Aware Attention mechanism and an EmotionGraphFormer, preserving the integrity of graph-structured information. Experiments on multiple datasets demonstrate that HyperEmo-RAG significantly outperforms existing methods.

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.

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.

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, a WSI classification framework that leverages both geometric spaces. To model long-range dependencies among thousands of patches, we employ a structured state space sequence model (S4) backbone that encodes patch sequences with linear computational complexity. Furthermore, to account for regional heterogeneity, we introduce a chunk-level mixture-of-experts (MoE) module that groups patches into regions and dynamically routes them to specialized subnetworks, improving representational capacity while reducing redundant computation. Extensive experiments on seven WSI datasets spanning six cancer types demonstrate that BatMIL consistently outperforms state-of-the-art MIL approaches in slide-level classification tasks. These results indicate that geometry-aware representation learning offers a promising direction for next-generation computational pathology.

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.

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.

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.

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.

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.