hyperbolic prototype learning

Existing hyperspherical prototype learning (HPL) methods lack theoretical foundations and are constrained to fixed dimensions, impeding simultaneous geometric controllability and scale invariance. Method: We propose the first principled HPL optimization framework, rigorously proving its global optimality. To overcome dimensional limitations, we construct highly separable class prototypes on arbitrary-dimensional unit hyperspheres via linear group codes, integrating spherical coding theory, convex optimization, and hyperspherical geometric modeling. Contribution/Results: Our framework establishes complete theoretical characterizations—both achievability and converse bounds—for prototype separation. The resulting prototype layouts are provably near-optimal, significantly enhancing inter-class separation and classification robustness across diverse dimensions. Empirical results align closely with theoretical guarantees, demonstrating consistent improvements in both synthetic and real-world benchmarks.

Existing hyperbolic learning methods employ fixed curvature and uniform distance metrics, limiting their ability to model heterogeneous hierarchical structures prevalent in real-world data. To address this, we propose a geometry-aware dynamic hyperbolic distance metric that adaptively generates personalized projections and local curvatures for each data pair—enabling the first input-dependent parameterization of hyperbolic distance functions. Our method incorporates low-rank parameterization to reduce computational overhead and integrates hard-negative mining to enhance discriminability. We further establish a theoretical upper bound on estimation error via Talagrand’s concentration inequality. Extensive experiments demonstrate state-of-the-art performance across image classification, hierarchical classification, and few-shot learning tasks: on mini-ImageNet, our approach achieves over 5% absolute accuracy improvement. Visualizations confirm sharper class boundaries and higher prototype separation, validating improved geometric expressivity.

研究了在树结构原型网络中，使用双曲几何而非欧氏几何作为潜在流形，能否更好地保持层次分类模型的局部与全局结构。

This work challenges the prevailing assumption in hyperbolic graph representation learning that “hyperbolic is superior to Euclidean.” Through systematic reproduction and fair empirical comparison, the authors identify methodological flaws in prior studies—specifically, inconsistent baseline configurations, unjustified modeling assumptions, and inadequate geometric quantification. They formally articulate three critical issues, introduce a controllable family of tree-structured benchmark datasets, and propose a novel evaluation paradigm grounded in Gromov δ-hyperbolicity and geometric suitability. Under rigorously controlled training frameworks and hyperparameters, they compare mainstream hyperbolic models (e.g., HGCN, HypER) against their Euclidean GNN counterparts. Results demonstrate that, on highly δ-hyperbolic data (e.g., perfect trees), properly tuned Euclidean models match or even surpass hyperbolic models in performance—thereby undermining foundational theoretical premises and widely held practical consensus in the field.

To address dimensionality collapse caused by feature aggregation in hyperbolic graph contrastive learning, this paper proposes the first hierarchical contrastive learning framework tailored for the Poincaré ball model. Methodologically, it formally defines uniformity requirements at both leaf-level and height-level hierarchies, and jointly optimizes a hierarchical alignment loss with an isotropic uniformity regularizer—constrained by a differentiable annular density penalty—to overcome limitations of Euclidean contrastive learning paradigms. The contributions are threefold: (1) theoretical modeling of hierarchical uniformity in hyperbolic space; (2) design of a differentiable annular density constraint to mitigate dimensionality collapse; and (3) consistent and significant performance gains across multiple hierarchical graph benchmarks, enhancing both feature space utilization and representation discriminability in downstream tasks.

This work addresses a key limitation in existing generalized category discovery (GCD) methods, which typically perform clustering in Euclidean space despite learning representations on hyperspheres, thereby failing to exploit the inherent hierarchical structure of data. To overcome this, we propose HC-GCD, the first end-to-end framework for GCD that operates entirely in hyperbolic space. Our approach leverages the Lorentz model to learn hyperbolic embeddings and directly applies hyperbolic K-Means for clustering, eliminating the need for any projection into Euclidean space. Evaluated on the Semantic Shift Benchmark, HC-GCD achieves state-of-the-art performance, significantly improving clustering accuracy for unseen categories and demonstrating enhanced robustness to variations in label granularity.

为解决双曲几何在欧氏空间中难以直观展示的问题，本文通过将双曲表面离散化为网格，并优化变形能量以匹配双曲平面中的边长，从而在欧氏空间中嵌入双曲表面。

This work addresses the lack of a unified and reproducible evaluation framework in existing hyperbolic graph representation learning methods, which hinders systematic comparison and practical deployment. We propose an open-source, standardized framework that integrates multiple state-of-the-art hyperbolic graph embedding algorithms, offering consistent training pipelines, visualization tools, and evaluation interfaces for downstream tasks such as link prediction and node classification. The framework seamlessly interoperates with widely used network analysis libraries. Comprehensive experiments on real-world networks not only validate the predictive performance of various methods but also uncover their respective strengths and limitations, thereby providing empirical guidance for method selection. This significantly enhances reproducibility and practical efficiency in hyperbolic graph learning research.

This study addresses the numerical instability and angle collapse encountered during gradient-based optimization of hyperbolic graph embeddings by proposing a family of preconditioners based on Euclidean tangent parameterization. Methodologically, the metric scaling factor is revealed as an optional design choice, enabling the construction of a preconditioner family whose curvature varies continuously from −1 to 0, thereby overcoming the limitations of a single fixed metric. Furthermore, a two-stage strategy combining distinct curvatures decouples low-distortion embedding from efficient optimization. Experimental results demonstrate that the proposed approach reduces the loss by 46% to 74% compared to the best single-curvature baseline on real-world tree-structured data, significantly improving representation quality.

This study addresses the challenges of cumulative interference-induced catastrophic forgetting and unreliable module matching during inference in class-incremental learning. To this end, it proposes HyPro, a novel framework that assigns independent LoRA experts to individual tasks to achieve isolated representation learning. Furthermore, this work pioneers the projection of routing features onto a Poincaré ball, enabling reliable task-level discrimination through geodesic nearest-prototype matching. By integrating parameter-efficient fine-tuning, a Mixture-of-LoRA-Experts architecture, and hyperbolic geometric computation, the proposed approach demonstrates substantial improvements over existing strong baselines. Extensive experiments on both standard and few-shot class-incremental learning benchmarks confirm that HyPro yields significantly superior average and final accuracy.