audit hyperbolic geometry

It remains unclear whether existing hyperbolic vision–language models genuinely leverage hyperbolic geometry for hierarchical modeling. This work proposes a diagnostic framework centered on the dimensionless operating point $\sqrt{c}\rho$, integrating entailment cone activation detection, radial depth sensitivity analysis, and angle–curvature disentanglement experiments to systematically audit models such as MERU, HyCoCLIP, and PHyCLIP. The study reveals that these models predominantly operate in near-Euclidean regimes, with their purported hierarchical capabilities stemming not from hyperbolic geometric mechanisms but rather from angular distances or composite supervisory signals. The root cause lies in curvature collapse induced by low curvature and overly wide cone configurations, which create shortcut pathways. This work is the first to demonstrate that the geometric machinery of hyperbolic models remains largely inactive and provides a reproducible evaluation paradigm.

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 investigates the geometric foundations of ROC and PR curves in binary classification, aiming to unify the understanding of curve morphology and classifier behavior through a geometric lens. Methodologically, it introduces the composite function (G = F_p circ F_n^{-1}) as a core modeling framework—where (F_p) and (F_n) denote the CDFs of positive and negative class score distributions—and rigorously establishes a geometric mapping between ROC/PR curve shapes and the underlying distributional geometry. It reveals that (G) quantifies inter-class leakage and admits interpretation via KL divergence. Furthermore, it derives geometric criteria for classifier dominance and interpretability grounded in differential geometry, statistical inference, and CDF transformation theory. The contributions include: (i) a principled, geometrically interpretable framework for threshold selection; (ii) robust, distribution-agnostic tools for classifier comparison; and (iii) enhanced reliability and adaptability in cost-sensitive deployment—particularly under class imbalance and distributional overlap.

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.

本文提出一种基于几何的船体形式可制造性早期筛选框架，通过无量纲总偏差、曲率类面积分数等描述符来评估船体表面特征。

This work addresses the lack of effective support for Kendall’s 3D shape space in existing Python libraries—such as Geomstats—which has hindered the practical application of Riemannian geometry in three-dimensional shape analysis. We present the first systematic implementation of an efficient and user-friendly Python toolkit tailored specifically to Kendall’s 3D shape space, enabling scale-, position-, and orientation-invariant shape modeling and statistical analysis. By providing a ready-to-use, open-source solution for shape statistics on manifolds, this contribution fills a critical software gap in advanced 3D shape analysis, substantially lowering the barrier to entry for researchers, improving computational efficiency, and enhancing the reproducibility of results.

This work investigates the construction of a divergence measure between Gaussian distributions in hyperbolic space that remains invariant under Möbius transformations. Building upon the Poincaré disk model, the study establishes a geometric duality between an L² embedding and the spherical squared Hellinger distance, thereby uncovering an intrinsic connection between this divergence and hyperbolic isometric invariants. Leveraging this insight, the authors propose a novel Gaussian divergence that is rigorously invariant under the action of the Möbius group. This contribution not only ensures robustness under hyperbolic isometries but also offers a fresh perspective and new geometric tools for information-theoretic divergence analysis by integrating principles from hyperbolic geometry.

This study addresses the inconsistent feature attribution and lack of geometry-aware interpretability caused by geometric operations in hyperbolic neural networks. We propose the LRP-radial-all rule, which leverages the Poincaré–Lorentz logarithmic map and inter-layer relevance propagation to handle the geometric scaling and signal branching within radial modules. By introducing geometric representation invariance (GRI) and a zero-curvature consistency criterion, our method overcomes the failure of traditional conservative propagation under equivalent computations, thereby ensuring relevance conservation and equivalence decomposition invariance. Experiments demonstrate that the proposed approach achieves high attribution fidelity on MNIST, sEEG, and CIFAR-10, with computational efficiency significantly surpassing Integrated Gradients and existing baseline methods.

This work addresses the limitations of conventional gradient-based optimization methods, which struggle to adapt to dynamic changes in length, curvature, and preconditioning implicitly induced by internal states under fixed geometric assumptions. The authors formulate optimization as a coupled system involving parameter trajectories, particle distributions, and a time-evolving Riemannian metric, explicitly distinguishing immutable obstacles from remediable geometric mismatches. They introduce the notion of “dynamic geometric complexity” and establish the first lower bound on geometric optimization difficulty based on affine-invariant distance. By leveraging gauge-invariant observables and Morse saddle-point flux analysis, they precisely characterize this complexity—in the setting of strongly convex quadratic objectives with a fully positive-definite metric oracle—as the affine-invariant distance from the relative logarithmic spectrum to the set of well-conditioned metrics.

本文解决了约束运动规划问题，通过引入诱导黎曼度量来统一不同表示方法下的路径长度计算，从而在采样规划和轨迹优化中实现一致的几何处理。