发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在分层分类模型中，对比庞加莱球与欧氏空间作为树结构原型网络的潜在流形，发现双曲流形能更好保留近邻图拓扑，在局部检索任务上优于欧氏流形，为两类潜在几何提供经验区分。
AI 中文摘要
我们研究分层分类模型中类原型布局上的树结构正则化项，并探究原型的潜在流形选择（欧氏空间R^d与庞加莱球B^d_c）是否会影响在不扭曲数据似然的前提下满足该正则化项的程度。两种流形仅在体积增长上存在差异：双曲空间的体积随半径呈指数增长，且嵌入树时的失真度在匹配维度下远低于R^d，因此在B^d_c上满足该结构化正则化项的成本应更低。我们在WikiArt数据集（含27种风格、81446幅绘画，使用冻结的CLIP ViT-B/16特征）上，针对嵌入维度、曲率和正则化强度，开展了150次种子重复的正则化最大似然拟合实验，发现一个稳健效应：庞加莱原型能显著更好地保留潜在空间中近邻图的拓扑结构，相较于匹配维度的欧氏原型，其兄弟召回率@5提升8.7个百分点，表兄弟召回率提升15.2个百分点；配对t检验p值小于10^-4，符号一致性为0.94，且该差距在三种参考树定义（人工构建谱系、CLIP衍生、DINOv2衍生）中均存在。分类任务上，欧氏原型与基于原始编码器特征的逻辑回归表现相当，表明潜在几何未产生可检测的贡献；仅双曲拟合在局部检索任务上优于k-NN编码器基线。全局树保真度比较在不同参考树间不稳定，故不判定胜负。这些结果在真实分层分类问题上，为类结构化正则化项的两种自然潜在几何提供了经验性区分。
英文摘要
We study a tree-structured regularizer over class-prototype layouts in a hierarchical-classification model and ask whether the choice of latent manifold for the prototypes (Euclidean R^d vs. the Poincare ball B^d_c) affects how well that regularizer can be satisfied without distorting the data likelihood. The two manifolds differ only in their volume growth: hyperbolic space grows exponentially with radius and embeds trees with provably lower distortion than R^d of matched dimension, so the structured regularizer should be cheaper to satisfy on B^d_c. Across 150 seed-replicated regularized maximum-likelihood fits spanning embedding dimension, curvature, and regularizer strength on WikiArt (27 styles, 81,446 paintings, frozen CLIP ViT-B/16 features), we find a single robust effect: Poincare prototypes preserve the topology of the nearest-neighbor graph in latent space substantially better than matched Euclidean prototypes (sibling recall@5 +8.7 pp, cousin recall +15.2 pp; paired-t p < 10^-4, sign agreement 0.94), and the gap holds across three reference-tree definitions (hand-built lineage, CLIP-derived, and DINOv2-derived). On classification, Euclidean prototypes are tied with logistic regression on raw encoder features, indicating no detectable contribution from the latent geometry; only the hyperbolic fit improves on a k-NN encoder baseline for local retrieval. Global tree-fidelity comparisons are unstable across reference trees and we do not claim a winner. The results give an empirical separation, on a real hierarchical-classification problem, between two natural latent geometries for a class-structured regularizer.
Comments12 pages, 13 figures, 1 table (including appendix). Code, sweep CSVs, and figure-generation scripts: https://github.com/pgrindehollevik-harvard/hyperbolic
