用于分层二分数据的双曲潜在位置模型
Hyperbolic Latent Position Models for Hierarchical Bipartite Data
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中文总结 AI 辅助
针对分层二分数据的特性，提出HELPI双曲潜在位置模型，分离几何交互与加性倾向，在模拟及两类真实数据上验证了模型的有效性与可解释性。
中文摘要 AI 辅助
二分二元数据常结合行与列的异质性及分层交互，例如学生按先修要求完成练习，或立法者对嵌套政策领域投票。固定维欧氏空间的体积仅随半径呈多项式增长，而分支分层结构呈指数增长，双曲空间契合这种增长特性，其根格罗莫夫乘积可度量共享亲缘关系。我们提出双曲二分交互数据潜在位置模型（HELPI），将两类对象置于双曲空间，同时分离几何交互与加性倾向。在无约束主效应下，双曲距离预测器与根格罗莫夫乘积预测器似然等价；径向深度被加性项吸收，留下根不变的投影分层信号作为识别的几何目标。我们建立了模型识别与后验收缩结果，并开发了针对二元结果的增广变分方法。模拟实验支持分层交互的恢复，并阐明了弱分支机制。在骏毅学院（Junyi Academy）数据中，课程锚点生成可解释的几何结构，时间预测接近项目反应基准，同时区分已尝试练习与无条件掌握情况；在美国众议院投票数据中，灵活的HELPI变体优于加性模型和两参数逻辑基准，且在拟合过程中无需使用政党标签即可恢复政党与政策结构。
英文摘要
Binary bipartite data often combine row and column heterogeneity with hierarchical interaction, as when students answer exercises organized by prerequisites or legislators vote on nested policy areas. Fixed-dimensional Euclidean volume grows only polynomially with radius, whereas branching hierarchies expand exponentially. Hyperbolic space matches this growth, and its rooted Gromov product measures shared ancestry. We propose the \emph{HypErbolic Latent Position model for bipartite Interaction data} (HELPI), which places both object types in hyperbolic space while separating geometric interaction from additive propensities. With unrestricted main effects, a hyperbolic-distance predictor is likelihood-equivalent to a rooted Gromov-product predictor. Radial depth is absorbed by additive terms, leaving a root-invariant projected hierarchy signal as the identified geometric target. We establish identification and posterior contraction results and develop augmented variational procedures for binary outcomes. Simulations support recovery of hierarchical interaction and clarify weak-branch regimes. In Junyi Academy data, curriculum anchors yield an interpretable geometry with temporal prediction close to item-response benchmarks, while distinguishing attempted exercises from unconditional mastery. In U.S. House roll calls, flexible HELPI variants improve on additive and two-parameter logistic benchmarks and recover party and policy structure without using party labels during fitting.
