球面上加权经验测度的几何信息分解
Geometric Information Decomposition for Weighted Empirical Measures on the Sphere
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中文总结 AI 辅助
研究单位球面上加权概率测度的方向不确定性，传统方法用von Mises-Fisher分布不完整，引入几何信息分解（GID），通过球面特征拟合最大熵投影序列，证明相关性质，实验展示不同情况下GID作用。
中文摘要 AI 辅助
我们研究当数据已表示单位球面上的加权概率测度时的方向不确定性，如在重要性样本、求积规则或注意力加权嵌入中。一种标准方法拟合von Mises-Fisher分布并报告其集中度或熵。这是有原则的但不完整，因为vMF仅使用平均方向信息，可能错过对映、轴向、带状或多峰结构。我们引入几何信息分解（GID），它使用球面特征拟合最大熵投影的嵌套序列，并报告在每个级别添加的熵差距。第一个差距恢复vMF信息，第二个捕捉Fisher-Bingham/Bingham型各向异性，后续差距捕捉更精细的角度结构。我们证明了不变性、一致性、远离零差距时的渐近正态性，以及用于确定新级别是否携带信息的二次型零校准。在圆形和球形示例上的实验、校准研究以及查询加权数字投影展示了vMF不确定性何时足够，以及高阶差距何时揭示隐藏结构。
英文摘要
Weighted observations on the unit sphere arise in importance sampling, quadrature, and attention-weighted embeddings. Directional uncertainty is often summarized through a von Mises-Fisher (vMF) fit and its concentration or entropy. This summary uses only mean-direction information. It can miss antipodal, axial, girdle-like, or multimodal structure. We introduce geometric information decomposition (GID), which fits nested maximum-entropy projections to spherical features. Each gap measures the entropy reduction contributed by one feature level. The first gap is the fitted vMF distribution's KL divergence from uniformity. The second measures residual quadratic information, including Fisher-Bingham anisotropy. Later gaps describe finer angular structure. We establish invariance, consistency, alternative-regime asymptotic normality, and quadratic-form null calibration. Circular and spherical experiments include importance-weight calibration and a query-weighted digit projection. The results separate settings where vMF uncertainty is adequate from settings with higher-order structure.
