发表机构
Peking University; Wuhan University（北京大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用积分表示推导Bingham分布归一化常数及其导数的渐近级数，分析矩与熵的渐近行为，并证明熵分解为对数主项与Lipschitz修正项，揭示高维退化现象。
AI 中文摘要
Bingham分布广泛用于建模具有对跖对称性的方向数据，其核心分析对象是归一化常数$Z$。尽管对于有界或中等参数值存在高效算法，但其在一般维度下的奇异行为在文献中仍未得到充分探索。我们利用一个众所周知的积分表示，获得了$Z$及其导数的渐近级数，进而得到其矩的渐近轮廓。随后，我们还研究了Bingham熵的渐近行为，并证明其可分解为一个显式的对数主项和一个Lipschitz连续修正项。高维Bingham分布退化为低维对应分布的现象在渐近级数和熵中均被观察到。
英文摘要
The Bingham distribution is widely used to model directional data with antipodal symmetry, and its normalizing constant $Z$ and moments play a central role in statistical inference and closure models. Their behavior becomes singular when one or more eigenvalue gaps of the parameter matrix become unbounded. In this work, we develop a uniform asymptotic framework for the Bingham distribution in arbitrary dimensions. Starting from an inverse-Laplace integral representation, we derive asymptotic expansions with explicit remainder estimates that are uniform with respect to arbitrary relative scales among multiple diverging eigenvalues. In a particular regime, the asymptotic series becomes absolutely convergent with an exponentially small remainder. These results yield precise asymptotic profiles of the Bingham moments and characterize the degeneration of higher-dimensional distributions to lower-dimensional counterparts. As an application to the Bingham closure, we prove that the entropy admits a decomposition into an explicit logarithmic leading term and a residual that is uniformly Lipschitz continuous on the moment simplex. Moreover, on each boundary face, the residual agrees with its lower-dimensional version up to an explicit additive constant. The results provide a unified description of the singular structure of the Bingham distribution and have implications for numerical closure and $Q$-tensor models of nematic liquid crystals.
