高维矩阵Bingham分布与矩阵Langevin分布归一化常数的完全渐近展开
Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions
AI总结：
本文在高维极限下推导了Stiefel流形上矩阵Bingham分布与矩阵Langevin分布归一化常数的完全渐近展开，证明了截断精度随维数和项数递增，并给出下界与收敛速率，推广了高维Bingham分布的已有结果。
摘要：
对于满足 $d \ge p$ 的正整数 $d$ 和 $p$，令 $\mathbb{R}^{d \times p}$ 表示 $d \times p$ 实矩阵的集合，$I_p$ 为 $p$ 阶单位矩阵，$V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ 为 $\mathbb{R}^{d \times p}$ 中的Stiefel流形。本文在 $d \to \infty$ 时，得到了 $V_{d,p}$ 上矩阵Bingham概率分布和矩阵Langevin概率分布的归一化常数的完全渐近展开。每个截断展开的精度随 $d$ 严格递增；此外，当 $d$ 足够大时，精度随截断展开中项数 $m$ 严格递增。当矩阵Bingham分布的矩阵参数为正定矩阵、矩阵Langevin分布的矩阵参数为满秩矩阵时，得到了截断展开的下界。这些结果被应用于获得当 $d \to \infty$ 和 $p \to \infty$ 同时成立时渐近展开的收敛速率。利用众多数据集中出现的 $d$ 和 $p$ 值，说明了截断近似随 $d$ 或 $m$ 增大时的收敛速率。这些结果推广了最近得到的高维Bingham分布归一化常数的渐近展开。
英文摘要：
For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
