前k个样本特征向量的联合密度与主子空间推断

中文摘要 AI 辅助

本文针对任意p×p总体协方差矩阵Σ、样本量n>p-1以及1≤k≤p的情形，推导了前k个样本特征向量的精确联合密度。利用对称函数（如zonal多项式）以及I. G. Macdonald的对偶求和恒等式，结果以微分算子行列式的级数形式表达。矩阵Kummer变换将所得的局部交错展开转化为在整个正定矩阵空间上全局绝对收敛的级数。当k=2时，有序特征值积分可化简为Gauss超几何函数₂F₁，且在n=p+1时具有简单闭式形式。此前，显式公式仅适用于标量矩阵Σ=σ²I_p或一般矩阵Σ且p=2或k=1的情形，分别由T. W. Anderson和T. Sugiyama获得。该框架定律还导出了一个精确的Grassmann密度，为主子空间推断提供了有限样本基准。

英文摘要

In this paper, the exact joint density of the top $k$ sample eigenvectors is derived for any $p\times p$ population covariance matrix $\varSigma$, sample size $n > p -1$, and $1 \le k \le p$. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For $k=2$, the ordered eigenvalue integrals reduce to the Gauss hypergeometric function ${}_2F_1$ with a simple closed form when $n=p+1$. Previously, explicit formulas were only available for a scalar matrix $\varSigma = σ^2 I_p$ or a general matrix $\varSigma$ with $p=2$ or $k=1$, obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law also induces an exact Grassmann density that provides a finite-sample benchmark for principal subspace inference.

发表机构

• Tokyo University of Science（东京科学大学）
• Nankai University（南开大学）

机构由 AI 辅助整理，请以论文原文为准。