高维中的相关矩阵：作为样本相关集合的椭圆体（elliptope）
Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes
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中文总结 AI 辅助
该研究分析高维相关矩阵集合（椭圆体）的体积衰减特性，刻画均匀样本的分布特征，推导LKJ分布下的两两独立性，证明极端相关性点过程极限，确定影响谱和Frobenius距离的尺度，并给出投影到最近相关矩阵的平方修复成本下界。
中文摘要 AI 辅助
n×n相关矩阵的集合被称为椭圆体（elliptope），其体积以超指数速率exp{-¼n²log n}衰减。我们刻画了这种消失的体积集中的位置。均匀抽取的样本在元素层面接近单位矩阵，但在全局层面远离单位矩阵且近乎奇异：其最大绝对相关性的阶为√(log n / n)，其Frobenius距离渐近于√n，其经验谱分布收敛到比率为1的Marchenko-Pastur定律，其最小特征值具有精确的Beta(1,d)分布，其中d = n(n-1)/2，因此其阶为n⁻²。更一般地，在每个LKJ(η)分布下，不同的非对角元素恰好是两两独立的。对于均匀分布，这得到了极端相关性点过程极限的Chen-Stein证明，以及有限维超出计数相对于具有精确有限n均值的泊松定律的O(n⁻¹)总变差界。我们还确定了两个不同的尺度：ηₙ ≍ n会改变极限谱，而ηₙ ≍ n²是保持Frobenius距离有界所必需的。最后，对于有界、中心化的独立同分布非对角指定，投影到最近的相关矩阵会产生的平方修复成本渐近至少为其非对角部分的平方Frobenius范数的一半。
英文摘要
The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. Distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(η)$ law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.
发表机构
- University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。
