发表机构
University of California, Los Angeles; University of California, Santa Barbara（加州大学洛杉矶分校; 加州大学圣塔芭芭拉分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明在高维低样本量尖峰协方差模型中，尽管行格拉姆矩阵可逐模型一致估计，但完整的特征向量对齐矩阵无法在嵌套路径上实现均匀可靠的几乎必然恢复。
AI 中文摘要
在尖峰协方差模型中，设$\mathcal{H}$和$\mathcal{B}$分别为选定的样本和总体特征向量矩阵。它们的对齐矩阵$\Xi=\mathcal{H}^\top\mathcal{B}$是超越子空间重叠的方向特定对应的潜在神谕诊断量。我们在一个嵌套的高维低样本量实验中研究其均匀逐路径恢复，其中样本量固定而维度增长。在每个固定模型下，一个可观测的谱对比度能一致地估计行格拉姆矩阵$\Xi\Xi^\top$。然而，对于两个或更多尖峰，不存在任何可测的估计量序列能在具有共同尖峰强度和共同信号子空间但不同有序特征框架的模型族上，对$\Xi$实现均匀可靠的几乎必然逐路径恢复。对于每个$0<\epsilon<1/\sqrt{2}$，一个有限轨道迫使每个估计量在某个轨道点上，其近似路径恢复概率至多为一个变分界$u(\epsilon)$，其中当$\epsilon$趋于零时$u(\epsilon)$趋于零。在可数轨道上的最坏模型精确恢复概率为零。因此，格拉姆摘要可以逐模型恢复，而完整的对齐矩阵无法沿嵌套路径均匀恢复。
英文摘要
In a spiked covariance model, let $\mathcal{H}$ and $\mathcal{B}$ be selected sample and population eigenvector matrices. Their alignment matrix $Ξ=\mathcal{H}^\top\mathcal{B}$ is a latent oracle diagnostic of direction-specific correspondence beyond subspace overlap. We study its uniform pathwise recovery in a nested high-dimensional, low-sample-size experiment with fixed sample size and growing dimension. At every fixed model, an observable spectral contrast consistently estimates the row Gram matrix $ΞΞ^\top$. For two or more spikes, however, no measurable estimator sequence achieves uniformly reliable almost-sure path recovery of $Ξ$ over a family having common spike strengths and a common signal subspace but different ordered eigenframes. For every $0<ε<1/\sqrt{2}$, a finite orbit forces each estimator to have some orbit point at which the approximate path-recovery probability is at most a variational bound $u(ε)$, where $u(ε)$ tends to zero as $ε$ tends to zero. The worst-model exact-recovery probability over a countable orbit is zero. Thus the Gram summary is recoverable model by model, while the full alignment matrix is not uniformly recoverable along the nested path.
Comments24 pages. Presented at the SIAM Conference on Financial Mathematics and Engineering (FM25), Florida, 2025
