论文标题
应对小本本差距：异质噪声下的细粒特征向量估计和推断
Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise
论文作者
论文摘要
本文旨在解决在特征向量估计中引起的两个基本挑战，并从嘈杂的观察值中推断出低级别矩阵的推断：（1）如何估计当特征性gap时（即相关特征值和其他样品的其余部分之间的间隔）尤其小； （2）如何对特征向量的线性功能进行估计和推断 - 一种“细粒”统计推理远远超出了通常的$ \ ell_2 $分析。我们研究了如何在未知$ n \ times n $矩阵对称的环境中解决这些挑战，而添加噪声矩阵包含独立（和非对称）条目。基于非对称数据矩阵的本征分解，我们提出了未知特征向量的估计和不确定性量化程序，这进一步使我们能够理解未知特征向量的线性函数。提出的程序和随附的理论享有一些重要特征：（1）无分布（即不需要关于噪声分布的事先了解）； （2）适应异性噪声； （3）在高斯噪声下最佳最佳。在此过程中，我们建立了最佳程序，以构建未知特征值的置信区间。即使存在一个小的特征差距（最多$ o（\ sqrt {n/\ mathrm {poly} \ log（n）}）$次，这一切也比先前理论中的要求小得多），也可以保证所有这些。
This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: (1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; (2) how to perform estimation and inference on linear functionals of an eigenvector -- a sort of "fine-grained" statistical reasoning that goes far beyond the usual $\ell_2$ analysis. We investigate how to address these challenges in a setting where the unknown $n\times n$ matrix is symmetric and the additive noise matrix contains independent (and non-symmetric) entries. Based on eigen-decomposition of the asymmetric data matrix, we propose estimation and uncertainty quantification procedures for an unknown eigenvector, which further allow us to reason about linear functionals of an unknown eigenvector. The proposed procedures and the accompanying theory enjoy several important features: (1) distribution-free (i.e. prior knowledge about the noise distributions is not needed); (2) adaptive to heteroscedastic noise; (3) minimax optimal under Gaussian noise. Along the way, we establish optimal procedures to construct confidence intervals for the unknown eigenvalues. All this is guaranteed even in the presence of a small eigen-gap (up to $O(\sqrt{n/\mathrm{poly}\log (n)})$ times smaller than the requirement in prior theory), which goes significantly beyond what generic matrix perturbation theory has to offer.
