L-Infinity Eigenvector Perturbation Bound and Robust Factor Covariance Estimation

A deterministic entrywise ($\ell_\infty$) eigenvector and singular-vector perturbation theory and heavy-tailed factor covariance estimation framework (introduced by Jianqing Fan, Weichen Wang, and Yiqiao Zhong in 2018). For a rank-$r$ symmetric matrix $\mathbf{A} = \mathbf{V}\boldsymbol{\Lambda}\mathbf{V}^T \in \mathbb{R}^{d \times d}$ with orthonormal eigenvectors $\mathbf{V} = [\mathbf{v}_1, \dots, \mathbf{v}_r]$, coherence $\mu(\mathbf{V}) = (d/r)\max_i \sum_{j=1}^r V_{ij}^2$, eigengap $\gamma_0 = \min\{|\lambda_i - \lambda_j| : 1 \le i \ne j \le r+1\}$, and symmetric perturbation $\widetilde{\mathbf{A}} = \mathbf{A} + \mathbf{E}$ satisfying $\gamma_0 - \|\mathbf{E}\|_\infty = \Omega(r^3\mu^2\|\mathbf{E}\|_\infty)$, where $\|\mathbf{E}\|_\infty = \max_i \sum_{j=1}^d |E_{ij}|$ is the matrix $\ell_\infty$ operator norm, Theorem 1 (and Theorem 2.1) establishes the coordinatewise delocalized perturbation bound $\max_{1 \le k \le r}\|\widetilde{\mathbf{v}}_k - \operatorname{sgn}(\widetilde{\mathbf{v}}_k^T\mathbf{v}_k)\mathbf{v}_k\|_\infty = O\left(r^{5/2}\mu^{3/2}\,\frac{\|\mathbf{E}\|_\infty}{\gamma_0\sqrt{d}}\right)$, improving upon the Davis–Kahan $\ell_2$ bound by a factor of $1/\sqrt{d}$, while Theorems 2.2–2.4 extend the bound to rectangular low-rank matrices $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^T \in \mathbb{R}^{d_1 \times d_2}$ via asymmetric singular-vector norms $\|\mathbf{E}\|_{2,\infty}$ and approximate low-rank decompositions $\mathbf{A} = \mathbf{A}_r + \mathbf{A}_{\rm rem}$. Applying this $\ell_\infty$ eigenvector perturbation bound to a spiked approximate factor model $\mathbf{y}_i = \mathbf{B}\mathbf{f}_i + \boldsymbol{\varepsilon}_i$ with covariance $\boldsymbol{\Sigma} = \mathbf{B}\mathbf{B}^T + \boldsymbol{\Sigma}_u$, the framework constructs two robust high-dimensional factor covariance estimators under heavy tails: (1) an entrywise Catoni M-estimator $\widehat{\Sigma}_{jk}$ solving $\sum_{i=1}^n \psi(\alpha(Y_{ij}Y_{ik} - \widehat{\Sigma}_{jk})) = 0$ with $\psi(x) = \operatorname{sgn}(x)\log(1 + |x| + x^2/2)$ under finite fourth moments, followed by top-$r$ eigen-extraction and adaptive thresholding of the residual covariance (Section 3, Theorems 3.1–3.2); and (2) an elliptical-distribution estimator combining the eigenvectors $\widetilde{\mathbf{V}}^K$ of the spatial Kendall's tau matrix $\widetilde{\boldsymbol{\Sigma}}^K = \binom{n}{2}^{-1}\sum_{i < i'}(\mathbf{y}_i - \mathbf{y}_{i'})(\mathbf{y}_i - \mathbf{y}_{i'})^T / \|\mathbf{y}_i - \mathbf{y}_{i'}\|_2^2$ with Catoni variance and principal-eigenvalue estimators without requiring finite fourth moments for the eigenvector directions (Section 4, Theorems 4.1–4.2).