谱隙对非线性特征向量问题逆迭代收敛性的依赖关系

Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise

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 …

Optimal Convergence Rate of Self-Consistent Field Iteration for Solving Eigenvector-dependent Nonlinear Eigenvalue Problems

We present a comprehensive convergence analysis for Self-Consistent Field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector-dependency (NEPv). Using a tangent-angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that optimally characterize the local convergence of the plain SCF: the local contraction factor and the local average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an appli…

Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations

Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $γ\in (0,1)$. We show that with probability $0.99$, $A + γG_n$ has all of its eigenvalue condition numbers bounded by $O\left(n^{5/2}/γ^{3/2}\right)$ and eigenvector condition number bounded by $O\left(n^3 /γ^{3/2}\right)$. Furthermore, we show that for any $s > 0$, the probability that $A + γG_n$ has two eigenvalues within distance at most $s$ of each other is $O\left(n^4 s^{1/3}/γ^{5/2}\right).$ In fact, we show the above statements hold in the mo…

随机扰动下的特征向量：一种非渐近的雷利-施鲁德inger理论

Eigenvector Under Random Perturbation: A Nonasymptotic Rayleigh-Schrödinger Theory

本文提出非渐近的雷利-施鲁德inger理论，研究随机矩阵扰动下特征向量的性质，改进了戴维斯-卡汉定理。

Re-Analyze Gauss: Bounds for Private Matrix Approximation via Dyson Brownian Motion

Given a symmetric matrix $M$ and a vector $λ$, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating $M$ by a matrix whose spectrum is $λ$, under $(\varepsilon,δ)$-differential privacy. Our bounds depend on both $λ$ and the gaps in the eigenvalues of $M$, and hold whenever the top $k+1$ eigenvalues of $M$ have sufficiently large gaps. When applied to the problems of private rank-$k$ covariance matrix approximation and subspace recovery, our bounds yield improvements over previous bounds. Our bounds are obtained by viewing the addition of Gaussian n…

Rosenbrock系统的特征值向后误差与Rayleigh商之和的优化

Eigenvalue backward errors of Rosenbrock systems and optimization of sums of Rayleigh quotient

本文将Rosenbrock系统矩阵在块扰动下的特征值向后误差归结为SRQ2优化，通过联合数值域与NEPv刻画实现更高效可靠的求解。