Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvector Under Random Perturbation: A Nonasymptotic
  Rayleigh-Schr\"{o}dinger Theory   Rayleigh-Schr\"{o}dinger perturbation theory is a well-known theory in
quantum mechanics and it offers useful characterization of eigenvectors of a
perturbed matrix. Suppose $A$ and perturbation $E$ are both Hermitian matrices,
$A^t = A + tE$, $\{\lambda_j\}_{j=1}^n$ are eigenvalues of $A$ in descending
order, and $u_1, u^t_1$ are leading eigenvectors of $A$ and $A^t$.
Rayleigh-Schr\"{o}dinger theory shows asymptotically, $\langle u^t_1, u_j
\rangle \propto t / (\lambda_1 - \lambda_j)$ where $ t = o(1)$. However, the
asymptotic theory does not apply to larger $t$; in particular, it fails when $
t \| E \|_2 > \lambda_1 - \lambda_2$. In this paper, we present a nonasymptotic
theory with $E$ being a random matrix. We prove that, when $t = 1$ and $E$ has
independent and centered subgaussian entries above its diagonal, with high
probability, \begin{equation*} | \langle u^1_1, u_j \rangle | = O(\sqrt{\log n}
/ (\lambda_1 - \lambda_j)), \end{equation*} for all $j>1$ simultaneously, under
a condition on eigenvalues of $A$ that involves all gaps $\lambda_1 -
\lambda_j$. This bound is valid, even in cases where $\| E \|_2 \gg \lambda_1 -
\lambda_2$. The result is optimal, except for a log term. It also leads to an
improvement of Davis-Kahan theorem.

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The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph