Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on outlier eigenvectors for sparse non-Hermitian perturbations

Miltiadis Galanis, Michail Louvaris

Abstract

We consider a sparse i.i.d.\ non-Hermitian random matrix model $X_n$ (with sparsity parameter $K_n$) and a deterministic finite-rank perturbation $E_n$. Assuming biorthogonality for $E_n$ and a growth condition on $K_n$, we outline a finite-rank resolvent reduction leading to asymptotics for the overlap between an outlier eigenvector of $Y_n:=X_n+E_n$ and the corresponding spike eigenspace. In particular, for an outlier spike $μ$ with $|μ|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|μ|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.

A note on outlier eigenvectors for sparse non-Hermitian perturbations

Abstract

We consider a sparse i.i.d.\ non-Hermitian random matrix model (with sparsity parameter ) and a deterministic finite-rank perturbation . Assuming biorthogonality for and a growth condition on , we outline a finite-rank resolvent reduction leading to asymptotics for the overlap between an outlier eigenvector of and the corresponding spike eigenspace. In particular, for an outlier spike with , the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to . Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.
Paper Structure (7 sections, 11 theorems, 81 equations)

This paper contains 7 sections, 11 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notation
  4. Model
  5. Tools from Linear Algebra
  6. Results on bilinear forms of the resolvent of $X_n$.
  7. Proof of Theorem \ref{['main_thm']}

Key Result

Theorem 2.1

[Theorem 1.2 of hachem2026extreme] Assume that and that Assumption ass:E-bounded holds true. Define and let $m_n=|\sigma^+(E_n)|$. Then For each sequence $(n')$ with $n'\to\infty$ and $m_{n'}>0$ for all $n'$, with the convention $d_{\boldsymbol H}(\emptyset,\sigma^+(E^{n'}))=\infty$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 3.1: Finite-rank reduction: kernel--eigenspace bijection
  • proof
  • Corollary 3.2: Closed-form representation of the unit outlier eigenvector
  • ...and 15 more