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Davis-Kahan Theorem under a moderate gap condition

Phuc Tran, Van Vu

TL;DR

This work addresses the perturbation of eigenspaces for a real symmetric matrix under a moderate spectral gap, where the classical Davis–Kahan bound becomes conservative. It introduces a contour bootstrapping approach that reduces the perturbation estimate to a tractable contour-integral term F1, yielding a sharp bound that explicitly depends on the perturbation magnitude ||E||, the leading eigenvalue |λ_p|, the gap δ_p, and the interaction x = max_{i,j≤r} |u_i^T E u_j|, with sharpness up to a logarithmic factor. The main result, Theorem main1, provides a concrete bound ||̃Π_p − Π_p|| ≤ 24 [ (||E||/|λ_p|) log(6 σ1/δ_p) + (r^2 x)/δ_p ], and extensions to leading singular spaces and non-symmetric settings are discussed. The paper also demonstrates a practical application to fast computation under random sparsification, showing improved leading-eigenvector accuracy when the sparsified matrix preserves key spectral properties, and it links the theory to stability under random perturbations with explicit probabilistic bounds. Overall, the methodology broadens perturbation analysis in moderate-gap regimes and offers actionable insights for data-science tasks involving noisy, large-scale matrices.

Abstract

The classical Davis-Kahan theorem provides an efficient bound on the perturbation of eigenspaces of a matrix under a large (eigenvalue) gap condition. In this paper, we consider the case when the gap is moderate. Using a bootstrapping argument, we obtain a new bound which is efficient when the perturbation matrix is uncorrelated to the ground matrix. We believe that this bound is sharp up to a logarithmic term.

Davis-Kahan Theorem under a moderate gap condition

TL;DR

This work addresses the perturbation of eigenspaces for a real symmetric matrix under a moderate spectral gap, where the classical Davis–Kahan bound becomes conservative. It introduces a contour bootstrapping approach that reduces the perturbation estimate to a tractable contour-integral term F1, yielding a sharp bound that explicitly depends on the perturbation magnitude ||E||, the leading eigenvalue |λ_p|, the gap δ_p, and the interaction x = max_{i,j≤r} |u_i^T E u_j|, with sharpness up to a logarithmic factor. The main result, Theorem main1, provides a concrete bound ||̃Π_p − Π_p|| ≤ 24 [ (||E||/|λ_p|) log(6 σ1/δ_p) + (r^2 x)/δ_p ], and extensions to leading singular spaces and non-symmetric settings are discussed. The paper also demonstrates a practical application to fast computation under random sparsification, showing improved leading-eigenvector accuracy when the sparsified matrix preserves key spectral properties, and it links the theory to stability under random perturbations with explicit probabilistic bounds. Overall, the methodology broadens perturbation analysis in moderate-gap regimes and offers actionable insights for data-science tasks involving noisy, large-scale matrices.

Abstract

The classical Davis-Kahan theorem provides an efficient bound on the perturbation of eigenspaces of a matrix under a large (eigenvalue) gap condition. In this paper, we consider the case when the gap is moderate. Using a bootstrapping argument, we obtain a new bound which is efficient when the perturbation matrix is uncorrelated to the ground matrix. We believe that this bound is sharp up to a logarithmic term.
Paper Structure (10 sections, 14 theorems, 77 equations, 1 algorithm)

This paper contains 10 sections, 14 theorems, 77 equations, 1 algorithm.

Table of Contents

  1. The classical Davis-Kahan Theorem
  2. New results
  3. The contour bootstrapping argument and a key lemma
  4. The Contour Bootstrapping Argument
  5. Estimating $F_1$ and the proof of Theorem \ref{['main1']}
  6. An application: Fast computation with random sparsification
  7. Computing the leading eigenvector
  8. Proofs of the lemmas
  9. Proofs of Lemma \ref{['F1f1M2bound']} and Lemma \ref{['F1f1M3bound']}
  10. Proof of Lemma \ref{['F1f1M1bound']}

Key Result

Theorem 1.1

Suppose that $\text{dist}\,(\Lambda_S, \tilde{\Lambda}_{S^c}) = \delta > 0$. Then, Here $\text{dist}\,(\Lambda_S, \tilde{\Lambda}_{S^c}):= \min_{\lambda_i \in \Lambda_S, \tilde{\lambda}_j \in \tilde{\Lambda}_{S^c}} |\lambda_i- \tilde{\lambda}_j|$.

Theorems & Definitions (20)

  • Theorem 1.1: Davis-Kahan DKoriginalBook1
  • Remark 1.0.1
  • Corollary 1.1
  • Corollary 1.2
  • Remark 1.0.2
  • Theorem 2.1
  • Remark 2.0.1
  • Lemma 2.0.1
  • Corollary 2.1
  • Theorem 2.2
  • ...and 10 more