# Unperturbed: spectral analysis beyond Davis-Kahan

arXiv: https://arxiv.org/abs/1706.06516 · HTML: https://mlsanity.com/paper/1706.06516/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2017.

## Built on by (most cited first, 16 of 16)

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- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization](https://mlsanity.com/paper/2112.13199/) (2021)
- [Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA](https://mlsanity.com/paper/2506.12655/) (2025)
- [Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models](https://mlsanity.com/paper/2306.06845/) (2023)
- [On the Error-Propagation of Inexact Hotelling's Deflation for Principal Component Analysis](https://mlsanity.com/paper/2310.04283/) (2023)
- [On the Power of SVD in the Stochastic Block Model](https://mlsanity.com/paper/2309.15322/) (2023)
- [Multi-Irreducible Spectral Synchronization for Robust Rotation Averaging](https://mlsanity.com/paper/2311.16544/) (2023)
- [Entrywise error bounds for low-rank approximations of kernel matrices](https://mlsanity.com/paper/2405.14494/) (2024)
- [On Kernel Eigen-alignments of KRR: Reconstruction and Generalization](https://mlsanity.com/paper/2605.15240/) (2026)

## Abstract (arXiv)

Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the typical case. In this paper, we present perturbation bounds which consider the nature of the perturbation and its interaction with the unperturbed structure in order to obtain significant improvements over the classical theory in many scenarios, such as when the perturbation is random. We demonstrate the utility of these new results by analyzing perturbations in the stochastic blockmodel where we derive much tighter bounds than provided by the classical theory. We use our new perturbation theory to show that a very simple and natural clustering algorithm -- whose analysis was difficult using the classical tools -- nevertheless recovers the communities of the blockmodel exactly even in very sparse graphs.