# Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

arXiv: https://arxiv.org/abs/1811.12804 · HTML: https://mlsanity.com/paper/1811.12804/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2018.

## Built on by (most cited first, 10 of 10)

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- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Communication-Efficient Federated Low-Rank Update Algorithm and its Connection to Implicit Regularization](https://mlsanity.com/paper/2409.12371/) (2024)
- [Enhancing Accuracy in Deep Learning Using Random Matrix Theory](https://mlsanity.com/paper/2310.03165/) (2023)
- [Community detection with the Bethe-Hessian](https://mlsanity.com/paper/2411.02835/) (2024)
- [A simpler spectral approach for clustering in directed networks](https://mlsanity.com/paper/2102.03188/) (2021)
- [Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices](https://mlsanity.com/paper/2203.00821/) (2022)
- [On the Error-Propagation of Inexact Hotelling's Deflation for Principal Component Analysis](https://mlsanity.com/paper/2310.04283/) (2023)
- [Exploiting Low-Rank Objective Structure in Discrete Quadratic Optimization](https://mlsanity.com/paper/2602.20376/) (2026)

## Abstract (arXiv)

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix Mstar∈ ℝn× n, yet only a randomly perturbed version M is observed. The noise matrix M-Mstar is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of Mstar and arrange them into an em asymmetric data matrix M. The aim is to estimate the leading eigenvalue and eigenvector of Mstar. We demonstrate that the leading eigenvalue of the data matrix M can be O(√n) times more accurate --- up to some log factor --- than its (unadjusted) leading singular value in eigenvalue estimation. Further, the perturbation of any linear form of the leading eigenvector of M --- say, entrywise eigenvector perturbation --- is provably well-controlled. This eigen-decomposition approach is fully adaptive to heteroscedasticity of noise without the need of careful bias correction or any prior knowledge about the noise variance. We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be beneficial.