# Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps

arXiv: https://arxiv.org/abs/2104.03298 · HTML: https://mlsanity.com/paper/2104.03298/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 61 references that are in the map.
- First posted on arXiv 7 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors](https://mlsanity.com/paper/1512.02337/) (2015)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Near-optimal bounds for phase synchronization](https://mlsanity.com/paper/1703.06605/) (2017)
- [Spectral algorithms for tensor completion](https://mlsanity.com/paper/1612.07866/) (2016)
- [Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries](https://mlsanity.com/paper/1711.04934/) (2017)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [High-dimensional principal component analysis with heterogeneous missingness](https://mlsanity.com/paper/1906.12125/) (2019)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise](https://mlsanity.com/paper/2001.04620/) (2020)
- [An ℓp theory of PCA and spectral clustering](https://mlsanity.com/paper/2006.14062/) (2020)
- [Confidence Region of Singular Subspaces for Low-rank Matrix Regression](https://mlsanity.com/paper/1805.09871/) (2018)
- [Partial Recovery for Top-k Ranking: Optimality of MLE and Sub-Optimality of Spectral Method](https://mlsanity.com/paper/2006.16485/) (2020)

## Built on by (most cited first, 1 of 1)

- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)

## Abstract (arXiv)

Eigenvector perturbation analysis plays a vital role in various data science applications. A large body of prior works, however, focused on establishing ℓ2 eigenvector perturbation bounds, which are often highly inadequate in addressing tasks that rely on fine-grained behavior of an eigenvector. This paper makes progress on this by studying the perturbation of linear functions of an unknown eigenvector. Focusing on two fundamental problems -- matrix denoising and principal component analysis -- in the presence of Gaussian noise, we develop a suite of statistical theory that characterizes the perturbation of arbitrary linear functions of an unknown eigenvector. In order to mitigate a non-negligible bias issue inherent to the natural ``plug-in'' estimator, we develop de-biased estimators that (1) achieve minimax lower bounds for a family of scenarios (modulo some logarithmic factor), and (2) can be computed in a data-driven manner without sample splitting. Noteworthily, the proposed estimators are nearly minimax optimal even when the associated eigen-gap is em substantially smaller than what is required in prior statistical theory.