mlsanityData to 7 Oct 2026

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

arXiv 2104.03298 ↗First posted 7 Apr 2021math.ST cs.IT cs.LGcited by 1

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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.

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

Builds on

14 of 61 references.

References · 14

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2 more
  1. 2018Confidence Region of Singular Subspaces for Low-rank Matrix Regressioncited by 1
  2. 2020Partial Recovery for Top-k Ranking: Optimality of MLE and Sub-Optimality of Spectral Methodcited by 1

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Cited by year

Citing papers per year: 2021: 1.

1 paper.

  1. 2021Inference for Heteroskedastic PCA with Missing Datacited by 21