Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimax Estimation of Linear Functions of Eigenvectors in the Face of
  Small Eigen-Gaps   Eigenvector perturbation analysis plays a vital role in various statistical
data science applications. A large body of prior works, however, focused on
establishing $\ell_{2}$ 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 substantially smaller than what is
required in prior theory.

34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio