Title: Bootstrap confidence sets for spectral projectors of sample covariance,
Abstract: Let $X_{1},\ldots,X_{n}$ be i.i.d. sample in $\mathbb{R}^{p}$ with zero mean
and the covariance matrix $\mathbf{\Sigma}$. The problem of recovering the
projector onto an eigenspace of $\mathbf{\Sigma}$ from these observations
naturally arises in many applications. Recent technique from [Koltchinskii,
Lounici, 2015] helps to study the asymptotic distribution of the distance in
the Frobenius norm $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ between
the true projector $\mathbf{P}_r$ on the subspace of the $r$-th eigenvalue and
its empirical counterpart $\widehat{\mathbf{P}}_r$ in terms of the effective
rank of $\mathbf{\Sigma}$. This paper offers a bootstrap procedure for building
sharp confidence sets for the true projector $\mathbf{P}_r$ from the given
data. This procedure does not rely on the asymptotic distribution of $\|
\mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ and its moments. It could be
applied for small or moderate sample size $n$ and large dimension $p$. The main
result states the validity of the proposed procedure for finite samples with an
explicit error bound for the error of bootstrap approximation. This bound
involves some new sharp results on Gaussian comparison and Gaussian
anti-concentration in high-dimensional spaces. Numeric results confirm a good
performance of the method in realistic examples.