Normal Approximation and Confidence Region of Singular Subspaces

TL;DR

This paper develops a non-asymptotic normal approximation framework for singular subspaces under i.i.d. noise, providing explicit formulas, bias corrections, and simulation validation, applicable to high-dimensional matrices without eigen-gap conditions.

Contribution

It introduces a novel explicit spectral projector formula, analyzes the expected projection distance with higher-order approximations, and establishes non-asymptotic normality under minimal conditions.

Findings

Explicit spectral projector formula valid for deterministic perturbations

Normal approximation of projection distance with bias corrections

Applicable to diverging rank and no eigen-gap requirement

Abstract

This paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate the expected projection distance between the empirical singular subspaces and true singular subspaces. Our method allows obtaining arbitrary $k$-th order approximation of the expected projection distance. Third, we prove the non-asymptotical normal approximation of the projection distance with different levels of bias corrections. By the $\lceil \log(d_1+d_2)\rceil$-th order bias corrections, the asymptotical normality holds under optimal signal-to-noise ration (SNR) condition where $d_1$ and $d_2$ denote the matrix sizes. In addition, it shows that higher order approximations are unnecessary when $|d_1-d_2|=O((d_1+d_2)^{1/2})$. Finally, we provide comprehensive simulation results to merit our theoretic discoveries. Unlike the existing results, our approach is non-asymptotical and the convergence rates are established. Our method allows the rank $r$ to diverge as fast as $o((d_1+d_2)^{1/3})$. Moreover, our method requires no eigen-gap condition (except the SNR) and no constraints between $d_1$ and $d_2$.