Normal approximation and confidence region of singular subspaces

This paper is on the normal approximation of singular subspaces when noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula empirical spectral projectors. The neat and holds for deterministic perturbations. Second, calculate expected projection distance between true subspaces. method allows obtaining arbitrary k-th order distance. Third, prove non-asymptotical with different levels bias corrections. By ?log(d1+d2)?-th corrections, asymptotical normality under optimal signal-to-noise ratio (SNR) condition where d1 d2 denote sizes. In addition, it shows that higher approximations unnecessary |d1?d2|=O((d1+d2)1?2). Finally, provide comprehensive simulation results to merit our theoretic discoveries. Unlike existing results, approach convergence rates established. rank r diverge as fast o((d1+d2)1?3). Moreover, requires no eigen-gap (except SNR) constraints d2.