Bayesian inference for spectral projectors of covariance matrix

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 classic principal component analysis estimates the projector $\mathbf{P}^*_{\mathcal{J}}$ onto the direct sum of some eigenspaces of $\mathbf{\Sigma}^*$ by its empirical counterpart $\widehat{\mathbf{P}}_{\mathcal{J}}$. Recent papers [Koltchinskii, Lounici, 2017], [Naumov et al., 2017] investigate the asymptotic distribution of the Frobenius distance between the projectors $\| \widehat{\mathbf{P}}_{\mathcal{J}} - \mathbf{P}^*_{\mathcal{J}} \|_2$. The problem arises when one tries to build a confidence set for the true projector effectively. We consider the problem from a Bayesian perspective and derive an approximation for the posterior distribution of the Frobenius distance between projectors. The derived theorems hold true for non-Gaussian data: the only assumption that we impose is the concentration of sample covariance $\widehat{\mathbf{\Sigma}}$ in a vicinity of $\mathbf{\Sigma}^*$. The obtained results are applied to construction of sharp confidence sets for the true projector. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in quite challenging regimes.