Bayesian inference for spectral projectors of covariance matrix
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Let X_1, ..., X_n be i.i.d. sample in R^p with zero mean and the covariance matrix Σ^*. The classic principal component analysis estimates the projector P^*_J onto the direct sum of some eigenspaces of Σ^* by its empirical counterpart P_J. Recent papers [Koltchinskii, Lounici (2017)], [Naumov et al. (2017)] investigate the asymptotic distribution of the Frobenius distance between the projectors P_J - P^*_J_2. The problem arises when one tries to build a confidence set for the true projector effectively. We consider the problem from 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 the sample covariance Σ in a vicinity of Σ^*. 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 regime.
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