Spectral norm posterior contraction in Bayesian sparse spiked covariance matrix model

Abstract

Posterior contraction rates with regard to non-intrinsic metrics have been a long-standing challenge in the Bayesian analysis of high-dimensional models. This paper establishes the minimax-optimal posterior contraction rates of the Bayesian sparse spiked covariance matrix model under the spectral norm, a non-intrinsic metric for the Gaussian covariance matrix model. Our proof technique relies on the recent advance in the geometric properties of Euclidean representation for subspaces and low-rank matrices, a local asymptotic normality argument, and the distributional approximation to the asymptotic posterior distribution of the sparse spiked covariance matrix.