  In this paper, we propose a new Bayesian inference method for a
high-dimensional sparse factor model that allows both the factor dimensionality
and the sparse structure of the loading matrix to be inferred. The novelty is
to introduce a certain dependence between the sparsity level and the factor
dimensionality, which leads to adaptive posterior concentration while keeping
computational tractability. We show that the posterior distribution
asymptotically concentrates on the true factor dimensionality, and more
importantly, this posterior consistency is adaptive to the sparsity level of
the true loading matrix and the noise variance. We also prove that the proposed
Bayesian model attains the optimal detection rate of the factor dimensionality
in a more general situation than those found in the literature. Moreover, we
obtain a near-optimal posterior concentration rate of the covariance matrix.
Numerical studies are conducted and show the superiority of the proposed method
compared with other competitors.
