  We study the sparse high-dimensional Gaussian mixture model when the number
of clusters is allowed to grow with the sample size. A minimax lower bound for
parameter estimation is established, and we show that a constrained maximum
likelihood estimator achieves the minimax lower bound. However, this
optimization-based estimator is computationally intractable because the
objective function is highly nonconvex and the feasible set involves discrete
structures. To address the computational challenge, we propose a Bayesian
approach to estimate high-dimensional Gaussian mixtures whose cluster centers
exhibit sparsity using a continuous spike-and-slab prior. Posterior inference
can be efficiently computed using an easy-to-implement Gibbs sampler. We
further prove that the posterior contraction rate of the proposed Bayesian
method is minimax optimal. The mis-clustering rate is obtained as a by-product
using tools from matrix perturbation theory. The proposed Bayesian sparse
Gaussian mixture model does not require pre-specifying the number of clusters,
which can be adaptively estimated via the Gibbs sampler. The validity and
usefulness of the proposed method is demonstrated through simulation studies
and the analysis of a real-world single-cell RNA sequencing dataset.
