  High-dimensional compositional data arise naturally in many applications such
as metagenomic data analysis. The observed data lie in a high-dimensional
simplex, and conventional statistical methods often fail to produce sensible
results due to the unit-sum constraint. In this article, we address the problem
of covariance estimation for high-dimensional compositional data, and introduce
a composition-adjusted thresholding (COAT) method under the assumption that the
basis covariance matrix is sparse. Our method is based on a decomposition
relating the compositional covariance to the basis covariance, which is
approximately identifiable as the dimensionality tends to infinity. The
resulting procedure can be viewed as thresholding the sample centered log-ratio
covariance matrix and hence is scalable for large covariance matrices. We
rigorously characterize the identifiability of the covariance parameters,
derive rates of convergence under the spectral norm, and provide theoretical
guarantees on support recovery. Simulation studies demonstrate that the COAT
estimator outperforms some naive thresholding estimators that ignore the unique
features of compositional data. We apply the proposed method to the analysis of
a microbiome dataset in order to understand the dependence structure among
bacterial taxa in the human gut.
