  Compositional data arise in many areas of research in the natural and
biomedical sciences. One prominent example is in the study of the human gut
microbiome, where one can measure the relative abundance of many distinct
microorganisms in a subject's gut. Often, practitioners are interested in
learning how the dependencies between microbes vary across distinct populations
or experimental conditions. In statistical terms, the goal is to estimate a
covariance matrix for the (latent) log-abundances of the microbes in each of
the populations. However, the compositional nature of the data prevents the use
of standard estimators for these covariance matrices. In this article, we
propose an estimator of multiple covariance matrices which allows for
information sharing across distinct populations of samples. Compared to some
existing estimators, which estimate the covariance matrices of interest
indirectly, our estimator is direct, ensures positive definiteness, and is the
solution to a convex optimization problem. We compute our estimator using a
proximal-proximal gradient descent algorithm. Asymptotic properties of our
estimator reveal that it can perform well in high-dimensional settings. Through
simulation studies, we demonstrate that our estimator can outperform existing
estimators. We show that our method provides more reliable estimates than
competitors in an analysis of microbiome data from subjects with chronic
fatigue syndrome.
