  High-dimensional compositional data are prevalent in many applications. The
simplex constraint poses intrinsic challenges to inferring the conditional
dependence relationships among the components forming a composition, as encoded
by a large precision matrix. We introduce a precise specification of the
compositional precision matrix and relate it to its basis counterpart, which is
shown to be asymptotically identifiable under suitable sparsity assumptions. By
exploiting this connection, we propose a composition adaptive regularized
estimation (CARE) method for estimating the sparse basis precision matrix. We
derive rates of convergence for the estimator and provide theoretical
guarantees on support recovery and data-driven parameter tuning. Our theory
reveals an intriguing trade-off between identification and estimation, thereby
highlighting the blessing of dimensionality in compositional data analysis. In
particular, in sufficiently high dimensions, the CARE estimator achieves
minimax optimality and performs as well as if the basis were observed. We
further discuss how our framework can be extended to handle data containing
zeros, including sampling zeros and structural zeros. The advantages of CARE
over existing methods are illustrated by simulation studies and an application
to inferring microbial ecological networks in the human gut.
