CARE: Large Precision Matrix Estimation for Compositional Data

Abstract

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 tradeoff 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Keywords:

• Blessing of dimensionality
• Graphical modeling
• High-dimensional data
• Identifiability
• Microbiome

Supplementary Materials

The supplementary materials contain the proofs of theoretical results, additional discussion and numerical results, and R code and data.

Acknowledgments

The authors thank the Associate Editor and two reviewers for valuable comments that have led to an improved article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Lin’s research was supported by National Natural Science Foundation of China (12171012, 12292980, and 12292981). Zhang’s research was supported by National Natural Science Foundation of China (12201111 and 12371264) and the Fundamental Research Funds for the Central Universities in UIBE (21QD13).