CARE: Large Precision Matrix Estimation for Compositional Data

TL;DR

This paper introduces CARE, a novel method for estimating sparse precision matrices in high-dimensional compositional data, providing theoretical guarantees and demonstrating superior performance in simulations and microbial network analysis.

Contribution

We propose a new compositional precision matrix estimation method, CARE, with theoretical guarantees and optimal convergence rates, extending to zero-inflated data.

Findings

CARE achieves minimax optimality in high dimensions.

Theoretical support recovery guarantees are established.

Simulation and real data show CARE outperforms existing methods.

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 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.