# CARE: Large Precision Matrix Estimation for Compositional Data

arXiv: https://arxiv.org/abs/2309.06985 · HTML: https://mlsanity.com/paper/2309.06985/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 57 references that are in the map.
- First posted on arXiv 13 Sep 2023.

## Builds on (most cited first, 6 of 6)

- [Asymptotic normality and optimalities in estimation of large Gaussian graphical models](https://mlsanity.com/paper/1309.6024/) (2013)
- [Large Covariance Estimation for Compositional Data via Composition-Adjusted Thresholding](https://mlsanity.com/paper/1601.04397/) (2016)
- [High-dimensional Log-Error-in-Variable Regression with Applications to Microbial Compositional Data Analysis](https://mlsanity.com/paper/1811.11709/) (2018)
- [Diagonal Nonlinear Transformations Preserve Structure in Covariance and Precision Matrices](https://mlsanity.com/paper/2107.04136/) (2021)
- [Innovated scalable efficient estimation in ultra-large Gaussian graphical models](https://mlsanity.com/paper/1605.03313/) (2016)
- [Zero-inflated Poisson Factor Model with Application to Microbiome Absolute Abundance Data](https://mlsanity.com/paper/1910.11985/) (2019)

## Abstract (arXiv)

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.