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Microbial communities analysis is drawing growing attention due to the rapiddevelopment of high-throughput sequencing techniques nowadays. The observeddata has the following typical characteristics: it is high-dimensional,compositional (lying in a simplex) and even would be leptokurtic and highlyskewed due to the existence of overly abundant taxa, which makes theconventional correlation analysis infeasible to study the co-occurrence andco-exclusion relationship between microbial taxa. In this article, we addressthe challenges of covariance estimation for this kind of data. Assuming thebasis covariance matrix lying in a well-recognized class of sparse covariancematrices, we adopt a proxy matrix known as centered log-ratio covariance matrixin the literature, which is approximately indistinguishable from the real basiscovariance matrix as the dimensionality tends to infinity. We construct aMedian-of-Means (MOM) estimator for the centered log-ratio covariance matrixand propose a thresholding procedure that is adaptive to the variability ofindividual entries. By imposing a much weaker finite fourth moment conditioncompared with the sub-Gaussianity condition in the literature, we derive theoptimal rate of convergence under the spectral norm. In addition, we alsoprovide theoretical guarantee on support recovery. The adaptive thresholdingprocedure of the MOM estimator is easy to implement and gains robustness whenoutliers or heavy-tailedness exist. Thorough simulation studies are conductedto show the advantages of the proposed procedure over some state-of-the-artsmethods. At last, we apply the proposed method to analyze a microbiome datasetin human gut. The R script for implementing the method is available athttps://github.com/heyongstat/RCEC.

Robust Covariance Estimation for High-dimensional Compositional Data with Application to Microbial Communities Analysis