A new robust covariance matrix estimation for high-dimensional microbiome data

Microbiome data typically lie in a high-dimensional simplex. One of the key questions in metagenomic analysis is to exploit the covariance structure for this kind of data. In this paper, a framework called approximate-estimate-threshold (AET) is developed for the robust basis covariance estimation for high-dimensional microbiome data. To be specific, we first construct a proxy matrix $\Gamma$, which is almost indistinguishable from the real basis covariance matrix $\sum$. Then, any estimator $\hat{\Gamma }$ satisfying some conditions can be used to estimate $\Gamma$. Finally, we impose a thresholding step on $\hat{\Gamma }$ to obtain the final estimator $\hat{\sum }$. In particular, this paper applies a Huber-type estimator $\hat{\Gamma }$, and achieves robustness by only requiring the boundedness of 2+$ϵ$ moments for some $ϵ\in (0,2]$. We derive the convergence rate of $\hat{\sum }$ under the spectral norm, and provide theoretical guarantees on support recovery. Extensive simulations and a real example are used to illustrate the empirical performance of our method.