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$$ \boldsymbol{\Gamma} $$, which is almost indistinguishable from the real basis covariance matrix & sum;$$ \boldsymbol{\Sigma} $$. Then, any estimator Gamma<^>$$ \hat{\boldsymbol{\Gamma}} $$ satisfying some conditions can be used to estimate Gamma$$ \boldsymbol{\Gamma} $$. Finally, we impose a thresholding step on Gamma<^>$$ \hat{\boldsymbol{\Gamma}} $$ to obtain the final estimator & sum;<^>$$ \hat{\boldsymbol{\Sigma}} $$. In particular, this paper applies a Huber-type estimator Gamma<^>$$ \hat{\boldsymbol{\Gamma}} $$, and achieves robustness by only requiring the boundedness of 2+& varepsilon;$$ \epsilon $$ moments for some & varepsilon;is an element of(0,2]$$ \epsilon \in \left(0,2\right] $$. We derive the convergence rate of & sum;<^>$$ \hat{\boldsymbol{\Sigma}} $$ 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.