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This paper is motivated by the recent interest in the analysis of high dimen-sional microbiome data. A key feature of this data is the presence of`structural zeros' which are microbes missing from an observation vector due toan underlying biological process and not due to error in measurement. Typicalnotions of missingness are insufficient to model these structural zeros. Wedefine a general framework which allows for structural zeros in the model andpropose methods of estimating sparse high dimensional covariance and precisionmatrices under this setup. We establish error bounds in the spectral andfrobenius norms for the proposed esti- mators and empirically support them witha simulation study. We also apply the proposed methodology to the global humangut microbiome data of Yatsunenko (2012).

Analysis of High Dimensional Compositional Data Containing Structural Zeros with Applications to Microbiome Data