  Metagenomics sequencing is routinely applied to quantify bacterial abundances
in microbiome studies, where the bacterial composition is estimated based on
the sequencing read counts. Due to limited sequencing depth and DNA dropouts,
many rare bacterial taxa might not be captured in the final sequencing reads,
which results in many zero counts. Naive composition estimation using count
normalization leads to many zero proportions, which tend to result in
inaccurate estimates of bacterial abundance and diversity. This paper takes a
multi-sample approach to the estimation of bacterial abundances in order to
borrow information across samples and across species. Empirical results from
real data sets suggest that the composition matrix over multiple samples is
approximately low rank, which motivates a regularized maximum likelihood
estimation with a nuclear norm penalty. An efficient optimization algorithm
using the generalized accelerated proximal gradient and Euclidean projection
onto simplex space is developed. The theoretical upper bounds and the minimax
lower bounds of the estimation errors, measured by the Kullback-Leibler
divergence and the Frobenius norm, are established. Simulation studies
demonstrate that the proposed estimator outperforms the naive estimators. The
method is applied to an analysis of a human gut microbiome dataset.
