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Motivated by the problem of estimating the bacterial growth rates for genomeassemblies from shotgun metagenomic data, we consider the permuted monotonematrix model $Y=\Theta\Pi+Z$, where $Y\in \mathbb{R}^{n\times p}$ is observed,$\Theta\in \mathbb{R}^{n\times p}$ is an unknown approximately rank-one signalmatrix with monotone rows, $\Pi \in \mathbb{R}^{p\times p}$ is an unknownpermutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. Thispaper studies the estimation of the extreme values associated to the signalmatrix $\Theta$, including its first and last columns, as well as theirdifference. Treating these estimation problems as compound decision problems,minimax rate-optimal estimators are constructed using the spectral columnsorting method. Numerical experiments through simulated and syntheticmicrobiome metagenomic data are presented, showing the superiority of theproposed methods over the alternatives. The methods are illustrated bycomparing the growth rates of gut bacteria between inflammatory bowel diseasepatients and normal controls.

Optimal Estimation of Bacterial Growth Rates Based on Permuted Monotone Matrix