  Motivated by recent research on quantifying bacterial growth dynamics based
on genome assemblies, we consider a permuted monotone matrix model
$Y=\Theta\Pi+Z$, where the rows represent different samples, the columns
represent contigs in genome assemblies and the elements represent log-read
counts after preprocessing steps and Guanine-Cytosine (GC) adjustment. In this
model, $\Theta$ is an unknown mean matrix with monotone entries for each row,
$\Pi$ is a permutation matrix that permutes the columns of $\Theta$, and $Z$ is
a noise matrix. This paper studies the problem of estimation/recovery of $\Pi$
given the observed noisy matrix $Y$. We propose an estimator based on the best
linear projection, which is shown to be minimax rate-optimal for both exact
recovery, as measured by the 0-1 loss, and partial recovery, as quantified by
the normalized Kendall's tau distance. Simulation studies demonstrate the
superior empirical performance of the proposed estimator over alternative
methods. We demonstrate the methods using a synthetic metagenomics dataset of
45 closely related bacterial species and a real metagenomic dataset to compare
the bacterial growth dynamics between the responders and the non-responders of
the IBD patients after 8 weeks of treatment.
