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Abstract:
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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 $\Theta$ 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 data set 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.
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