Optimal estimation of bacterial growth rates based on a permuted monotone matrix

Summary Motivated by the problem of estimating bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model Y=ΘΠ+Z where Y∈ Rn× p is observed, Θ∈ Rn× p is an unknown approximately rank-one signal matrix with monotone rows, Π ∈ Rp× p is an unknown permutation matrix, and Z∈ Rn× p is the noise matrix. In this article we study estimation of the extreme values associated with the signal matrix Θ, including its first and last columns and their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column-sorting method. Numerical experiments on simulated and synthetic microbiome metagenomic data are conducted, demonstrating the superiority of the proposed methods over existing alternatives. The methods are illustrated by comparing the growth rates of gut bacteria in inflammatory bowel disease patients and control subjects.