Optimal Estimation of Bacterial Growth Rates Based on Permuted Monotone Matrix

TL;DR

This paper develops minimax rate-optimal estimators for bacterial growth rates from permuted monotone matrix models, demonstrating their effectiveness through simulations and real microbiome data analysis.

Contribution

It introduces a spectral column sorting method for optimal estimation of extreme values in permuted monotone matrices, specifically applied to bacterial growth rate estimation.

Findings

Proposed estimators outperform existing methods in simulations.

Effective in distinguishing bacterial growth rates between patient groups.

Validated on real microbiome datasets.

Abstract

Motivated by the problem of estimating the bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix 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 signal matrix with monotone rows, $\Pi \in \mathbb{R}^{p\times p}$ is an unknown permutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. This paper studies the estimation of the extreme values associated to the signal matrix $\Theta$, including its first and last columns, as well as their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column sorting method. Numerical experiments through simulated and synthetic microbiome metagenomic data are presented, showing the superiority of the proposed methods over the alternatives. The methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls.