Optimal and Adaptive Estimation of Extreme Values in the Permuted Monotone Matrix Model
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Motivated by applications in metagenomics, we consider the permuted monotone matrix model Y=ΘΠ+Z, where Y∈R^n× p is observed, Θ∈R^n× p is an unknown signal matrix with monotone rows, Π∈R^p× p is an unknown permutation matrix, and Z∈R^n× p is the noise matrix. This paper studies the estimation of the extreme values associated to the signal matrix Θ, including its first and last columns, as well as their difference (the range vector). Treating these estimation problems as compound decision problems, minimax rate-optimal and adaptive estimators are constructed using spectral column sorting. Novel techniques that can be effective in estimating an arbitrary high-dimensional nonlinear operator are developed to establish minimax lower bounds, including generalized Le Cam's method and Fano's method. Numerical experiments using 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.
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