Minimax Structured Normal Means Inference

TL;DR

This paper develops a unified theoretical framework for structured normal means problems, providing tight bounds and optimality certificates for error probabilities across various structured Gaussian inference tasks.

Contribution

It establishes nearly matching minimax bounds and introduces an optimality certificate for maximum likelihood estimators applicable to many structured Gaussian problems.

Findings

Derived tight minimax bounds for structure recovery

Provided an optimality certificate for MLE in structured settings

Developed an algorithm for optimal experimental design

Abstract

We provide a unified treatment of a broad class of noisy structure recovery problems, known as structured normal means problems. In this setting, the goal is to identify, from a finite collection of Gaussian distributions with different means, the distribution that produced some observed data. Recent work has studied several special cases including sparse vectors, biclusters, and graph-based structures. We establish nearly matching upper and lower bounds on the minimax probability of error for any structured normal means problem, and we derive an optimality certificate for the maximum likelihood estimator, which can be applied to many instantiations. We also consider an experimental design setting, where we generalize our minimax bounds and derive an algorithm for computing a design strategy with a certain optimality property. We show that our results give tight minimax bounds for many structure recovery problems and consider some consequences for interactive sampling.