Minimax Structured Normal Means Inference
This work provides a unified theoretical framework for structure recovery problems, which is incremental as it generalizes prior results on specific cases like sparse vectors and biclusters.
The paper tackles the problem of identifying the correct Gaussian distribution from a set with different means in structured normal means problems, establishing nearly matching upper and lower bounds on the minimax probability of error and deriving an optimality certificate for the maximum likelihood estimator.
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.