Minimaxity in Structured Normal Means Inference
We provide a unified treatment of a broad class of noisy structure recovery problems, known as structured normal means problems. In these problems, the goal is to identify, from a finite collection of Gaussian distributions with different means, the distribution that produced the 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. We also derive an optimality certificate for the maximum likelihood estimator, which can be applied to many instantiations. We 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.