Minimax Risk Upper Bounds Based on Shell Analysis of a Quantized Maximum Likelihood Estimator

This paper develops a unified framework for upper bounding the minimax risk in high-dimensional parameter estimation problems. To this end, we study a quantized maximum likelihood estimator, where the estimator computes the likelihood for all points within a discrete cover, and outputs the candidate with the maximal likelihood. While this concept is straightforward, our analysis is quite delicate. It splits the competing candidates in the cover to small shells, and controls the number of candidates in each shell, as well as the probability that a candidate in the shell outscores a candidate which is close to the true parameter. We demonstrate the utility of our bounds by applying them to different Gaussian problems, and showing that they recover the optimal minimax rate for the Gaussian location model and the spiked Wigner Model. For the multi-reference alignment problem we obtain a novel minimax upper bound, which essentially places no assumptions on the signal of interest.