Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model

We investigate the problem of center estimation in the high dimensional binary sub-Gaussian Mixture Model with Hidden Markov structure on the labels. We first study the limitations of existing results in the high dimensional setting and then propose a minimax optimal procedure for the problem of center estimation. Among other findings, we show that our procedure reaches the optimal rate that is of order $\sqrt{\delta d/n} + d/n$ instead of $\sqrt{d/n} + d/n$ where $\delta \in(0,1)$ is a dependence parameter between labels. Along the way, we also develop an adaptive variant of our procedure that is globally minimax optimal. In order to do so, we rely on a more refined and localized analysis of the estimation risk. Overall, leveraging the hidden Markovian dependence between the labels, we show that it is possible to get a strict improvement of the rates adaptively at almost no cost.