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

TL;DR

This paper introduces a minimax optimal and adaptive procedure for center estimation in high-dimensional hidden Markov sub-Gaussian mixture models, improving rate bounds by leveraging label dependence.

Contribution

It proposes a new adaptive estimation method that achieves minimax optimal rates, accounting for label dependence in high-dimensional settings.

Findings

Achieves the optimal rate of ab7ab4 + d/n for center estimation.

Develops an adaptive procedure that is globally minimax optimal.

Shows that exploiting label dependence improves estimation rates.

Abstract

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.