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

• URL: http://arxiv.org/abs/2406.12446v1
• Date: Tue, 18 Jun 2024 09:48:21 GMT
• Title: Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model
• Authors: Vahe Karagulyan, Mohamed Ndaoud,
• Abstract summary: 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. We show that our procedure reaches the optimal rate that is of order $sqrtdelta d/n + d/n$ instead of $sqrtd/n + d/n$ where $delta in(0,1)$ is a dependence parameter between labels.
• Score: 2.07180164747172
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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