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

arXiv: https://arxiv.org/abs/2406.12446 · HTML: https://mlsanity.com/paper/2406.12446/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 10 of 10)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Statistical and Computational Guarantees of Lloyd's Algorithm and its Variants](https://mlsanity.com/paper/1612.02099/) (2016)
- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [Randomly initialized EM algorithm for two-component Gaussian mixture achieves near optimality in O(√n) iterations](https://mlsanity.com/paper/1908.10935/) (2019)
- [Partial recovery bounds for clustering with the relaxed Kmeans](https://mlsanity.com/paper/1807.07547/) (2018)
- [Sharp Analysis of Expectation-Maximization for Weakly Identifiable Models](https://mlsanity.com/paper/1902.00194/) (2019)
- [Cutoff for exact recovery of Gaussian mixture models](https://mlsanity.com/paper/2001.01194/) (2020)
- [Statistical Guarantees for Estimating the Centers of a Two-component Gaussian Mixture by EM](https://mlsanity.com/paper/1608.02280/) (2016)
- [Fundamental limits for learning hidden Markov model parameters](https://mlsanity.com/paper/2106.12936/) (2021)
- [Clustering risk in Non-parametric Hidden Markov and I.I.D. Models](https://mlsanity.com/paper/2309.12238/) (2023)

## Abstract (arXiv)

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 √δ d/n + d/n instead of √d/n + d/n where δ ∈(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.