Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929390878720000 |
|---|---|
| author | Karagulyan, Vahe Ndaoud, Mohamed |
| author_facet | Karagulyan, Vahe Ndaoud, Mohamed |
| contents | 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 $\sqrt{δd/n} + d/n$ instead of $\sqrt{d/n} + d/n$ where $δ\in(0,1)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model Karagulyan, Vahe Ndaoud, Mohamed Statistics Theory Machine Learning 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 $\sqrt{δd/n} + d/n$ instead of $\sqrt{d/n} + d/n$ where $δ\in(0,1)$ 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. |
| title | Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2406.12446 |