Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models

• URL: http://arxiv.org/abs/2206.02455v2
• Date: Tue, 7 Jun 2022 10:56:10 GMT
• Title: Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models
• Authors: Yihan Zhang, Nir Weinberger
• Abstract summary: We consider a high-dimensional mean estimation problem over a binary hidden Markov model. We establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $|theta_*|,delta,d,n$.
• Score: 12.746888269949407
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an estimator observes $n$ samples of a $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$, multiplied by a random sign $ S_i $ ($1\le i\le n$), and corrupted by isotropic standard Gaussian noise. The sequence of signs $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with flip probability $\delta\in[0,1/2]$. As $\delta$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $\delta=0$ and the Gaussian Mixture Model for which $\delta=1/2$. Assuming that the estimator knows $\delta$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|\theta_{*}\|,\delta,d,n$. We then provide an upper bound to the case of estimating $\delta$, assuming a (possibly inaccurate) knowledge of $\theta_{*}$. The bound is proved to be tight when $\theta_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $\theta_{*}$ with $\delta$ unknown a priori, and theoretical guarantees on its error are stated.

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