Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model

• URL: http://arxiv.org/abs/2007.05557v1
• Date: Fri, 10 Jul 2020 18:25:38 GMT
• Title: Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model
• Authors: Yingyu Liang and Hui Yuan
• Abstract summary: We study mean estimation for entangled single-sample Gaussians with a common mean but different unknown variances. We show that the method achieves error $O left(fracsqrtnln nmright)$ with high probability when $m=Omega(sqrtnln n)$. We further prove lower bounds, showing that the error is $Omegaleft(left(fracnm4right)1/6right)$ when $m$ is between $Omega(n
• Score: 28.839136703139225
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of $n$ distributions, given one single sample from each distribution. This paper studies mean estimation for entangled single-sample Gaussians that have a common mean but different unknown variances. We propose the subset-of-signals model where an unknown subset of $m$ variances are bounded by 1 while there are no assumptions on the other variances. In this model, we analyze a simple and natural method based on iteratively averaging the truncated samples, and show that the method achieves error $O \left(\frac{\sqrt{n\ln n}}{m}\right)$ with high probability when $m=\Omega(\sqrt{n\ln n})$, matching existing bounds for this range of $m$. We further prove lower bounds, showing that the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/2}\right)$ when $m$ is between $\Omega(\ln n)$ and $O(n^{1/4})$, and the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/6}\right)$ when $m$ is between $\Omega(n^{1/4})$ and $O(n^{1 - \epsilon})$ for an arbitrarily small $\epsilon>0$, improving existing lower bounds and extending to a wider range of $m$.

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