论文标题
在标志模型中学习纠缠的单样本高斯人
Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model
论文作者
论文摘要
在纠缠单样本分布的设置中，目标是估算一个$ n $分布的家族共享的一些共同参数，鉴于每个分布中的一个样本。本文研究的平均估计是对具有共同均值但未知方差不同的纠缠单样本高斯人的估计。我们提出了信号模型，其中$ m $方差的未知子集由1界限，而其他方差没有假设。在此模型中，我们根据迭代平均截断的样品分析了一种简单自然的方法，并表明该方法达到了错误$ o \ left（\ frac {\ sqrt {\ sqrt {n \ ln n}}} {m}} {m} {m} \ rightsibal，当$ m = m =ω（\ s \ sqrt {n}时， $ m $。我们进一步证明了下限，表明该错误是$ω\ left（\ left（\ frac {n} {m^4} \ right）^{1/2} \ right）$当$ m $在$ω（\ ln n）$和$ o（n^{1/4}）$之间，并且错误是错误的。 $ω\ left（\ left（\ frac {n} {m^4} \ right）^{1/6} \ right）$当$ m $在$ω（n^{1/4}）$和$ o（n^{1/4}）$和$ o（n^{1 -ε}）$的$> $ $ $ 0 $ $> 0 $ $ $ $ $ $上，并将范围改进范围的范围a的范围a a a a n^n^{1/4}）$（n^{1/4}）$（n^{1/4}）$。
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 $m=Ω(\sqrt{n\ln n})$, matching existing bounds for this range of $m$. We further prove lower bounds, showing that the error is $Ω\left(\left(\frac{n}{m^4}\right)^{1/2}\right)$ when $m$ is between $Ω(\ln n)$ and $O(n^{1/4})$, and the error is $Ω\left(\left(\frac{n}{m^4}\right)^{1/6}\right)$ when $m$ is between $Ω(n^{1/4})$ and $O(n^{1 - ε})$ for an arbitrarily small $ε>0$, improving existing lower bounds and extending to a wider range of $m$.
