论文标题
通过傅立叶分析学习球形高斯的混合物
Learning Mixtures of Spherical Gaussians via Fourier Analysis
论文作者
论文摘要
假设我们从混合物中获得了独立的，相同分布的样品$ x_l $，$μ$不超过$ k $ $ d $ d $ d $ d $ d $ d $ d $ $μ_i$，$ 1 $，以便在两个不同的中心$ y_l $ y_l $ y_l $ y_l $ y_ y_j $ $ y__ $ y_ y_j $ \ fe \ leq之间， δ$，其中$ c \ in（0,1）$是一个小的正通用常数。我们开发了一种随机算法，该算法在$ \ ell_2 $范围内学习了$ us的$ y_l $的中心，$Δ<\ frac {δ\ sqrt {d}} {2}} {2} {2} $以及$ w_l $在$ cw_ {min} in cw_ {min} $ cobabilitia $ exp $ exp $ exp $ 1-- c.- cw_ to -exp $ 1-- c。样品数量和计算时间以上是$ poly（k，d，\ frac {1}δ）$的界限。当$ω（1）\ leq d \ leq o（\ log k）$时，以前在样本上的绑定和计算复杂性是未知的。当$ d = o（1）$时，这是从Regev和Vijayaraghavan的工作中开始的。这些作者还表明，在$ d $ dimensions中学习高卢斯$θ（\ sqrt {d}）$随机混合的样本复杂性，当$ d $是$ d $ as $θ（\ log log k）$至少是$ poly（k，\ frac {1}Δ）$，显示了我们的结果。
Suppose that we are given independent, identically distributed samples $x_l$ from a mixture $μ$ of no more than $k$ of $d$-dimensional spherical gaussian distributions $μ_i$ with variance $1$, such that the minimum $\ell_2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $\sqrt{d} Δ$ for some $c \leq Δ$, where $c\in (0,1)$ is a small positive universal constant. We develop a randomized algorithm that learns the centers $y_l$ of the gaussians, to within an $\ell_2$ distance of $δ< \frac{Δ\sqrt{d}}{2}$ and the weights $w_l$ to within $cw_{min}$ with probability greater than $1 - \exp(-k/c)$. The number of samples and the computational time is bounded above by $poly(k, d, \frac{1}δ)$. Such a bound on the sample and computational complexity was previously unknown when $ω(1) \leq d \leq O(\log k)$. When $d = O(1)$, this follows from work of Regev and Vijayaraghavan. These authors also show that the sample complexity of learning a random mixture of gaussians in a ball of radius $Θ(\sqrt{d})$ in $d$ dimensions, when $d$ is $Θ( \log k)$ is at least $poly(k, \frac{1}δ)$, showing that our result is tight in this case.
