We develop a new approach for clustering non-spherical (i.e., arbitrary component covariances) Gaussian mixture models via a subroutine, based on the sum-of-squares method, that finds a low-dimensional separation-preserving projection of the input data. Our method gives a non-spherical analog of the classical dimension reduction, based on singular value decomposition, that, among several other applications, forms a key component of the celebrated spherical clustering algorithm of Vempala and Wang [VW04]. As applications, we obtain an algorithm to (1) cluster an arbitrary total-variation separated mixture of $k$ centered (i.e., zero-mean) Gaussians with $n\geq \operatorname{poly}(d) f(w_{\min}^{-1})$ samples and $\operatorname{poly}(n)$ time, and (2) cluster an arbitrary total-variation separated mixture of $k$ Gaussians with identical but arbitrary unknown covariance with $n \geq d^{O(\log w_{\min}^{-1})} f(w_{\min}^{-1})$ samples and $n^{O(\log w_{\min}^{-1})}$ time. Here, $w_{\min}$ is the minimum mixing weight of the input mixture, and $f$ does not depend on the dimension $d$. Our algorithms naturally extend to tolerating a dimension-independent fraction of arbitrary outliers. Before this work, the techniques in the state-of-the-art non-spherical clustering algorithms needed $d^{O(k)} f(w_{\min}^{-1})$ samples and time for clustering such mixtures. Our results may come as a surprise in the context of the $d^{Ω(k)}$ statistical query and sum-of-squares lower bounds [DKS17, DKPP24] for clustering non-spherical Gaussian mixtures. While these results are usually thought to rule out $d^{o(k)}$ cost algorithms for the problem, our results show that the lower bounds can in fact be circumvented for a remarkably general class of Gaussian mixtures.

翻译：我们提出了一种新的非球形（即任意分量协方差）高斯混合模型聚类方法，该方法通过一个基于平方和技术的子程序，寻找输入数据的低维保分离投影。该方法为经典基于奇异值分解的降维方法提供了非球形模拟，后者在众多应用中构成Vempala和Wang [VW04] 经典球形聚类算法的关键组件。作为应用，我们实现了以下算法：(1) 对任意全变差分离的$k$个中心化（即零均值）高斯混合模型进行聚类，所需样本量为$n\geq \operatorname{poly}(d) f(w_{\min}^{-1})$，时间复杂度为$\operatorname{poly}(n)$；(2) 对任意全变差分离的$k$个具有相同但未知协方差的高斯混合模型进行聚类，所需样本量为$n \geq d^{O(\log w_{\min}^{-1})} f(w_{\min}^{-1})$，时间复杂度为$n^{O(\log w_{\min}^{-1})}$。其中$w_{\min}$为输入混合分布的最小混合权重，$f$与维度$d$无关。我们的算法天然支持容忍与维度无关比例的任意异常值。在此工作之前，最先进非球形聚类算法需要$d^{O(k)} f(w_{\min}^{-1})$的样本量和时间复杂度才能完成此类聚类。我们的结果可能出乎意料，因为针对非球形高斯混合聚类的$d^{Ω(k)}$统计查询和平方和下界[DKS17, DKPP24]通常被认为排除了该问题的$d^{o(k)}$复杂度算法，而我们的研究表明，对于一类非常广泛的高斯混合模型，这些下界实际上可以被规避。