We consider mixtures of $k\geq 2$ Gaussian components with unknown means and unknown covariance (identical for all components) that are well-separated, i.e., distinct components have statistical overlap at most $k^{-C}$ for a large enough constant $C\ge 1$. Previous statistical-query [DKS17] and lattice-based [BRST21, GVV22] lower bounds give formal evidence that even distinguishing such mixtures from (pure) Gaussians may be exponentially hard (in $k$). We show that this kind of hardness can only appear if mixing weights are allowed to be exponentially small, and that for polynomially lower bounded mixing weights non-trivial algorithmic guarantees are possible in quasi-polynomial time. Concretely, we develop an algorithm based on the sum-of-squares method with running time quasi-polynomial in the minimum mixing weight. The algorithm can reliably distinguish between a mixture of $k\ge 2$ well-separated Gaussian components and a (pure) Gaussian distribution. As a certificate, the algorithm computes a bipartition of the input sample that separates a pair of mixture components, i.e., both sides of the bipartition contain most of the sample points of at least one component. For the special case of colinear means, our algorithm outputs a $k$-clustering of the input sample that is approximately consistent with the components of the mixture. We obtain similar clustering guarantees also for the case that the overlap between any two mixture components is lower bounded quasi-polynomially in $k$ (in addition to being upper bounded polynomially in $k$). A key technical ingredient is a characterization of separating directions for well-separated Gaussian components in terms of ratios of polynomials that correspond to moments of two carefully chosen orders logarithmic in the minimum mixing weight.

翻译：我们考虑 $k\geq 2$ 个高斯成分的混合物，其均值未知且协方差（所有成分相同）未知，但成分之间充分分离，即不同成分的统计重叠最多为 $k^{-C}$，其中 $C\ge 1$ 是一个足够大的常数。先前的统计查询 [DKS17] 和基于格 [BRST21, GVV22] 的下界给出了形式化证据，表明即使区分此类混合物与（纯）高斯分布也可能在 $k$ 上呈指数难度。我们证明，这种困难性仅当混合权重允许呈指数小时才会出现，而对于多项式下界的混合权重，在拟多项式时间内可实现非平凡算法保证。具体而言，我们开发了一种基于平方和方法（运行时间关于最小混合权重呈拟多项式）的算法。该算法能够可靠地区分 $k\ge 2$ 个充分分离高斯成分的混合物与（纯）高斯分布。作为认证，该算法计算输入样本的一个二分划分，将一对混合物成分分离，即二分两边各包含至少一个成分的大部分样本点。对于均值共线的特例，我们的算法输出输入样本的一个 $k$-聚类，该聚类与混合物成分大致一致。当任意两个混合物成分之间的重叠在 $k$ 上具有拟多项式下界（同时具有多项式上界）时，我们也获得类似的聚类保证。一个关键技术要素是：通过精心选择与最小混合权重对数相关的两个阶次的矩的多项式比值，刻画充分分离高斯成分的分离方向。