鲁棒地学习任意高斯混合
Robustly Learning any Clusterable Mixture of Gaussians
摘要：
我们研究了高维高斯混合在异常鲁棒设置，其中一个小的常数部分的数据是不利损坏的有效的可学习性。我们解决了这个问题的多项式学习时，组件是成对分离的总变化距离。具体地，我们提供了一种算法，该算法对于任何恒定数量的分量$k$，在多项式时间内运行，并且在$\tilde{O}(ε)$的信息理论上接近最优误差内学习$ε$损坏的$k$混合物的分量，假设任何一对分量$P_i, P_j$之间的重叠（即，量$1-TV(P_i, P_j)$）由$\mathrm{poly}(ε)$限定。我们的分离条件是定性上最弱的假设，在此条件下，样本的准确聚类是可能的。特别是，它允许具有任意协方差的分量和具有相同均值的分量，只要它们的协方差充分不同。我们的算法是解决这个问题的第一个多项式时间算法，甚至是针对$k=2$的。我们的算法遵循基于平方和的算法方法的证明。我们的主要技术贡献是一个新的强大的可识别性证明的集群从高斯混合，它可以被捕获的常数度平方和证明系统。这个证明的关键成分是一个新的使用SOS认证的反浓度和一个新的特征对高斯小（维数无关）重叠的参数距离。
英文摘要：
We study the efficient learnability of high-dimensional Gaussian mixtures in the outlier-robust setting, where a small constant fraction of the data is adversarially corrupted. We resolve the polynomial learnability of this problem when the components are pairwise separated in total variation distance. Specifically, we provide an algorithm that, for any constant number of components $k$, runs in polynomial time and learns the components of an $ε$-corrupted $k$-mixture within information theoretically near-optimal error of $\tilde{O}(ε)$, under the assumption that the overlap between any pair of components $P_i, P_j$ (i.e., the quantity $1-TV(P_i, P_j)$) is bounded by $\mathrm{poly}(ε)$. Our separation condition is the qualitatively weakest assumption under which accurate clustering of the samples is possible. In particular, it allows for components with arbitrary covariances and for components with identical means, as long as their covariances differ sufficiently. Ours is the first polynomial time algorithm for this problem, even for $k=2$. Our algorithm follows the Sum-of-Squares based proofs to algorithms approach. Our main technical contribution is a new robust identifiability proof of clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system. The key ingredients of this proof are a novel use of SoS-certifiable anti-concentration and a new characterization of pairs of Gaussians with small (dimension-independent) overlap in terms of their parameter distance.
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