Settling the Robust Learnability of Mixtures of Gaussians

Settling the Polynomial Learnability of Mixtures of Gaussians

Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has a running time, and data requirement polynomial in the dimension and the inverse of the desired accuracy, with provably minimal assumptions on the Gaussians. As simple consequences of our learning algorithm, we can perform near-optimal clustering of the sample points and density estimation for mixtures of k Gaussians, efficiently. The building blocks of our algorithm are based on the work Kalai et al. [STOC 2010] that gives …

利用公共数据进行私有分布学习：来自样本压缩的视角

Private Distribution Learning with Public Data: The View from Sample Compression

本文从样本压缩与列表学习角度刻画公共-私有分布学习，给出 Gauss 混合等样本复杂度上界及封闭性质，并证明 Gauss 分布需至少 $d$ 个公共样本。

Polynomial Learning of Distribution Families

The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what w…

Robustly Learning any Clusterable Mixture of Gaussians

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 o…

超越高斯分布的高效反集中度证书

Efficient Certificates of Anti-Concentration Beyond Gaussians

本文提出新的反集中度表述，通过规范整数规划的平方和松弛与多项式再加权分析，给出适用于有界乘积分布、$L_p$ 球均匀分布等非高斯分布的拟多项式时间可验证证书，并扩展稳健学习与聚类结果。